Extending the reals
David R Tribble
of an extension to the reals, defining real-like numbers that are
larger than any regular real.
I've written an article exploring the effects of adding a few axioms to
standard arithmetic to extend the reals with what I call, for lack of
a better term, "h-numbers". Briefly, an h-number has a magnitude
greater than any real. The first axiom states that n1 (eta_1), a
primitive h-number constant, exists, and that x < n1 for all x in R.
>From there, an entire set H of h-numbers is defined as containing other
h-numbers, being sums and products of reals and n1.
The article goes on to explore arithmetic for the h-numbers and ends
up defining an entire hierarchy of such numbers (n1, n2, etc.) and
their multiplicative inverses (e1 = 1/n1, e2, etc.).
The article (which requires a browser capable of rendering certain
mathematical HTML characters) is at:
http://david.tribble.com/text/hnumbers.html
Comments and suggestions are welcome. I'm curious to know if
something like this has been done before, or whether it's
mathematically inconsistent.
-drt
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Denis Feldmann
> Over the last few months I've been noodling around with the concept
> of an extension to the reals, defining real-like numbers that are
> larger than any regular real.
>
> I've written an article exploring the effects of adding a few axioms to
> standard arithmetic to extend the reals with what I call, for lack of
> a better term, "h-numbers". Briefly, an h-number has a magnitude
> greater than any real. The first axiom states that n1 (eta_1), a
> primitive h-number constant, exists, and that x < n1 for all x in R.
>>From there, an entire set H of h-numbers is defined as containing other
> h-numbers, being sums and products of reals and n1.
>
> The article goes on to explore arithmetic for the h-numbers and ends
> up defining an entire hierarchy of such numbers (n1, n2, etc.) and
> their multiplicative inverses (e1 = 1/n1, e2, etc.).
>
> The article (which requires a browser capable of rendering certain
> mathematical HTML characters) is at:
> http://david.tribble.com/text/hnumbers.html
>
> Comments and suggestions are welcome. I'm curious to know if
> something like this has been done before, or whether it's
> mathematically inconsistent.
Look at hyperreals, non-standard analysis (and also, maybe, at surreal
numbers, non-archimedian fields...)
cbr...@cbrownsystems.com
> Over the last few months I've been noodling around with the concept
> of an extension to the reals, defining real-like numbers that are
> larger than any regular real.
>
<snip>
> The article (which requires a browser capable of rendering certain
> mathematical HTML characters) is at:
> http://david.tribble.com/text/hnumbers.html
>
> Comments and suggestions are welcome. I'm curious to know if
> something like this has been done before, or whether it's
> mathematically inconsistent.
>
Nice job. In conversations with Tony Orlow, I also thought of a similar
system (essentially, an ordered field extension of R using a polynomial
basis over some "unit infinity" B). The unanswered question (to me) in
your "Loose ends" section is defining a topology on the h-numbers so
that limits can be expressed. Will think about this on the drive home
from Xmas Festivities...
(Holiday) Cheers - Chas
David R Tribble
>> Over the last few months I've been noodling around with the concept
>> of an extension to the reals, defining real-like numbers that are
>> larger than any regular real.
>>
>> mathematical HTML characters) is at:
>> http://david.tribble.com/text/hnumbers.html
>>
>> Comments and suggestions are welcome. I'm curious to know if
>> something like this has been done before, or whether it's
>> mathematically inconsistent.
>
Chas Brown wrote:
> Nice job. In conversations with Tony Orlow, I also thought of a similar
> system (essentially, an ordered field extension of R using a polynomial
> basis over some "unit infinity" B).
I was hoping this conversation would go quite a bit further
before Tony's name was mentioned. His "unit infinity" is an
extermely flawed and inconsistent concept from the get go.
Oh well.
For the record, I was inspired by some concepts of NSA for
creating my h-numbers.
> The unanswered question (to me) in
> your "Loose ends" section is defining a topology on the h-numbers so
> that limits can be expressed. Will think about this on the drive home
> from Xmas Festivities...
Yes, I would like to refine the h-numbers (which I can probably
start calling the "suprareals") to define analytic functions, limits,
etc., over them. I'd also like to know if and how they are related
to the nonstandard reals of NSA. I'm no expert in these areas,
though, so feedback from others would be appreciated.
Ross A. Finlayson
Ha ha ha ha ha ha. Troll! Heh.
I applaud your notion, to be honest.
Your "seed", or eta_1 or what have you of your h-numbers, does look
quite similar to Tony's H-riffics' "unit infinity", or Yaroslav
Sergeyev's grossone or (1), or the unit scalar infinity. However, it
appears they are more similar to the Robinso(h)nian hyperreals. Tony's
rule would seem to apply, or not.
The blackboard bold H is used for quaternions, the set and division
algebra. You might want to select a different symbol to avoid
ambiguity, although it is clear in context.
Do you see any applications of your system? Are you able to formalize
otherwise inaccessible or what are deemed paradoxical true results
using your system? If so, do they apply to the real numbers and thus
demand that all numbers on the real number line are real numbers,
demanding reevaluation of sufficiency and necessity in representing the
real numbers? What's the point?
Good luck,
Ross F.
David R Tribble
>> Over the last few months I've been noodling around with the concept
>> of an extension to the reals, defining real-like numbers that are
>> larger than any regular real.
>>
>> The article (which requires a browser capable of rendering certain
>> mathematical HTML characters) is at:
>> http://david.tribble.com/text/hnumbers.html
>>
Chas Brown wrote:
>> Nice job. In conversations with Tony Orlow, I also thought of a similar
>> system (essentially, an ordered field extension of R using a polynomial
>> basis over some "unit infinity" B).
>
David R Tribble wrote:
>> I was hoping this conversation would go quite a bit further
>> before Tony's name was mentioned. His "unit infinity" is an
>> extermely flawed and inconsistent concept from the get go.
>> Oh well.
>>
>> For the record, I was inspired by some concepts of NSA for
>> creating my h-numbers.
>
Ross A. Finlayson wrote:
> Ha ha ha ha ha ha. Troll! Heh.
>
> I applaud your notion, to be honest.
>
> Your "seed", or eta_1 or what have you of your h-numbers, does look
> quite similar to Tony's H-riffics' "unit infinity", or Yaroslav
> Sergeyev's grossone or (1), or the unit scalar infinity. However, it
> appears they are more similar to the Robinso(h)nian hyperreals. Tony's
> rule would seem to apply, or not.
They have nothing in common. My n1 (eta_1) is in almost all respects
another kind of real, obeying relations such as 1+n1>n1. Infinite
numbers don't act that way, and I make it quite clear that n1 is no
such thing. They are similar to Robinson's illimited numbers, and
I'd like to know just how closely related.
> The blackboard bold H is used for quaternions, the set and division
> algebra. You might want to select a different symbol to avoid
> ambiguity, although it is clear in context.
Such details can be worked out later. "H" would appear to reflect the
name "Hamiltonian". My "H" is more properly pronounced "Eta".
I originally started to call them "T-numbers", but I thought "eta"
looked like a better choice for "number". But these are minor details
that can be adjusted later.
> Do you see any applications of your system?
A strange question to ask of abstract mathematics.
> Are you able to formalize otherwise inaccessible or what are
> deemed paradoxical true results using your system?
Such as?
> If so, do they apply to the real numbers and thus
> demand that all numbers on the real number line are real numbers,
> demanding reevaluation of sufficiency and necessity in representing the
> real numbers?
As you can see at the end of the article, the reals R are a subset of
the suprareals H, designated H0. The entire hierarchy of H is a field.
I also make it quite clear that the h-numbers are an extension to the
reals, requiring additional axioms for their existence. So what is
true of the reals is not necessarily true of the h-numbers.
> What's the point?
Good question. What IS the point of the real number line?
Ross A. Finlayson
Well, I wonder, can you partition the unit interval into eta_1 many
equal-sized intervals?
About application, and where your numbers are a field that makes them
very similar to the hyperreals, where via the transfer principle the
large part of true statements about the hyperreals is true about the
standard reals and vice versa, and thus they're indistinguishable at a
distance, and interchangeable in practice, I wonder if there are any
what you would see as true facts about the real numbers that you can
verify via properties of the h-numbers or what have you that you don't
see verifiable from the standard properties of the "standard" real
numbers.
I'm glad you interpreted the question about the point in that way.
What's the point? Do you find it conceivable that the points and
variously all the lines in some ultimate space don't need to be defined
but that the existence of objects with those properties is a
consequence of the existence of space? How about space as a
consequence of existence of anything, even nothing?
Also, what's the point of your h-numbers, as in, why have you extended
effort to describe what you see as abstract? What perceived
insufficiency in simpler language about the real numbers do you see as
justifying the effort in inventing a notation and description of these
things? If you can't prove statements about the real numbers not
already proven by those of the standard real numbers, how is your
treatment of these objects any different than relabelling the
properties of the real numbers?
If the reals are complete in their total ordering there aren't any
numbers in that ordering not in it.
I thought it was funny to write "Troll!", maybe you can understand that
many perceive something in terms of numbers and number theories beyond
the standard, as you do, not all who troll are trolls.
Why bother addending to the standard real numbers?
Do these h-numbers have any geometric meaning?
For the record, I was inspired by the numbers.
The point is polydimensional.
Ross
hagman
David R Tribble schrieb:
The easiest, most obvious and literally universal method to extend the
reals by adding new numbers is to look at R[X].
In fact, that is what you did; additionally you defined an order
relation on HuR = R[X] by declaring all non-zero-polynomials with
positive leading coefficient as positive.
It is trivially verified that for this set P we have P*P subset P, P+P
subset P and R[X] is the disjoint union of P and -P and {0}.
Later you construct HuRuL in a way that essentially boils down to
R[X,Y]/(XY-1) and claim that this is a field.
However, Theorem 18c. fails to be true (at least with the first version
of Axiom 8; the second version is not equivalent):
In R[X,Y]/(XY-1), X+1 has no inverse!
Instead, You should have used R(X), the field of rational functions in
one variable.
The order on R[X] induces an order on R(X), i.e. f/g is positive if f
and g are both positive or both negative.
(Again, it is trivially verified that for this set P we have P*P subset
P, P+P subset P and R[X] is the disjoint union of P and -P and {0}).
To obtain nested hierarchies, repeat the step, i.e. use more variables:
Let S be a totally ordered set (of variables, i.e. viewed as disjoint
with R etc.; if you want to use S=R or the like, use some standard
trick to obtain a disjoint copy)
Then R[S] is a ring and R(S) is a field.
To define an order relation on R[S] (and thus on R(S)), declare a
non-zero element of R[S] as positive if the leading coefficient is a
positive real.
Note that each element of R[S] is in fact an element of R[X1,...,Xn]
for a finite number of variables X1<...<Xn in S. Since
R[X1,..,Xn]=R[X1,...,X{n-1}][Xn], the leading coefficient method can be
applied step by step.
This works for any totally ordered set S, be it finite, countable,
continuum-sized or whatever.
cbr...@cbrownsystems.com
Also, for example, 18a claims that u + v in H u R u L; but given h in
H, h + 1/h is not in any of H, R, or L.
>
> Instead, You should have used R(X), the field of rational functions in
> one variable.
This is actually what I /thought/ the OP had defined (so much for my
careful reading!).
> The order on R[X] induces an order on R(X), i.e. f/g is positive if f
> and g are both positive or both negative.
> (Again, it is trivially verified that for this set P we have P*P subset
> P, P+P subset P and R[X] is the disjoint union of P and -P and {0}).
>
> To obtain nested hierarchies, repeat the step, i.e. use more variables:
> Let S be a totally ordered set (of variables, i.e. viewed as disjoint
> with R etc.; if you want to use S=R or the like, use some standard
> trick to obtain a disjoint copy)
> Then R[S] is a ring and R(S) is a field.
> To define an order relation on R[S] (and thus on R(S)), declare a
> non-zero element of R[S] as positive if the leading coefficient is a
> positive real.
> Note that each element of R[S] is in fact an element of R[X1,...,Xn]
> for a finite number of variables X1<...<Xn in S. Since
> R[X1,..,Xn]=R[X1,...,X{n-1}][Xn], the leading coefficient method can be
> applied step by step.
>
> This works for any totally ordered set S, be it finite, countable,
> continuum-sized or whatever.
Which leads us back to the question of what topology to use for these
numbers, and how/whether limits are to be defined.
Using open intervals seems the most "natural" basis for a topology for
this space; the sub-space topology on R is then the usual topology.
This gives us a Hausdorff space (I think, regular Hausdorff?).
Then, for example, the set of "finite" suprareals is an open set as is
the set of "infinite" suprareals, so we get the lack of connectedness
implied in the drawings the OP made. (Many other
examples of the disconnectedness of this system can be found.) Thus,
this space is not normal (the above two sets are also closed and
disjoint, but are not separated).
Let us denote the suprareals as H^. We can define lim (x->c) {f(x)} = L
as:
L in H^ is the limit of f(x) as x approaches c if, for every 0 <
epsilon in H^, there exists a 0 < delta in H^ such that for all x such
that |x-c| < delta, |f(x) -L| < epsilon.
It doesn't seem like this yields a complete space; but I can't
construct any counterexamples off the top of my head.
Cheers - Chas
David R Tribble
>> Over the last few months I've been noodling around with the concept
>> of an extension to the reals, defining real-like numbers that are
>> larger than any regular real.
>>
>> I've written an article exploring the effects of adding a few axioms to
>> standard arithmetic to extend the reals with what I call, for lack of
>> a better term, "h-numbers". Briefly, an h-number has a magnitude
>> greater than any real. The first axiom states that n1 (eta_1), a
>> primitive h-number constant, exists, and that x < n1 for all x in R.
>> >From there, an entire set H of h-numbers is defined as containing other
>> h-numbers, being sums and products of reals and n1.
>
Thank you very much for the feedback. I confess that, as an
amateur, this is out of my depth. But I'm willing to learn. Any
good books to recommend that cover R(X), R[X], etc.?
-drt
Tony Orlow
> David R Tribble wrote:
>> David R Tribble wrote:
>>>> Over the last few months I've been noodling around with the concept
>>>> of an extension to the reals, defining real-like numbers that are
>>>> larger than any regular real.
>>>>
>>>> The article (which requires a browser capable of rendering certain
>>>> mathematical HTML characters) is at:
>>>> http://david.tribble.com/text/hnumbers.html
>>>>
>> Chas Brown wrote:
>>>> Nice job. In conversations with Tony Orlow, I also thought of a similar
>>>> system (essentially, an ordered field extension of R using a polynomial
>>>> basis over some "unit infinity" B).
>> David R Tribble wrote:
>>>> I was hoping this conversation would go quite a bit further
>>>> before Tony's name was mentioned. His "unit infinity" is an
>>>> extermely flawed and inconsistent concept from the get go.
>>>> Oh well.
Did you honestly think that was a reasonable expectation? You seem to
have agreed that "larger than any finite" is a reasonable definition for
"infinite" That's a good start, a unit infinity. Welcome to the club.
>>>>
>>>> For the record, I was inspired by some concepts of NSA for
>>>> creating my h-numbers.
>> Ross A. Finlayson wrote:
>>> Ha ha ha ha ha ha. Troll! Heh.
>>>
>>> I applaud your notion, to be honest.
>>>
>>> Your "seed", or eta_1 or what have you of your h-numbers, does look
>>> quite similar to Tony's H-riffics' "unit infinity", or Yaroslav
>>> Sergeyev's grossone or (1), or the unit scalar infinity. However, it
>>> appears they are more similar to the Robinso(h)nian hyperreals. Tony's
>>> rule would seem to apply, or not.
>> They have nothing in common. My n1 (eta_1) is in almost all respects
>> another kind of real, obeying relations such as 1+n1>n1. Infinite
>> numbers don't act that way, and I make it quite clear that n1 is no
>> such thing. They are similar to Robinson's illimited numbers, and
>> I'd like to know just how closely related.
>>
>>
David's actually right. Despite the fact that he used "h" for his
numbers, after considering "t", his numbers correspond to the T-riffics
rather than the H-riffics, which are based on nested powers of 2 to
enumerate the reals. The T-riffics are a simple extension of the normal
digital numbers system, based on sums of powers of 2, or any number
base, preferably prime. They also correspond to Robinson's illimited
numbers, uncountably distant from each other, with countable rational
neighborhoods of digits surrounding limit points. In any case, your
"new" idea is not much different, at first glance, from my old and
"extremely flawed" concept. I'll print out and peruse your theory, and
then give more detailed comments, okay? Good job!
>>> The blackboard bold H is used for quaternions, the set and division
>>> algebra. You might want to select a different symbol to avoid
>>> ambiguity, although it is clear in context.
>> Such details can be worked out later. "H" would appear to reflect the
>> name "Hamiltonian". My "H" is more properly pronounced "Eta".
>> I originally started to call them "T-numbers", but I thought "eta"
>> looked like a better choice for "number". But these are minor details
>> that can be adjusted later.
>>
>>
I already used H and T. :) Pick another letter. How about D?
>>> Do you see any applications of your system?
>> A strange question to ask of abstract mathematics.
>>
>>
Not really.
>>> Are you able to formalize otherwise inaccessible or what are
>>> deemed paradoxical true results using your system?
>> Such as?
>>
>>
Oh, can you resolve the Continuum Hypothesis, or bring the subset
relation in line with infinite set size measures? Can you resolve the
difference between Galilean models of bijection and notions of set
density? You know, stuff like that.....
>>> If so, do they apply to the real numbers and thus
>>> demand that all numbers on the real number line are real numbers,
>>> demanding reevaluation of sufficiency and necessity in representing the
>>> real numbers?
>> As you can see at the end of the article, the reals R are a subset of
>> the suprareals H, designated H0. The entire hierarchy of H is a field.
>>
>> I also make it quite clear that the h-numbers are an extension to the
>> reals, requiring additional axioms for their existence. So what is
>> true of the reals is not necessarily true of the h-numbers.
>>
>>
And, what, precisely are the differences? I'll read in detail. No need
to answer until I have a specific question. :)
>>> What's the point?
>> Good question. What IS the point of the real number line?
>
> Well, I wonder, can you partition the unit interval into eta_1 many
> equal-sized intervals?
That might follow....
>
> About application, and where your numbers are a field that makes them
> very similar to the hyperreals, where via the transfer principle the
> large part of true statements about the hyperreals is true about the
> standard reals and vice versa, and thus they're indistinguishable at a
> distance, and interchangeable in practice, I wonder if there are any
> what you would see as true facts about the real numbers that you can
> verify via properties of the h-numbers or what have you that you don't
> see verifiable from the standard properties of the "standard" real
> numbers.
In other words, what are the differences between the reals and suprareals...
>
> I'm glad you interpreted the question about the point in that way.
> What's the point? Do you find it conceivable that the points and
> variously all the lines in some ultimate space don't need to be defined
> but that the existence of objects with those properties is a
> consequence of the existence of space? How about space as a
> consequence of existence of anything, even nothing?
Metrics, as Chas brought up....
>
> Also, what's the point of your h-numbers, as in, why have you extended
> effort to describe what you see as abstract? What perceived
> insufficiency in simpler language about the real numbers do you see as
> justifying the effort in inventing a notation and description of these
> things? If you can't prove statements about the real numbers not
> already proven by those of the standard real numbers, how is your
> treatment of these objects any different than relabelling the
> properties of the real numbers?
>
Again, what are the differences?
> If the reals are complete in their total ordering there aren't any
> numbers in that ordering not in it.
>
> I thought it was funny to write "Troll!", maybe you can understand that
> many perceive something in terms of numbers and number theories beyond
> the standard, as you do, not all who troll are trolls.
David's certainly divvied out enough in my direction. He can take it. :)
>
> Why bother addending to the standard real numbers?
Appending? Because it's there?
>
> Do these h-numbers have any geometric meaning?
Good question.
>
> For the record, I was inspired by the numbers.
>
> The point is polydimensional.
>
> Ross
>
Points have no dimension whatsoever, without context. But, every point
has a context, no?
Tony
David R Tribble
>> I've written an article exploring the effects of adding a few axioms to
>> standard arithmetic to extend the reals with what I call, for lack of
>> a better term, "h-numbers". Briefly, an h-number has a magnitude
>> greater than any real. The first axiom states that n1 (eta_1), a
>> primitive h-number constant, exists, and that x < n1 for all x in R.
>> >From there, an entire set H of h-numbers is defined as containing other
>> h-numbers, being sums and products of reals and n1.
>
hagman wrote:
>> The easiest, most obvious and literally universal method to extend the
>> reals by adding new numbers is to look at R[X].
>> In fact, that is what you did; additionally you defined an order
>> relation on HuR = R[X] by declaring all non-zero-polynomials with
>> positive leading coefficient as positive.
>> It is trivially verified that for this set P we have P*P subset P, P+P
>> subset P and R[X] is the disjoint union of P and -P and {0}.
>
I think I follow most of what you are saying from a more thorough
reading of your post...
>> Later you construct HuRuL in a way that essentially boils down to
>> R[X,Y]/(XY-1) and claim that this is a field.
>> However, Theorem 18c. fails to be true (at least with the first version
>> of Axiom 8; the second version is not equivalent):
>> In R[X,Y]/(XY-1), X+1 has no inverse!
>
I don't see why they are not equivalent. And why is there no inverse?
>> Instead, You should have used R(X), the field of rational functions in
>> one variable.
>> The order on R[X] induces an order on R(X), i.e. f/g is positive if f
>> and g are both positive or both negative.
>> (Again, it is trivially verified that for this set P we have P*P subset
>> P, P+P subset P and R[X] is the disjoint union of P and -P and {0}).
>>
>> To obtain nested hierarchies, repeat the step, i.e. use more variables:
>> Let S be a totally ordered set (of variables, i.e. viewed as disjoint
>> with R etc.; if you want to use S=R or the like, use some standard
>> trick to obtain a disjoint copy)
>> Then R[S] is a ring and R(S) is a field.
>> To define an order relation on R[S] (and thus on R(S)), declare a
>> non-zero element of R[S] as positive if the leading coefficient is a
>> positive real.
>> Note that each element of R[S] is in fact an element of R[X1,...,Xn]
>> for a finite number of variables X1<...<Xn in S. Since
>> R[X1,..,Xn]=R[X1,...,X{n-1}][Xn], the leading coefficient method can be
>> applied step by step.
>>
>> This works for any totally ordered set S, be it finite, countable,
>> continuum-sized or whatever.
>
Why does the leading coefficient determine the order (or perhaps I
don't understand what "leading" means)? Given suprareals h and g,
where
h = 2 - 3n1^1 + 4n1^2
g = 3 + 4n1^1 - 5n1^2
then g < h because the highest power of n1 is -5n1^2 for g and
is +4n1^2 for h, and because n1^p < n1^q for p < q.
In other words, shouldn't it be that the coefficient for the largest
power of n1 determines the order? (Or is that what you meant?)
Also bear in mind that for higher orders of suprareals, the
polynomials over n2 (n3, n4, etc.) has coefficients that are not
limited to reals but to all the suprareals of lower orders.
For example, for h in H2,
h = x0 + x1 n2^1 + x2 n2^2 + ... + xn n2^n
where each coefficient x0, x1, x2, ..., xn can itself be a suprareal
in H1, so that they are polynomials over n1, e.g.:
x0 = y0 + y1 n1^1 + y2 n1^2 + ... + ym n1^m
David R Tribble
>> Over the last few months I've been noodling around with the concept
>> of an extension to the reals, defining real-like numbers that are
>> larger than any regular real.
>
Chas Brown wrote:
>> Nice job. In conversations with Tony Orlow, I also thought of a similar
>> system (essentially, an ordered field extension of R using a polynomial
>> basis over some "unit infinity" B).
>
David R Tribble wrote:
>> I was hoping this conversation would go quite a bit further
>> before Tony's name was mentioned. His "unit infinity" is an
>> extermely flawed and inconsistent concept from the get go.
>> Oh well.
>
Tony Orlow wrote:
> Did you honestly think that was a reasonable expectation? You seem to
> have agreed that "larger than any finite" is a reasonable definition for
> "infinite" That's a good start, a unit infinity. Welcome to the club.
Please read the article. My suprareals are not infinite numbers.
They are numbers that are simply larger than the standard reals.
In most other respects, they act just like reals. See Theorem I
(infinity) in particular:
-oo < h < +oo for all h in H.
David R Tribble wrote:
>> For the record, I was inspired by some concepts of NSA for
>> creating my h-numbers.
>
Ross A. Finlayson wrote:
>> Your "seed", or eta_1 or what have you of your h-numbers, does look
>> quite similar to Tony's H-riffics' "unit infinity", or Yaroslav
>> Sergeyev's grossone or (1), or the unit scalar infinity. However, it
>> appears they are more similar to the Robinso(h)nian hyperreals. Tony's
>> rule would seem to apply, or not.
>
David R Tribble wrote:
>> They have nothing in common. My n1 (eta_1) is in almost all respects
>> another kind of real, obeying relations such as 1+n1>n1. Infinite
>> numbers don't act that way, and I make it quite clear that n1 is no
>> such thing. They are similar to Robinson's illimited numbers, and
>> I'd like to know just how closely related.
>
Tony Orlow wrote:
> David's actually right. Despite the fact that he used "h" for his
> numbers, after considering "t", his numbers correspond to the T-riffics
> rather than the H-riffics,
No they don't. "t" stands for something else, and I leave it to you
to guess what. "h" originally was supposed to mean "huge" or "hyper",
but "hyperreal" has already been used, so now I'm settling on
"suprareal". "H" is a capital "eta", from my seed constant eta_1 (n1).
I chose eta because it looks like "n", for "number".
I may end up changing the notation anyway.
> ... which are based on nested powers of 2 to
> enumerate the reals. The T-riffics are a simple extension of the normal
> digital numbers system, based on sums of powers of 2, or any number
> base, preferably prime. They also correspond to Robinson's illimited
> numbers, uncountably distant from each other, with countable rational
> neighborhoods of digits surrounding limit points.
I don't think you understand the hyperreals of NSA.
My suprareals resemble the hyperreals, or at least the suprareals
in H1 and L1. I'm not sure how they're related, and I don't think the
hyperreals are related to the rest of the suprareal hierachy (H2, L1,
H3, L3, etc.).
> In any case, your
> "new" idea is not much different, at first glance, from my old and
> "extremely flawed" concept. I'll print out and peruse your theory, and
> then give more detailed comments, okay? Good job!
>
It's completely different. For one, it uses axioms to define the
existence and properties of the suprareals. You don't have any.
Secondly, your "unit infinity" is supposed to be equal to the
cardinality of both [0,1] and N, which is inconsistent within standard
arithmetic. And there are more flaws, which have been pointed
out to you many times.
But please, this thread is supposed to be about my theory, not
yours. I'll be happy to join the thread you start for yours.
Ross A. Finlayson wrote:
>> Are you able to formalize otherwise inaccessible or what are
>> deemed paradoxical true results using your system?
>> Such as?
>
Tony Orlow wrote:
> Oh, can you resolve the Continuum Hypothesis, or bring the subset
> relation in line with infinite set size measures? Can you resolve the
> difference between Galilean models of bijection and notions of set
> density? You know, stuff like that.....
Those have all been resolved, regardless of your opinions about them.
The CH has been shown to be unresolvable within ZFC, so I would
speculate that the only way to "resolve" it would be to accept CH or
~CH as an additional axiom (which seems to be the usual approach)
or to try to add new axioms that force CH or ~CH.
At any rate, my little additional axioms won't do that.
Ross A. Finlayson
> Ross A. Finlayson wrote:
> >
> > The point is polydimensional.
> >
>
> Points have no dimension whatsoever, without context. But, every point
> has a context, no?
David, there's no place on the real number line for numbers that are
not real numbers to be on there. For example, there are only and
everywhere real numbers between zero and one, there couldn't be any
numbers that aren't real numbers there.
Where there is something like the hyperreals to extend the "complete"
ordered field, to satisfy notions of the existence of infinitesimals,
and their reciprocal infinities, as a description of the end-process of
infinite divisibility, the real numbers are the complete ordered field,
so if it is true that those things exist in the reals, they're real
numbers. Divide by zero. (That's done in ways that aren't a computer
programmer's pseudo-null pointer exception.) Sum the distances of Zeno
or the differentials of the unit, it's unity. For no finite
differential does the derivative or integral work.
Tony, that's an astute observation, yes the "geometrical"
(world-measure) point and its "definition" (place, fixation, in the
unbounded universe; un-/of "finition") has to do with where it is.
A point on a line is a point on each plane through that line, and it is
as well a point on each 3-D space containing those planes, and lines,
and 4-D space containing each of those spaces of less dimension, ad
infinitum. They're polydimensional in that at once they are
constituent elements of 1-D, 2-D, ..., n-D spaces, n E N, and N E N.
Then in looking at, considering, a point on a line, it is basically
defined from the left and from the right, on that left-to-right line,
it has two sides. Then, where the points are a contiguous sequence on
that line, ie integral iota multiples from zero translated by that
point, the points in the immediate neighborhood of that definite point
have only one side. That gets into a difference between a point _on_ a
line, and a point _in_ a line.
That's not very clear, in that, there's a lot of background to put into
place to then get into the nature of points vis-a-vis the continuum as
a variety of dimensioned spaces.
Where on a one-dimensional line it seems clear that given a known
point, by a standard real number, basically, and not a sequence like
.1, .01, .001, etcetera, that the numbers less than and greater than it
are obvious, it's got two sides. Then, in the atomic difference from
that definite real number, there are only and everwhere real numbers,
so even positing the existence of an iota value in the reals leads to
that it can't share all of what are generally recognized as standard
properties of the real numbers. Where it is a real number, as is
apparently a known counterexample to standard analysis, then the real
numbers themselves in what they are expected to be have restrictions on
some of their properties or that otherwise there is a reconciliation
that the standard reals are definite reals, in, say, only base two,
where there are nonstandard reals in the set of real numbers, in a
similar vein as the Russell set contains a set not requested in its
construction, then it doesn't. (ZF's universe is the Russell set, thus
ZF is inconsistent as it contains an irregular set.)
In a two-dimensional space, then there is the consideration that there
would be infinitely many lines through the point, instead of just lef
and right, and then some wondering about how many sides the point has.
It has four or five, or perhaps six, the indefinite: three.
Consider the Banach-Tarski 3-D ball decomposition "paradox". Take a
ball, topologically surgically decompose it into four or five specific
"nonmeasurable" sets. Recompose, through only translation and rotation
and no scaling, those partitions into two identical balls of equal
volume, equal to the initial volume. I see that as partitioning the
neighbor points, so no definite real-valued coordinate item is next to
one of its indefinite neighbors in a partition, nor any indefinite
point to one of its, when the indefinite points go from one- to
two-sided, in the recomposition as they have no definite address, they
neighbor to the nearest definite real and the result is
indistinguishable form the original.
Obviously where there are iota values there aren't any non-measurable
sets of reals, by the argument that is generally used, that there would
be ways to partition the unit interval into eta_1 or what-have-you many
equal-sized partitions were they to be measurable, thus they are
"non-measurable." I'd rather be able to sum the differentials between
zero and one and know the answer is one. If you could partition the
unit interval in that manner, there would not be non-measurable sets,
where as I understand it the existence of non-measurable sets hinges on
the Vitali and similar results. (Do you know any proofs of the
existence of non-measurable sets that aren't?) I seem to recall that
without non-measurable sets: CH in ZF. (ZF is inconsistent.)
Anyways, this polydimensionality of points is quite underdeveloped, but
I think that further development of it will lead to mathematics that is
useful for explaining point particles in nature, why the electron
deflects twice as much as the photon and etc.
Ross
Tony Orlow
> David R Tribble wrote:
>>> Over the last few months I've been noodling around with the concept
>>> of an extension to the reals, defining real-like numbers that are
>>> larger than any regular real.
>
> Chas Brown wrote:
>>> Nice job. In conversations with Tony Orlow, I also thought of a similar
>>> system (essentially, an ordered field extension of R using a polynomial
>>> basis over some "unit infinity" B).
>
> David R Tribble wrote:
>>> I was hoping this conversation would go quite a bit further
>>> before Tony's name was mentioned. His "unit infinity" is an
>>> extermely flawed and inconsistent concept from the get go.
>>> Oh well.
>
> Tony Orlow wrote:
>> Did you honestly think that was a reasonable expectation? You seem to
>> have agreed that "larger than any finite" is a reasonable definition for
>> "infinite" That's a good start, a unit infinity. Welcome to the club.
>
> Please read the article. My suprareals are not infinite numbers.
I have now read it. Yes, you are very careful not to call them
"infinite", and even to point out that, even though you illustrate them
as residing colinear with the reals, they are not really in that
relationship. I didn't see the point in tiptoeing around that,
personally. Your h-numbers are "larger than any finite", meaning farther
along the line from 0.
> They are numbers that are simply larger than the standard reals.
> In most other respects, they act just like reals. See Theorem I
> (infinity) in particular:
> -oo < h < +oo for all h in H.
>
>
Yes, I see that you made a distinction between absolute oo and the
h-numbers. I agree with that, just like infinitesimals are not absolute 0.
Okay, just as long as you don't start calling them Big'Uns and Lil'Uns.
Them's mines. ;)
>> ... which are based on nested powers of 2 to
>> enumerate the reals. The T-riffics are a simple extension of the normal
>> digital numbers system, based on sums of powers of 2, or any number
>> base, preferably prime. They also correspond to Robinson's illimited
>> numbers, uncountably distant from each other, with countable rational
>> neighborhoods of digits surrounding limit points.
>
> I don't think you understand the hyperreals of NSA.
>
> My suprareals resemble the hyperreals, or at least the suprareals
> in H1 and L1. I'm not sure how they're related, and I don't think the
> hyperreals are related to the rest of the suprareal hierachy (H2, L1,
> H3, L3, etc.).
>
>
Well, at least we both agree with Robinson that there's no smalleest
infinity in such a system.
>> In any case, your
>> "new" idea is not much different, at first glance, from my old and
>> "extremely flawed" concept. I'll print out and peruse your theory, and
>> then give more detailed comments, okay? Good job!
>>
>
> It's completely different. For one, it uses axioms to define the
> existence and properties of the suprareals. You don't have any.
> Secondly, your "unit infinity" is supposed to be equal to the
> cardinality of both [0,1] and N, which is inconsistent within standard
> arithmetic. And there are more flaws, which have been pointed
> out to you many times.
Big'un is the number of reals in (0,1] and the number of elements in *N,
the hypernaturals, both uncountable. I've made that clear multiple times.
>
> But please, this thread is supposed to be about my theory, not
> yours. I'll be happy to join the thread you start for yours.
>
>
Okay, just noting similarities.
> Ross A. Finlayson wrote:
>>> Are you able to formalize otherwise inaccessible or what are
>>> deemed paradoxical true results using your system?
>>> Such as?
>
> Tony Orlow wrote:
>> Oh, can you resolve the Continuum Hypothesis, or bring the subset
>> relation in line with infinite set size measures? Can you resolve the
>> difference between Galilean models of bijection and notions of set
>> density? You know, stuff like that.....
>
> Those have all been resolved, regardless of your opinions about them.
>
> The CH has been shown to be unresolvable within ZFC, so I would
> speculate that the only way to "resolve" it would be to accept CH or
> ~CH as an additional axiom (which seems to be the usual approach)
> or to try to add new axioms that force CH or ~CH.
>
> At any rate, my little additional axioms won't do that.
>
Oh well.
I had a few comments:
1. In the section "Even More Numbers," you say, "In fact it would appear
that every h-number can be represented as a polynomial over powers of
eta_1." Is that to say that one cannot have log_2(eta_1) and produce
another h-number using that function? How many bits are required to list
eta_1 elements?
2. When you speak of the h-numbers as being disconnected, with the
intervening set of standard reals between their negative and positive
elements, does it not occur to you that including the ih-numbers creates
the exact same situation for the standard reals, such that the positives
and negatives always have something between them? Does this make the
reals not a continuous set?
3. At the end of "Still Bigger Sets", you say, "Every element in this
set is either a real or an h-numbers (sic), an ih-number, a real plus an
ih-number, or an h-number plus an ih-number." Can it not also be a real
plus an h-number, or even a real plus an h-number plus an ih-number?
4. "An Uncountable Hierarchy" struck me as odd, coming from you, David.
On the one hand, you are enumerating a sequence of sets, each defined as
being the elements larger than all elements in the previous set (rather
like limit ordinals) but then you suggest that each set may be numbered
with a real. Are you suggesting an uncountable sequence of sets, which
you previously proclaimed to be an idea without any sense? What is the
set H_pi the set of, all elements larger than the element of H_x, where
x is the predecessor to pi in the natural order of the reals? I can see
how this might be defined, if each H_x uses an eta_x which is equal to
eta_1 to the x power, but I didn't see that defined there, and I imagine
you don't want to get that specific about the measure of the etas. I
could be wrong.
Anyways, that's my comments. Happy thinking!
Tony
cbr...@cbrownsystems.com
The second version of Axiom 8 states that there is an inverse for all h
in H; from which it follows immediately that there exists x such that
x*(n1 - 1) = 1.
The first version states only that there exists e1 such that e1*n1 = 1.
It must then be proven that, for example, there also exists x such that
x * (n1 - 1) = 1.
(I'll skip some rigor here). But if we try to find such an x by
(essentially) "long division", we get first that e1*(n1 - 1) = 1 - e1
(i.e., "1 divided by (n1 - 1) is e1 with a remainder of e1").
And (continuing our "long division"), then (e1 + e1^2)(n1 - 1) = 1 -
e1^2, and then that (e1 + e1^2 + e1^3)*(n1-1) = 1 - e1^3, and in
general (e1 + e1^2 + e1^3 + ... + e1^m)*(n1 - 1) = 1 - e1^m.
So we find that x = 1/(n1 - 1) = e1 + e1^2 + e1^3 + ... + e1^m + ... ;
but this is not a polynomial in e1 as it has an infinite number of
terms; so it is not in L.
(Side Bar: There are formulations of polynomials called "formal power
series" where such "infinite" polynomials are allowed. They come up in
combinatorics, amongst other places. Keywords: generating function,
formal power series).
Another (different) problem with your formulation is that (contrary to
theorem 18), H u R u L is not closed w.r.t to addition; because e1 + n1
is not in any of the sets H, R or L..
That's what he means; the "leading coefficient" is the coefficient of
the largest power of n1 which is non-zero.
> Also bear in mind that for higher orders of suprareals, the
> polynomials over n2 (n3, n4, etc.) has coefficients that are not
> limited to reals but to all the suprareals of lower orders.
> For example, for h in H2,
> h = x0 + x1 n2^1 + x2 n2^2 + ... + xn n2^n
> where each coefficient x0, x1, x2, ..., xn can itself be a suprareal
> in H1, so that they are polynomials over n1, e.g.:
> x0 = y0 + y1 n1^1 + y2 n1^2 + ... + ym n1^m
Usually, when we speak of polynomial rings such as K[n1, n2] (where K
is some ring), we include such polynomials as n1 + n1*n2 + n2 (i.e.,
where n1 and n2 are "mixed"). This can be thought of also as the
polynomial n1 + (1 + n1)*n2. In the former case, we have a polynomial
in n1 and n2 with coefficients in K; in the latter (equivalent) case,
it's a polynomial in n2, with coefficients in K[n1].
For suprareals, the latter approach is useful for defining the ordering
needed to guarantee that each extension is indeed an ordered field; but
other than that, the two formulations are equivalent.
Cheers - Chas
Ross A. Finlayson
> David R Tribble wrote:
> > David R Tribble wrote:
> > >> I've written an article exploring the effects of adding a few axioms to
> > >> standard arithmetic to extend the reals with what I call, for lack of
> > >> a better term, "h-numbers". Briefly, an h-number has a magnitude
> > >> greater than any real. The first axiom states that n1 (eta_1), a
> > >> primitive h-number constant, exists, and that x < n1 for all x in R.
> > >> >From there, an entire set H of h-numbers is defined as containing other
> > >> h-numbers, being sums and products of reals and n1.
> Usually, when we speak of polynomial rings such as K[n1, n2] (where K
> is some ring), we include such polynomials as n1 + n1*n2 + n2 (i.e.,
> where n1 and n2 are "mixed"). This can be thought of also as the
> polynomial n1 + (1 + n1)*n2. In the former case, we have a polynomial
> in n1 and n2 with coefficients in K; in the latter (equivalent) case,
> it's a polynomial in n2, with coefficients in K[n1].
>
> For suprareals, the latter approach is useful for defining the ordering
> needed to guarantee that each extension is indeed an ordered field; but
> other than that, the two formulations are equivalent.
>
> Cheers - Chas
Hello,
Browsing arXiv the other day I noticed there is a recent paper about
Non-Dedekindian numbers. The author claims in the abstract that all
types of hyperreals are the soi-disant "non-Dedekindian" numbers.
http://arxiv.org/abs/math.GM/0612590
The complete ordered field, is the complete ordered field. Where there
is only one of those, hyperreals are them. If there are infinitesimals
in the reals, they're reals. The real number are already the compete
ordered field.
Schmieden and Laugwitz' are not hyperreals.
Built on axiomatics of integers, each of the above systems (hyper-,
super-, supra-, partially ordered ring of with rather restricted
transfer principle, reals) is incomplete.
Ross
cbr...@cbrownsystems.com
Except for the fact that the suprareals don't form a "line"; they are
toplogically disconnected. By "disconnected", I mean that the
suprareals are the union of two non-empty disjoint open sets.
The real number line is /not/ the union of two non-empty disjoint open
sets. Lines are connected; the suprareals are not connected; therefore
the suprareals are not a line.
<snip>
> I had a few comments:
>
> 1. In the section "Even More Numbers," you say, "In fact it would appear
> that every h-number can be represented as a polynomial over powers of
> eta_1." Is that to say that one cannot have log_2(eta_1) and produce
> another h-number using that function? How many bits are required to list
> eta_1 elements?
(i) "2^eta_1" is not a polynomial in eta_1; so "2^eta_1" is not
meaningful as a suprareal. It follows that "log_2(eta_1)" is not
meaningful for suprareals.
(ii) eta_1 is not a cardinality; so it makes no sense to say "this set
has eta_1 elements"; just as it makes no sense to say "this set has
1/sqrt(2) elements".
One has at best "the cardinality of the suprareals <= eta_1 is the same
as the cardinality of the reals". Of course it is also true (as DT
proves) that "the cardinality of the suprareals > eta_1 is the same as
the cardinality of the reals"; and "the cardinality of the suprareals
<= eta_1 is the same as the cardinality of the suprareals <=
1/sqrt(2)".
(iii) There are as many suprareals as there are real numbers.
Therefore, you cannot in any case "list" the elements of the
suprareals, just as you cannot list the real numbers.
>
> 2. When you speak of the h-numbers as being disconnected, with the
> intervening set of standard reals between their negative and positive
> elements, does it not occur to you that including the ih-numbers creates
> the exact same situation for the standard reals, such that the positives
> and negatives always have something between them? Does this make the
> reals not a continuous set?
(i) There is a difference between saying "A is a connected subset of
B", "A is a dense subset of B", and "A is the image of a continuous
function of R -> B". These concepts are related, but not synonymous.
(ii) In the suprareals, a subset Y is disconnected iff Y is the union
of two disjoint, open sets under the subspace topology of the
suprareals (which I am assuming is the topology of open intervals).
That is to say, for a subset Y of the suprareals to be disconnected,
there must be two (non-empty) sets Y' and Y'', such that:
Y = Y' u Y'', and
Y' n Y'' is empty; and
there are open sets U, V in the suprareals such that Y' = U n Y, Y'' =
V n Y.
(iii) The subspace topology on R has as its basis the intersection of
open intervals in the suprareals and the set R. Let r be any element of
R and e1 = 1/n1. Then (r - e1, r + e1) is an open interval in the
suprareals, and (r - e1, r + e1) n R = r.
Therefore every point in R is open in the subspace topology, and every
point is also closed in the subspace topology; so R gets the discrete
topology. Thus R is a totally disconnected subset of the suprareals. (I
erred when I previously stated that R inherits the usual topology).
(iv) In the suprareals R u L is disconnected. Let Y' = {x : x in R u L,
x > 0 and 1/x in R u L}. Y' is an open set (it is the union of open
intervals (x, x+1) for x in Y'). Let Y'' = {x : x in R u L and (x<=0 or
1/x not in R u L)}. Y'' is an open set (it is the union of open
intervals (x-1, x) for x in Y''). R u L = Y' u Y''; Y' n Y'' is empty,
and thus R u L is a disconnected subset of the suprareals.
>
> 3. At the end of "Still Bigger Sets", you say, "Every element in this
> set is either a real or an h-numbers (sic), an ih-number, a real plus an
> ih-number, or an h-number plus an ih-number." Can it not also be a real
> plus an h-number, or even a real plus an h-number plus an ih-number?
>
(i) Suprareals in H u R are polynomials. H is the set of suprareals of
degree 1 or greater. R is the set of suprareals of degree 0. A
polynomial of degree 1 or greater plus a polynomial of degree 0 is,
again, a polynomial of degree 1 or greater. So any element of H plus
any element of R is again an element of H.
(ii) A better (and generally less confusing) formulation of the
suprareals is to refer to them as the closure of <H, R> under
multiplication, division, addition and subtraction; which is to say
that they are rational expressions of the form p/q, where p and q are
polynomials over some set of eta_'s with coefficients in R; e.g:
x = (1 + 2*eta_1)/(3 - eta_1 + 4*eta_1^2)
What we have been calling "H" is then the set of all polynomials p of
degree 1 or greater (i.e., where the "denominator" polynomial is 1).
"R" is the set of all polynomials of degree 0. What we have been
calling "L" is the set of expressions of the form r + 1/p, where r is a
real, and p is a polynomial of degree 1 or greater.
This scheme does not account for all suprareals (as noted elsewhere).
For example, it does not contain p + r + 1/q for polynomials p and q
and real r. However, it can be written as
(p*q + r*q + 1)/(q^2)
which /is/ of the above form (ratio of polynomials).
> 4. "An Uncountable Hierarchy" struck me as odd, coming from you, David.
> On the one hand, you are enumerating a sequence of sets, each defined as
> being the elements larger than all elements in the previous set (rather
> like limit ordinals) but then you suggest that each set may be numbered
> with a real. Are you suggesting an uncountable sequence of sets, which
> you previously proclaimed to be an idea without any sense?
(i) It's not an uncountable sequence; because a sequence is always, by
definition, countable (that's the part that is "an idea that doesn't
make any sense").
(ii) The numbers in H_R are (rational expressions of) polynomials in a
finite set of eta_'s with coefficients in R, with a finite number of
terms.
Say p and q are polynomials in H_R.
Then p is a polynomial over some finite set of eta_'s, call them eta_a,
eta_b, .., eta_m, where a, b, ..., m are real numbers; and having
coefficients in the real numbers, e.g:
p = 1 + pi*eta_a^2 - sqrt(2)*eta_b*eta_c + ... - 7*eta_m.
Note that using this terminology, eta_b * eta_c is NOT EQUAL TO
eta_(b*c). It remains the polynomial expression eta_b * eta_c. The real
number labels a, b, c, etc. are only there to indicate an ordering.
Suppose r is a polynomial of the form eta_r, eta_s, eta_t, ..., eta_z.
Then p - q is a polynomial in eta_a, eta_b, eta_c, ..., eta_m, eta_r,
eta_s, ..., eta_z.
Since there are only a finite number of eta_'s, there is a "largest"
eta_x, where x >= a, b, c, ..., z /using the usual ordering of the
reals/.
Then we say p > q if the coefficient of the largest power of eta_x in p
- q is > 0.
To extend this to rational expressions, we say that p/q > r/s iff p*s -
q*r > 0 (with a few modifications to handle multiplication when q or s
< 0).
> What is the
> set H_pi the set of, all elements larger than the element of H_x, where
> x is the predecessor to pi in the natural order of the reals?
(ii) Your statement makes no sense; because there is no such
"predecessor to pi" in the usual ordering of the reals.
(iii) Think of H_pi as being related to the set of all /finite/ subsets
of (0, pi] as a subset of R.
Each of these finite subsets then defines a finite set of eta_'s, over
which some set of polynomials is then defined over these eta_'s with
coefficients in R
H_pi is the union of all rational expressions of these polynomials,
taken over all finite subsets of (0,pi].
For example, {1, sqrt(2), 3} is a finite subset of (0, pi]. So
x = (1 + eta_(sqrt(2)) + (eta_3)^2)/(1 - eta_1)
is in H_pi; but
x = 1/eta_4
is not in H_pi, because 4 > pi, and thus 4 is not in (0, pi].
(iii) It follows that if 0 < x < y where x and y are real numbers, then
H_x is a proper subset of H_y.
So what he has described is a total order on sets, which is /not/
synonymous with "a sequence of sets".
> I can see
> how this might be defined, if each H_x uses an eta_x which is equal to
> eta_1 to the x power, but I didn't see that defined there, and I imagine
> you don't want to get that specific about the measure of the etas. I
> could be wrong.
>
(iv) You are.
Cheers - Chas
Tony Orlow
I'm sorry Chas, but that doesn't make sense to me. In the finite case it
does not hold. Say we have n1=4 and e1=1/4. 1/3 does not equal 1/4 with
a remainder of 1/4. 1/(n1-1)=((n1-1)+1)/(n1-1)-(n1-1)/(n1-1)= n1/(n1-1)-1.
Sorry, David, but I have to put this in terms of T-riffics.
Say we use base 2 T-riffics, and n1=1:000...000.000...000
n1-1=0:111...111.000...000. 1/(n-1)=0.000...001:000...001::::. That is,
in T-riffics, 1/(Big'un-1)= sum(n=-1->-oo: Big'un^n). This sum actually
holds for all bases.
>
> And (continuing our "long division"), then (e1 + e1^2)(n1 - 1) = 1 -
> e1^2, and then that (e1 + e1^2 + e1^3)*(n1-1) = 1 - e1^3, and in
> general (e1 + e1^2 + e1^3 + ... + e1^m)*(n1 - 1) = 1 - e1^m.
>
> So we find that x = 1/(n1 - 1) = e1 + e1^2 + e1^3 + ... + e1^m + ... ;
> but this is not a polynomial in e1 as it has an infinite number of
> terms; so it is not in L.
Actually that sounds like the result with the T-riffics: an unending sum
of subsequent infinitesimal levels.
>
> (Side Bar: There are formulations of polynomials called "formal power
> series" where such "infinite" polynomials are allowed. They come up in
> combinatorics, amongst other places. Keywords: generating function,
> formal power series).
>
> Another (different) problem with your formulation is that (contrary to
> theorem 18), H u R u L is not closed w.r.t to addition; because e1 + n1
> is not in any of the sets H, R or L..
>
Uh, yeah, the set of ih numbers needs to to be embedded in each real
unit interval, and likewise with each standard real "halo" about each
h-number. Any combination of the three should suffice as an element of
this universe.
Lexicographically!! :)
>> Also bear in mind that for higher orders of suprareals, the
>> polynomials over n2 (n3, n4, etc.) has coefficients that are not
>> limited to reals but to all the suprareals of lower orders.
>> For example, for h in H2,
>> h = x0 + x1 n2^1 + x2 n2^2 + ... + xn n2^n
>> where each coefficient x0, x1, x2, ..., xn can itself be a suprareal
>> in H1, so that they are polynomials over n1, e.g.:
>> x0 = y0 + y1 n1^1 + y2 n1^2 + ... + ym n1^m
>
> Usually, when we speak of polynomial rings such as K[n1, n2] (where K
> is some ring), we include such polynomials as n1 + n1*n2 + n2 (i.e.,
> where n1 and n2 are "mixed"). This can be thought of also as the
> polynomial n1 + (1 + n1)*n2. In the former case, we have a polynomial
> in n1 and n2 with coefficients in K; in the latter (equivalent) case,
> it's a polynomial in n2, with coefficients in K[n1].
>
> For suprareals, the latter approach is useful for defining the ordering
> needed to guarantee that each extension is indeed an ordered field; but
> other than that, the two formulations are equivalent.
>
> Cheers - Chas
>
Cheers
David R Tribble
>> Later you construct HuRuL in a way that essentially boils down to
>> R[X,Y]/(XY-1) and claim that this is a field.
>> However, Theorem 18c. fails to be true (at least with the first version
>> of Axiom 8; the second version is not equivalent):
>> In R[X,Y]/(XY-1), X+1 has no inverse!
>
David R Tribble wrote:
>> I don't see why they are not equivalent. And why is there no inverse?
>
Chas Brown wrote:
> The second version of Axiom 8 states that there is an inverse for all h
> in H; from which it follows immediately that there exists x such that
> x*(n1 - 1) = 1.
>
> The first version states only that there exists e1 such that e1*n1 = 1.
> It must then be proven that, for example, there also exists x such that
> x * (n1 - 1) = 1.
>
> (I'll skip some rigor here). But if we try to find such an x by
> (essentially) "long division", we get first that e1*(n1 - 1) = 1 - e1
> (i.e., "1 divided by (n1 - 1) is e1 with a remainder of e1").
>
> And (continuing our "long division"), then (e1 + e1^2)(n1 - 1) = 1 -
> e1^2, and then that (e1 + e1^2 + e1^3)*(n1-1) = 1 - e1^3, and in
> general (e1 + e1^2 + e1^3 + ... + e1^m)*(n1 - 1) = 1 - e1^m.
>
> So we find that x = 1/(n1 - 1) = e1 + e1^2 + e1^3 + ... + e1^m + ... ;
> but this is not a polynomial in e1 as it has an infinite number of
> terms; so it is not in L.
>
> (Side Bar: There are formulations of polynomials called "formal power
> series" where such "infinite" polynomials are allowed. They come up in
> combinatorics, amongst other places. Keywords: generating function,
> formal power series).
Yes, I had suspected something like that was the case, since 1/h for
an arbitrary suprareal h, being a polynomial, is a non-terminating
polynomial (a la Newton's expansion of 1/P(x), if I remember
correctly). I didn't follow up on it, though.
Limiting the suprareals to finite-power polynomials makes the
handling of their inverses tricky.
> Another (different) problem with your formulation is that (contrary to
> theorem 18), H u R u L is not closed w.r.t to addition; because e1 + n1
> is not in any of the sets H, R or L..
Yes, I got ahead of myself there. I was thinking of the total union
of all suprareal polynomials, including sums of h-numbers and
ih-numbers. This is one of the sections that needs reworking,
I suppose.
David R Tribble wrote:
>> In other words, shouldn't it be that the coefficient for the largest
>> power of n1 determines the order? (Or is that what you meant?)
>
Chas Brown wrote:
> That's what he means; the "leading coefficient" is the coefficient of
> the largest power of n1 which is non-zero.
Thanks, thought so.
David R Tribble wrote:
>> Also bear in mind that for higher orders of suprareals, the
>> polynomials over n2 (n3, n4, etc.) has coefficients that are not
>> limited to reals but to all the suprareals of lower orders.
>> For example, for h in H2,
>> h = x0 + x1 n2^1 + x2 n2^2 + ... + xn n2^n
>> where each coefficient x0, x1, x2, ..., xn can itself be a suprareal
>> in H1, so that they are polynomials over n1, e.g.:
>> x0 = y0 + y1 n1^1 + y2 n1^2 + ... + ym n1^m
>
Chas Brown wrote:
> Usually, when we speak of polynomial rings such as K[n1, n2] (where K
> is some ring), we include such polynomials as n1 + n1*n2 + n2 (i.e.,
> where n1 and n2 are "mixed"). This can be thought of also as the
> polynomial n1 + (1 + n1)*n2. In the former case, we have a polynomial
> in n1 and n2 with coefficients in K; in the latter (equivalent) case,
> it's a polynomial in n2, with coefficients in K[n1].
>
> For suprareals, the latter approach is useful for defining the ordering
> needed to guarantee that each extension is indeed an ordered field; but
> other than that, the two formulations are equivalent.
Yes, and it follows that when g is a suprareal in H2 having
coefficients in H1, the resulting terms are products of eta_1^n
and eta_2^m, as expected.
Thanks very much for the interest and the feedback. I hope to hammer
the article into better shape soon.
David R Tribble
>> Please read the article. My suprareals are not infinite numbers.
>
Tony Orlow wrote:
>> I have now read it. Yes, you are very careful not to call them
>> "infinite", and even to point out that, even though you illustrate them
>> as residing colinear with the reals, they are not really in that
>> relationship. I didn't see the point in tiptoeing around that,
>> personally. Your h-numbers are "larger than any finite", meaning farther
>> along the line from 0.
>
Chas Brown wrote:
> Except for the fact that the suprareals don't form a "line"; they are
> toplogically disconnected. By "disconnected", I mean that the
> suprareals are the union of two non-empty disjoint open sets.
>
> The real number line is /not/ the union of two non-empty disjoint open
> sets. Lines are connected; the suprareals are not connected; therefore
> the suprareals are not a line.
Exactly. (I appreciate the answers, Chas.)
I thought my diagrams made this clear.
Tony Orlow wrote:
>> I had a few comments:
>>
>> 1. In the section "Even More Numbers," you say, "In fact it would appear
>> that every h-number can be represented as a polynomial over powers of
>> eta_1." Is that to say that one cannot have log_2(eta_1) and produce
>> another h-number using that function? How many bits are required to list
>> eta_1 elements?
>
Chas Brown wrote:
> (i) "2^eta_1" is not a polynomial in eta_1; so "2^eta_1" is not
> meaningful as a suprareal. It follows that "log_2(eta_1)" is not
> meaningful for suprareals.
>
> (ii) eta_1 is not a cardinality; so it makes no sense to say "this set
> has eta_1 elements"; just as it makes no sense to say "this set has
> 1/sqrt(2) elements".
>
> One has at best "the cardinality of the suprareals <= eta_1 is the same
> as the cardinality of the reals". Of course it is also true (as DT
> proves) that "the cardinality of the suprareals > eta_1 is the same as
> the cardinality of the reals"; and "the cardinality of the suprareals
> <= eta_1 is the same as the cardinality of the suprareals <=
> 1/sqrt(2)".
>
> (iii) There are as many suprareals as there are real numbers.
> Therefore, you cannot in any case "list" the elements of the
> suprareals, just as you cannot list the real numbers.
Also, it makes no sense to talk about "how many bits" could be
used to encode a given suprareal, since eta_1 cannot be represented
in a binary notation, since it's not a real. It's kind of like asking
how many bits are required to encode i or w (omega).
Tony Orlow wrote:
>> 4. "An Uncountable Hierarchy" struck me as odd, coming from you, David.
>> On the one hand, you are enumerating a sequence of sets, each defined as
>> being the elements larger than all elements in the previous set (rather
>> like limit ordinals) but then you suggest that each set may be numbered
>> with a real. Are you suggesting an uncountable sequence of sets, which
>> you previously proclaimed to be an idea without any sense?
>
Chas Brown wrote:
> (i) It's not an uncountable sequence; because a sequence is always, by
> definition, countable (that's the part that is "an idea that doesn't
> make any sense").
Exactly.
> Note that using this terminology, eta_b * eta_c is NOT EQUAL TO
> eta_(b*c). It remains the polynomial expression eta_b * eta_c. The real
> number labels a, b, c, etc. are only there to indicate an ordering.
Exactly.
Tony Orlow wrote:
>> What is the
>> set H_pi the set of, all elements larger than the element of H_x, where
>> x is the predecessor to pi in the natural order of the reals?
>
Chas Brown wrote:
> (ii) Your statement makes no sense; because there is no such
> "predecessor to pi" in the usual ordering of the reals.
Exactly.
> (iii) It follows that if 0 < x < y where x and y are real numbers, then
> H_x is a proper subset of H_y.
>
> So what he has described is a total order on sets, which is /not/
> synonymous with "a sequence of sets".
Exactly.
(Thanks, Chas.)
cbr...@cbrownsystems.com
It depends on what context (usually a ring) we are working in.
We don't /usually/ say "40 divided by 7 is 11*(1/2) remainder 3*(1/2)";
but there are situations in which we might. For example, if we have 40
pairs of shoes, and we split them up evenly between 7 people, we can
give 11 individual shoes (each 1/2 of a pair) to each person, with
three shoes ("1/2 pairs") left over
(Disregard for the moment that usually individual shoes lack utility.
Perhaps these shoes have sentimental value for these 7 persons; or
perhaps they don't care if their shoes actually match).
In that case we might say "40 pairs of shoes divided by 7 is 11 shoes
each, with 3 shoes left over" to mean "7*11 shoes + 3 shoes = 40 pairs
of shoes", equivalently, 7*11/2 = 40 - 3/2.
"1/2" of a pair of shoes make at least some sense in this context (one
shoe). We wouldn't say that "each person should get 5 and 5/7th's pairs
of shoes", because the natural context here is either pairs of shoes,
or "1/2" pairs of shoes (a.k.a., individual shoes); but "1/7th" of a
pair of shoes has no useful meaning for us (in my made up example).
Similarly, /in a suitable context/, "1 divided by (4 - 1) is 1/4
remainder 1/4" simply means that 1/4*(4 - 1) + 1/4 = 1, equivalently
1/4*(4 - 1) = 1 - 1/4; or equivalently in your example, e1*(n1 - 1) = 1
- e1. This might be the case when numbers which can be represented as
some number of 1/4's are acceptable to us for some reason, but not,
say, 1/3's.
When the "numbers" we are dealing with are polynomials (as they are in
David's system), we might say things like "n1^2 + n1 + 1 divided by n1
+ 1 is n1 with remainder 1" to mean "n1*(n1+1) + 1 = n1^2 + n1 + 1".
This might be the case if 1/(n1+1) is not an acceptable "number" to us
(just as 1/7th of a pair of shoes was not acceptable to us), so the
statement "(n1^2 + n1 + 1)/(n1 + 1) = n1 + 1/(n1 + 1)" would not be
acceptable to us as an answer to "what 'number' do you mean by (n1^2 +
n1 + 1)/(n1 + 1)?"
> 1/(n1-1)=((n1-1)+1)/(n1-1)-(n1-1)/(n1-1)= n1/(n1-1)-1.
Remember that we are working in David's system here.
"1/(n1 -1)" is merely shorthand for "that Axiom8.1 suprareal number
which, when multiplied by (n1 - 1), yields 1".
But just because we can /write/ "1/(n1-1)" doesn't mean that such a
number actually exists in /his/ system.
If it exists, sure, it will follow that "1/(n1 -1) + 1/(n1 - 1) =
2/(n1-1)", and all sorts of other things are true. But this has nothing
to do with whether "1/(n1-1)" exists to start with.
Compare this with "let x be the natural number such that, when
mulitplied by 2, yields 1". We can certainly write '1/2' to /represent/
this natural number x; but that does not mean that there actually /is/
a /natural/ number x such that x*2 = 1.
> > So we find that x = 1/(n1 - 1) = e1 + e1^2 + e1^3 + ... + e1^m + ... ;
> > but this is not a polynomial in e1 as it has an infinite number of
> > terms; so it is not in L.
>
> Actually that sounds like the result with the T-riffics: an unending sum
> of subsequent infinitesimal levels.
>
It should even more "sound like" the "unending sum" 1/3 = 0.33333...,
as opposed to the "not unending sum" 1/25 = 0.04.
In the latter case, the "long-division-like" algorithm for finding the
digits in the decimal expansion at some point comes to a "stop" with a
"remainder" of 0; in former case this never happens. A similar thing is
happening in the Axiom 8.1 suprareals when we try to find an polynomial
expression for 1/(n1 - 1).
But in David's system, there are no "unending sums" allowed. /Every/
suprareal must be represented as the /finite/ sum of real numbers times
powers of n1 and real numbers times powers of e1.
Cheers - Chas
cbr...@cbrownsystems.com
> Thanks very much for the interest and the feedback. I hope to hammer
> the article into better shape soon.\
I continue to look on with interest.
WRT your original quest, I found the following very easy to understand
link regarding the hypereals:
http://mathforum.org/dr.math/faq/analysis_hyperreals.html
which made understanding an ultrafilter a piece of cake. It provides
equivalence classes which seem much more "subtle" than the equivalence
classes of the suprareals. Two suprareals p/q and r/s are equivalent
iff p*s = r*q; but two hyperreals are equivalent only if it pleases the
Great and Powerful Ultrafilter! (Pay no attention to that Axiom behind
the curtain...)
In general it seems like a "kewl" system, because we can easily talk
about non-standard versions of sin, cos, log, gamma, etc while
maintaining their values over an embedded version of R.; which is
currently impractical with the suprareals
Best of all, the limit of a sequence of hyperreals appears to be a
("large") sequence of limits of sequences of reals; so the problem of
limits is easily solved.
I still haven't thought of a good definition of limit of a sequence of
suprareals that yields a limit of (1, 1/2, 1/4, ..., 1/2^n, ...) = 0.
Cheers - Chas
Tony Orlow
> David R Tribble wrote:
>>> Please read the article. My suprareals are not infinite numbers.
>
> Tony Orlow wrote:
>>> I have now read it. Yes, you are very careful not to call them
>>> "infinite", and even to point out that, even though you illustrate them
>>> as residing colinear with the reals, they are not really in that
>>> relationship. I didn't see the point in tiptoeing around that,
>>> personally. Your h-numbers are "larger than any finite", meaning farther
>>> along the line from 0.
>
> Chas Brown wrote:
>> Except for the fact that the suprareals don't form a "line"; they are
>> toplogically disconnected. By "disconnected", I mean that the
>> suprareals are the union of two non-empty disjoint open sets.
>>
>> The real number line is /not/ the union of two non-empty disjoint open
>> sets. Lines are connected; the suprareals are not connected; therefore
>> the suprareals are not a line.
>
> Exactly. (I appreciate the answers, Chas.)
> I thought my diagrams made this clear.
>
>
And yet, when you introduce the ih-numbers as a non-real halo around 0,
then you have disconnected the positive reals from the negative reals,
as the positive h-numbers are separated by the relatively infinitesimal
expanse of standard reals between them. Where you include all standard
reals, all ih-numbers, and all h-numbers, they form a single line,
though in a sense it's a discontinuous line, since your system is
restricted to finite polynomials, which are countable.
> Tony Orlow wrote:
>>> I had a few comments:
>>>
>>> 1. In the section "Even More Numbers," you say, "In fact it would appear
>>> that every h-number can be represented as a polynomial over powers of
>>> eta_1." Is that to say that one cannot have log_2(eta_1) and produce
>>> another h-number using that function? How many bits are required to list
>>> eta_1 elements?
>
> Chas Brown wrote:
>> (i) "2^eta_1" is not a polynomial in eta_1; so "2^eta_1" is not
>> meaningful as a suprareal. It follows that "log_2(eta_1)" is not
>> meaningful for suprareals.
That follows from the h-numbers being restricted to polynomials over
eta1, but that restriction itself seems to be simply declared to
"appear" true, and I don't see why it follows. The fact that the inverse
of a finite polynomial is not generally a finite polynomial indicates
that this restriction makes some of the properties David desires in his
system impossible. I am suggesting that this restriction might not be
desirable. That's all.
>>
>> (ii) eta_1 is not a cardinality; so it makes no sense to say "this set
>> has eta_1 elements"; just as it makes no sense to say "this set has
>> 1/sqrt(2) elements".
eta_1 is not a fraction, but a number beyond all reals. While one could
consider a non-integral such number, one would do better to assume eta_1
to be integral, with fractional differences covered by its finite real
"halo".
>>
>> One has at best "the cardinality of the suprareals <= eta_1 is the same
>> as the cardinality of the reals". Of course it is also true (as DT
>> proves) that "the cardinality of the suprareals > eta_1 is the same as
>> the cardinality of the reals"; and "the cardinality of the suprareals
>> <= eta_1 is the same as the cardinality of the suprareals <=
>> 1/sqrt(2)".
Eh, cardinality. Do you want to be able to perform arithmetic on eta_1
or not?
>>
>> (iii) There are as many suprareals as there are real numbers.
>> Therefore, you cannot in any case "list" the elements of the
>> suprareals, just as you cannot list the real numbers.
>
> Also, it makes no sense to talk about "how many bits" could be
> used to encode a given suprareal, since eta_1 cannot be represented
> in a binary notation, since it's not a real. It's kind of like asking
> how many bits are required to encode i or w (omega).
>
>
The imaginaries are an analog of the reals, and omega's not a number
with any "value". If you have a number of elements, ceiling(log2) of
that number is the number of bits required for the list. You can have
infinite digital strings to the left, which represent infinite values,
ala p-adics or T-riffics.
> Tony Orlow wrote:
>>> 4. "An Uncountable Hierarchy" struck me as odd, coming from you, David.
>>> On the one hand, you are enumerating a sequence of sets, each defined as
>>> being the elements larger than all elements in the previous set (rather
>>> like limit ordinals) but then you suggest that each set may be numbered
>>> with a real. Are you suggesting an uncountable sequence of sets, which
>>> you previously proclaimed to be an idea without any sense?
>
> Chas Brown wrote:
>> (i) It's not an uncountable sequence; because a sequence is always, by
>> definition, countable (that's the part that is "an idea that doesn't
>> make any sense").
>
> Exactly.
>
>> Note that using this terminology, eta_b * eta_c is NOT EQUAL TO
>> eta_(b*c). It remains the polynomial expression eta_b * eta_c. The real
>> number labels a, b, c, etc. are only there to indicate an ordering.
>
> Exactly.
>
>
You first define the successive etas as being the set of objects larger
than all elements of the last. Of course, this means that each set
includes its successor, and you no longer have a hierarchy. I mean, how
do you have everything larger than everything larger than any finite?
Isn't that an empty set? So, the successive notion ultimately doesn't
make sense.
Then, you suggested an "uncountable hierarchy" using a continuous set to
index the levels. That relies on the notion of some infinitesimal unit,
or you don't have successive levels in your hierarchy.
Now, if you want to define an uncountable set of eta units, such that
each is a real power of some standard eta_1, then anything above eta_0
is beyond the finites, and any real difference in the eta number will
result in a value difference beyond any finite number. By introducing a
particular formula, you can get a system that works. But, don't listen
to me...
> Tony Orlow wrote:
>>> What is the
>>> set H_pi the set of, all elements larger than the element of H_x, where
>>> x is the predecessor to pi in the natural order of the reals?
>
> Chas Brown wrote:
>> (ii) Your statement makes no sense; because there is no such
>> "predecessor to pi" in the usual ordering of the reals.
>
> Exactly.
>
>
Then eta_pi is not a level in a hierarchy, is it?
>> (iii) It follows that if 0 < x < y where x and y are real numbers, then
>> H_x is a proper subset of H_y.
Or a proper superset, as I've said. If H_0 is all numbers greater than
all ih-numbers, that includes the finites AND the h-humbers. So, H_0
would include H_1.
>>
>> So what he has described is a total order on sets, which is /not/
>> synonymous with "a sequence of sets".
>
> Exactly.
>
> (Thanks, Chas.)
>
Yeah, thanks Chas.
cbr...@cbrownsystems.com
> David R Tribble wrote:
> > David R Tribble wrote:
> >>> Please read the article. My suprareals are not infinite numbers.
> >
> > Tony Orlow wrote:
> >>> I have now read it. Yes, you are very careful not to call them
> >>> "infinite", and even to point out that, even though you illustrate them
> >>> as residing colinear with the reals, they are not really in that
> >>> relationship. I didn't see the point in tiptoeing around that,
> >>> personally. Your h-numbers are "larger than any finite", meaning farther
> >>> along the line from 0.
> >
> > Chas Brown wrote:
> >> Except for the fact that the suprareals don't form a "line"; they are
> >> toplogically disconnected. By "disconnected", I mean that the
> >> suprareals are the union of two non-empty disjoint open sets.
> >>
> >> The real number line is /not/ the union of two non-empty disjoint open
> >> sets. Lines are connected; the suprareals are not connected; therefore
> >> the suprareals are not a line.
> >
> > Exactly. (I appreciate the answers, Chas.)
> > I thought my diagrams made this clear.
> >
> >
>
> And yet, when you introduce the ih-numbers as a non-real halo around 0,
> then you have disconnected the positive reals from the negative reals,
> as the positive h-numbers are separated by the relatively infinitesimal
> expanse of standard reals between them.
When I say "the line is connected; the suprareals are not connected", I
mean something quite specific: the surpareals are union of two
non-empty, disjoint open sets.
This very specific meaning of "disconnected" applies to the "usual"
meaning of disconnected - for example, if I remove the point 1 from the
real open interval (0,2), the resulting "broken line" is then the union
of (0, 1) and (1, 2); it has "two pieces" that "don't touch".
It turns out that not just the positive and negative reals are
disconnected in the suprareals; in fact, /every/ two disjoint sets R1
and R2 with R = R1 union R2 are disconnected (this is what "totally
disconnected" means).
> Where you include all standard
> reals, all ih-numbers, and all h-numbers, they form a single line,
I think what you mean to say is that they form a total order (sometimes
misleadingly called a linear order). The real line is one /example/ of
a total order; but there are many, many other types of total orders,
sometimes with quite bizarre properties which make them very much
"unlike" the real line.
> though in a sense it's a discontinuous line,
As I previously mentioned, "dense", "connected", and "continuous" have
different technical meanings. Usually, continuous is used to refer to
/functions/ which map one figure to another; for example mapping the
real line to a parabola. Continuous functions preserve interesting
features, one of which is connectedness.
Usually, we are talking about functions from the reals to the reals. So
when we say "y = x^2 is continuous", we really mean "the parabola is
just as connected as the real line". When we say "y = 1/x is not
continuous", we mean that the function y: R->R maps the real line
(which is connected) onto the hyperbola (which is not connected), so
the /function/ is not a continuous /function/ from R -> R
However, in our case we are not just considering functions from the
reals to the reals. We are in fact working in a system where there is
no continuous function from the real line to the suprareals (which is
not to say that there are no continuous functions from the suprareals
to the suprareals). So "continuous" becomes a very ambiguous thing to
say: do we mean a continuous function from the reals to the suprareals,
or a continuous function from the suprareals to the suprareals, or a
continuous function from the hyperreals to the suprareals, or what?
That's why I use "disconnected" rather than "discontinuous", so as to
be clear in my meaning. This isn't usually neccessary in the reals,
because the ideas are so closely connected when considering functions
from the reals to the reals, but we have left these safe confines, and
must be very careful with what we say.
> since your system is
> restricted to finite polynomials, which are countable.
>
No. Since the coefficients of the polynomials in question are real
numbers, there are, for example, an uncountable number of polynomials
of the form "a + b*eta_1".
BTW, the "reason" why the suprareals are not connected is not because
there are "too few" suprareals; it's because there are "too damned
many".
This is an example of where just relying on your "gut" can lead you
astray. Your "gut" may say that all we need to do is "insert" more
"suprareals" until "all the gaps are filled up", and then it will be
connected.
But that's what we just /did/ - we "inserted" a bunch of numbers to
"fill up" the reals to make the suprareals, and yet the result is not
"more connected", it's actually much /less/ connected.
> > Tony Orlow wrote:
> >>> I had a few comments:
> >>>
> >>> 1. In the section "Even More Numbers," you say, "In fact it would appear
> >>> that every h-number can be represented as a polynomial over powers of
> >>> eta_1." Is that to say that one cannot have log_2(eta_1) and produce
> >>> another h-number using that function? How many bits are required to list
> >>> eta_1 elements?
> >
> > Chas Brown wrote:
> >> (i) "2^eta_1" is not a polynomial in eta_1; so "2^eta_1" is not
> >> meaningful as a suprareal. It follows that "log_2(eta_1)" is not
> >> meaningful for suprareals.
>
> That follows from the h-numbers being restricted to polynomials over
> eta1, but that restriction itself seems to be simply declared to
> "appear" true, and I don't see why it follows.
I'm not sure what your objection is.
David is simply suggesting a particular mathematical object for our
perusal. Of course he "simply declares" that it has certain properties,
such as that it contains an element eta_1, that it is closed under
addition, multiplication, etc., so that we are sure that both he and we
are talking about the "same" mathematical object.
It indeed "follows" that the set of polynomials in eta_1 obeys these
properties, so it is an example of the kind of mathematical object he
is talking about. Since other people have studied polynomials
extensively, we can take advantage of their work to figure certain
things out (such as that we needed more than simply 1/eta_1 to
construct a field).
> The fact that the inverse
> of a finite polynomial is not generally a finite polynomial indicates
> that this restriction makes some of the properties David desires in his
> system impossible.
If David "desires" a property to be true, he can assert it so that we
know what property he means. Of course, adding such a property may
cause a contradiction. So it is wise to proceed step by step to see
what the implications of the properties he has already asserted are.
For example, he may have "wanted" to preserve connectedness by
"inserting" a bunch of infinitesimals; but it turns out that that
doesn't work.
(And "infinite polynomials", a.k.a, formal power series, have their own
problems - particularly when it comes to finding multiplicative
inverses.)
> I am suggesting that this restriction might not be
> desirable. That's all.
>
> >>
> >> (ii) eta_1 is not a cardinality; so it makes no sense to say "this set
> >> has eta_1 elements"; just as it makes no sense to say "this set has
> >> 1/sqrt(2) elements".
>
> eta_1 is not a fraction, but a number beyond all reals.
eta_1 is not a "number beyond all reals", except in a poetic sense.
It's a symbol; an abstraction.
> While one could
> consider a non-integral such number, one would do better to assume eta_1
> to be integral, with fractional differences covered by its finite real
> "halo".
One would do better to be specific about what one means, rather than
sling around mathematical sounding phrases without specific meaning.
"Fractional differences covered by its finite real "halo" " is hardly
comprehensible to me.
We are not considering "numbers" here in the usual sense. We are
considering mathematical objects which share many, but not all,
properties of what are usually called "numbers". We can define
something that acts very much like addition between the usual numbers;
so much so that we call it "adding two suprareals"; but it is important
to keep track of the fact that they are /not/ numbers as we usually
think of them - they are "abstract numbers".
As such, eta_1 is not a "quantity" in the sense that "5" is a
"quantity". And it's not a "measure" in the sense that we say "the
length of the diagonal of the unit square is sqrt(2)".
It's a symbol which represents an abstraction.
>
> >>
> >> One has at best "the cardinality of the suprareals <= eta_1 is the same
> >> as the cardinality of the reals". Of course it is also true (as DT
> >> proves) that "the cardinality of the suprareals > eta_1 is the same as
> >> the cardinality of the reals"; and "the cardinality of the suprareals
> >> <= eta_1 is the same as the cardinality of the suprareals <=
> >> 1/sqrt(2)".
>
> Eh, cardinality. Do you want to be able to perform arithmetic on eta_1
> or not?
>
The ability to "perform arithmetic" is different from the ability to
prove that there is a surjection from one set onto another set.
Cardinality is about the /existence/ (or non-existence) of functions
mapping sets onto other sets.
Arithmetic is about some /particular/ function(s) which map pairs of
elements in a set to single elements of a set, with very specific
constraints on that function. For example, for the function "+", we
usually require that a + 0 = a, and a + b = b + a.
The two concepts are quite different, particularly when we are talking
about "abstract numbers".
> >>
> >> (iii) There are as many suprareals as there are real numbers.
> >> Therefore, you cannot in any case "list" the elements of the
> >> suprareals, just as you cannot list the real numbers.
> >
> > Also, it makes no sense to talk about "how many bits" could be
> > used to encode a given suprareal, since eta_1 cannot be represented
> > in a binary notation, since it's not a real. It's kind of like asking
> > how many bits are required to encode i or w (omega).
> >
> >
>
> The imaginaries are an analog of the reals, and omega's not a number
> with any "value". If you have a number of elements, ceiling(log2) of
> that number is the number of bits required for the list.
This begs the question of what you mean by "a number of elements".
For example, we have already determined that ceiling(log2(eta_1)) is
meaningless; because there is no suprareal (and certainly no real!)
which is of the form "a^eta_1".
And even in the real numbers, there is no meaning to
"ceiling(log2(-50))", despite the fact that -50 is not a fraction, and
is indeed a number.
You are confusing the cardinals ("how many") with the reals ("how
much") because the reals contain a copy of the /finite/ cardinals (the
naturals). This allows us to make certain calculations regarding
/finite/ cardinalities "as if" cardinalities were real numbers; but
cardinalities are /not/ real numbers, and real numbers are /not/
cardinalities; and suprareals are another thing entirely.
> You can have
> infinite digital strings to the left, which represent infinite values,
> ala p-adics or T-riffics.
>
> > Tony Orlow wrote:
> >>> 4. "An Uncountable Hierarchy" struck me as odd, coming from you, David.
> >>> On the one hand, you are enumerating a sequence of sets, each defined as
> >>> being the elements larger than all elements in the previous set (rather
> >>> like limit ordinals) but then you suggest that each set may be numbered
> >>> with a real. Are you suggesting an uncountable sequence of sets, which
> >>> you previously proclaimed to be an idea without any sense?
> >
> > Chas Brown wrote:
> >> (i) It's not an uncountable sequence; because a sequence is always, by
> >> definition, countable (that's the part that is "an idea that doesn't
> >> make any sense").
> >
> > Exactly.
> >
> >> Note that using this terminology, eta_b * eta_c is NOT EQUAL TO
> >> eta_(b*c). It remains the polynomial expression eta_b * eta_c. The real
> >> number labels a, b, c, etc. are only there to indicate an ordering.
> >
> > Exactly.
> >
> >
>
> You first define the successive etas as being the set of objects larger
> than all elements of the last. Of course, this means that each set
> includes its successor, and you no longer have a hierarchy.
I think you are confusing two different steps in his development.
In the first step, he starts with a set, call it T_1. He then defines a
set, T_2, which contains T_1 as a proper subset. He then "succesively"
continues this process, to define a sequence of sets (T_1, T_2, ...,
T_n, ...), and proves by induction that they have the property that for
naturals n < m, T_n < T_m.
In his second step, he generalizes his ideas by defining a /different/
uncountable collection of sets, S = {S_x}. He defines an ordering on
these sets which is the same as the usual ordering of the reals; and
proves that they have certain properties. One of these properties is
that for real numbers x < y, the set S_x is a proper subset of S_y.
> I mean, how
> do you have everything larger than everything larger than any finite?
The same way that you "have" all natural numbers, and yet you "have" no
natural number which is larger than any other natural number. If you
want to "define" something to be larger than all naturals, be my guest;
but that thing will not be a natural number (just as eta_1 is not a
natural number).
> Isn't that an empty set? So, the successive notion ultimately doesn't
> make sense.
>
Sure it does. It makes just as much sense as generating "successive"
prime numbers, without end. It's simply a fact that there is no largest
prime number; the sequence of primes goes on forever, just as the
sequence (eta_1, eta_2, eta_3, ...) goes on forever.
(Remember: eta_1, eta_2, etc. are not real numbers, and they are not
cardinalities. They are /abstractions/.)
> Then, you suggested an "uncountable hierarchy" using a continuous set to
> index the levels.
He doesn't rely on /continuity/ in his use of the reals as an index
set. He relies on their /ordering/. That's all that is important about
his use of the reals.
> That relies on the notion of some infinitesimal unit,
> or you don't have successive levels in your hierarchy.
>
You're badly misunderstanding what he's doing here. Let's take it a
step at a time. I know it won't look at all relevant at first; but stay
with it, because it is basically what David is doing.
Given some real number x > 0, suppose U is a /finite/ set of real
numbers {u_1, u_2, ..., u_n} such that for all u_i, 0 < u_i <= x.
You can imagine that there are many, many such finite sets for a given
real number x. Given a real number x, let us call the set of /all/ such
/finite/ sets "S_x".
So if U is a member of S_x, it is a set (not a real number). U itself
is a finite set of real numbers; and each of these real numbers is less
than or equal to x.
The members of S_x (sets) have the property that if U and V are in S_x,
then their union is /also/ in S_x (the union of two finite sets is also
a finite set; and each element in that union will be <= x). And their
intersection is also in S_x; even if their intersection is empty - the
empty set is also a finite set of real numbers less than x (0 is a
finite number too!).
It would be wrong to think that x/2 is a member of S_x. x/2 is a real
number, not a set of real numbers - {x/2} would be a member of S_x, but
not x/2.
Take a moment to consider the members of S_x; the finite sets which are
members of S_x.
Now, suppose y is a real number, and x < y. Then if U is a member of
S_x, each real number in U is also less than or equal to y (actually,
just less than).
That means that U must /also/ be a member of S_y. U is a finite set of
real numbers, each of which is <= y.
On the other hand, there are certainly sets in S_y that are not members
of S_x. For example the set {y} is a finite set of real numbers, and y
<= y; so the set {y} is a member of S_y, but not a member of S_x
(because y > x).
So S_x is therefore a proper subset of S_y. And it follows that given
any two positive reals x and y, x < y implies that S_x < S_y. For
example, S_(3/2) < S_(pi).
Now let S be the set of /all/ such sets S_x, for /all/ positive real
numbers x.
Each member of S is a set.
For each member V of S, there is a real number x such that V = S_x.
And for any two members V, W of S, it follows that either V < W or W <
V or W = V.
That's what I call "trichotomy"! So the sets in S form "a line" in your
wording: the sets in S are totally ordered by inclusion.
Hopefully, you agree that the above set S "makes sense".
Now, where in this discussion did I invoke "succesive reals", or
"infinitesimal units"? I simply defined a collection of sets {S_x},
indexed by the reals, and showed that they are totally ordered by
inclusion, in the same way that the reals are totally ordered: S_x <
S_y iff x < y.
That's really all David has done. Honestly. His "completed" uncountable
collection of sets H = {H_x} is very much related to the "completed"
set S = {S_x}. I would be happy to explain this, but first I would want
to know that you "understand" what I mean by the set S, and that that
set "makes sense" to you.
> Now, if you want to define an uncountable set of eta units, such that
> each is a real power of some standard eta_1...
If he wanted to do that, he wouldn't be using polynomials; because in a
polynomial, the powers of eta_1 are natural numbers.
Instead, because he /wants/ to use polynomials, he defines an
uncountable set of eta_'s, ordered as the reals are ordered. If x < y,
then r*eta_x^n < eta_y for all real numbers r, and all natural numbers
n.
> , then anything above eta_0
> is beyond the finites, and any real difference in the eta number will
> result in a value difference beyond any finite number. By introducing a
> particular formula, you can get a system that works. But, don't listen
> to me...
>
Instead, I urge you to "listen to" David, so that you can see how one
defines and examines mathematical objects.
> > Tony Orlow wrote:
> >>> What is the
> >>> set H_pi the set of, all elements larger than the element of H_x, where
> >>> x is the predecessor to pi in the natural order of the reals?
> >
> > Chas Brown wrote:
> >> (ii) Your statement makes no sense; because there is no such
> >> "predecessor to pi" in the usual ordering of the reals.
> >
> > Exactly.
> >
> >
>
> Then eta_pi is not a level in a hierarchy, is it?
>
That does not follow. What follows is that the "heirarchy" does not
have the same ordering as the naturals; instead it has the ordering of
the reals.
> >> (iii) It follows that if 0 < x < y where x and y are real numbers, then
> >> H_x is a proper subset of H_y.
>
> Or a proper superset, as I've said. If H_0 is all numbers greater than
> all ih-numbers, that includes the finites AND the h-humbers. So, H_0
> would include H_1.
No, you have misunderstood his construction if you think that.
Cheers - Chas
David R Tribble
>> The real number line is /not/ the union of two non-empty disjoint open
>> sets. Lines are connected; the suprareals are not connected; therefore
>
Tony Orlow wrote:
>> And yet, when you introduce the ih-numbers as a non-real halo around 0,
>> then you have disconnected the positive reals from the negative reals,
>> as the positive h-numbers are separated by the relatively infinitesimal
>> expanse of standard reals between them. Where you include all standard
>> reals, all ih-numbers, and all h-numbers, they form a single line,
>
Chas Brown wrote:
> It turns out that not just the positive and negative reals are
> disconnected in the suprareals; in fact, /every/ two disjoint sets R1
> and R2 with R = R1 union R2 are disconnected (this is what "totally
> disconnected" means).
>
> I think what you mean to say is that they form a total order (sometimes
> misleadingly called a linear order). The real line is one /example/ of
> a total order; but there are many, many other types of total orders,
> sometimes with quite bizarre properties which make them very much
> "unlike" the real line.
Right. Introducing the i-suprareals (infinitesimals) does not make the
existing reals unconnected. Think of the suprareals and i-suprareals
as sets of "numbers" separate from the set of reals.
Which means that it's not proper to think of the union of the reals
and the suprareals as a single "number line", but more as separate
disconnected number lines. More properly, they are disconnected
sets.
Tony Orlow wrote:
>> though in a sense it's a discontinuous line, since your system is
>> restricted to finite polynomials, which are countable.
>
Chas Brown wrote:
> No. Since the coefficients of the polynomials in question are real
> numbers, there are, for example, an uncountable number of polynomials
> of the form "a + b*eta_1".
>
> BTW, the "reason" why the suprareals are not connected is not because
> there are "too few" suprareals; it's because there are "too damned
> many".
>
> This is an example of where just relying on your "gut" can lead you
> astray. Your "gut" may say that all we need to do is "insert" more
> "suprareals" until "all the gaps are filled up", and then it will be
> connected.
>
> But that's what we just /did/ - we "inserted" a bunch of numbers to
> "fill up" the reals to make the suprareals, and yet the result is not
> "more connected", it's actually much /less/ connected.
Exactly. Even the simplest construction, x+eta_1 for any real x,
produces an uncountable set.
What is curious (and which I will probably add to the next revision)
is the fact that
x0 + eta_1
x1 eta_1^1
x2 eta_1^2
...
are unconnected (uncountable) sets. Any member of one set in the
list is less than any member of the next set (assuming all positive
real x's).
Chas Brown wrote:
>> (i) "2^eta_1" is not a polynomial in eta_1; so "2^eta_1" is not
>> meaningful as a suprareal. It follows that "log_2(eta_1)" is not
>> meaningful for suprareals.
>
Tony Orlow wrote:
>> That follows from the h-numbers being restricted to polynomials over
>> eta1, but that restriction itself seems to be simply declared to
>> "appear" true, and I don't see why it follows.
>
Chas Brown wrote:
> I'm not sure what your objection is.
>
> David is simply suggesting a particular mathematical object for our
> perusal. Of course he "simply declares" that it has certain properties,
> such as that it contains an element eta_1, that it is closed under
> addition, multiplication, etc., so that we are sure that both he and we
> are talking about the "same" mathematical object.
>
> It indeed "follows" that the set of polynomials in eta_1 obeys these
> properties, so it is an example of the kind of mathematical object he
> is talking about. Since other people have studied polynomials
> extensively, we can take advantage of their work to figure certain
> things out (such as that we needed more than simply 1/eta_1 to
> construct a field).
Yes. I'm trying to work out what is meant by eta_1^p for non-integer
p, e.g., eta_1^(1/2) or sqrt(eta_1). It follows from the axioms that
x < eta_1^p for all x in R and p > 0.
As Chas points out, my current definition of suprareals are polynomials
(with integer exponents) over eta_i. Perhaps it could be extended to
include more, but these things require some care. I can't just declare
it to work without showing that it agrees with the previous axioms
and theorems.
Tony Orlow wrote:
>> The fact that the inverse
>> of a finite polynomial is not generally a finite polynomial indicates
>> that this restriction makes some of the properties David desires in his
>> system impossible.
>
Chas Brown wrote:
> If David "desires" a property to be true, he can assert it so that we
> know what property he means. Of course, adding such a property may
> cause a contradiction. So it is wise to proceed step by step to see
> what the implications of the properties he has already asserted are.
>
> For example, he may have "wanted" to preserve connectedness by
> "inserting" a bunch of infinitesimals; but it turns out that that
> doesn't work.
>
> (And "infinite polynomials", a.k.a, formal power series, have their own
> problems - particularly when it comes to finding multiplicative
> inverses.)
Yes. I have two choices:
a) allow suprareals to be polynomials with an infinite number of
terms;
b) accept that H_1 U L_1 is not a field.
Option (b) appears to be the more acceptable at this point, because
it's simpler and because it allows me to keep the more general
suprareal construction as being a polynomial with both positive and
negative integer powers of eta_i.
h = sum{i= -n to +m} x_i eta_1, x_i in R
Obviously, I need to rework some of my article.
Tony Orlow wrote:
>> eta_1 is not a fraction, but a number beyond all reals.
>
Chas Brown wrote:
> eta_1 is not a "number beyond all reals", except in a poetic sense.
> It's a symbol; an abstraction.
You could say that eta_1 is a "real-like number" that exists outside
the set of reals. I chose "suprareal" to emphasize the fact that they
are greater than the reals, instead of something like "extrareal" or
"unreal".
And the suprareals can be compared to reals, although I'm not sure
that's what Tony meant by "beyond". Again, the suprareals and the
reals do not reside within the same "number line" or connected set.
Tony Orlow wrote:
>> While one could
>> consider a non-integral such number, one would do better to assume eta_1
>> to be integral, with fractional differences covered by its finite real
>> "halo".
>
Chas Brown wrote:
> One would do better to be specific about what one means, rather than
> sling around mathematical sounding phrases without specific meaning.
> "Fractional differences covered by its finite real "halo" " is hardly
> comprehensible to me.
>
> We are not considering "numbers" here in the usual sense. We are
> considering mathematical objects which share many, but not all,
> properties of what are usually called "numbers". We can define
> something that acts very much like addition between the usual numbers;
> so much so that we call it "adding two suprareals"; but it is important
> to keep track of the fact that they are /not/ numbers as we usually
> think of them - they are "abstract numbers".
>
> As such, eta_1 is not a "quantity" in the sense that "5" is a
> "quantity". And it's not a "measure" in the sense that we say "the
> length of the diagonal of the unit square is sqrt(2)".
>
> It's a symbol which represents an abstraction.
Furthermore, eta_1
a. is not a real
b. is not a fraction
c. is not a sum of reals
d. is not prime
e. is not composite
Properties (a) and (c) mean that there is no possible decimal or binary
representation for eta_1.
Properties (d) and (e) apply to eta_1 the same way that they apply to
1 or i.
David R Tribble
>> Eh, cardinality. Do you want to be able to perform arithmetic on eta_1
>> or not?
>
Chas Brown wrote:
> The ability to "perform arithmetic" is different from the ability to
> prove that there is a surjection from one set onto another set.
>
> Cardinality is about the /existence/ (or non-existence) of functions
> mapping sets onto other sets.
>
> Arithmetic is about some /particular/ function(s) which map pairs of
> elements in a set to single elements of a set, with very specific
> constraints on that function. For example, for the function "+", we
> usually require that a + 0 = a, and a + b = b + a.
>
> The two concepts are quite different, particularly when we are talking
> about "abstract numbers".
Furthermore, you can perform arithmetic on cardinal numbers - it's
called cardinal arithmetic, and is defined in terms of sets.
You can perform arithmetic on supreals, but the rules are a little
different than pure real arithmetic. The ordering axioms and the fact
that some suprareal sets are not fields must be taken into account.
David R Tribble wrote:
>> Also, it makes no sense to talk about "how many bits" could be
>> used to encode a given suprareal, since eta_1 cannot be represented
>> in a binary notation, since it's not a real. It's kind of like asking
>> how many bits are required to encode i or w (omega).
>
Tony Orlow wrote:
>> The imaginaries are an analog of the reals, and omega's not a number
>> with any "value". If you have a number of elements, ceiling(log2) of
>> that number is the number of bits required for the list.
>
"Number of bits" only applies to numbers that can actually be
represented by a string of binary digits, i.e., the real numbers.
Other kinds of numbers (and there are many other kinds of
numbers in mathematics) simply can't be represented as bit strings.
Chas Brown wrote:
> This begs the question of what you mean by "a number of elements".
>
> For example, we have already determined that ceiling(log2(eta_1)) is
> meaningless; because there is no suprareal (and certainly no real!)
> which is of the form "a^eta_1".
And even if we were to successfully extend the suprareals to include
consistent definitions for log(eta_1), eta_1^(1/2), x^eta_1, etc., it's
certain that none of these would be representable as bit strings,
because they would not be real numbers.
Tony Orlow wrote:
>> You can have infinite digital strings to the left,
>> which represent infinite values, ala p-adics or T-riffics.
You forget that p-adics are not infinite numbers. They are
"alternative" representations of reals that may employ an
infinite number of digits, that's all.
There are no infinite numbers within the standard reals.
Tony Orlow wrote:
>> You first define the successive etas as being the set of objects larger
>> than all elements of the last. Of course, this means that each set
>> includes its successor, and you no longer have a hierarchy.
>
Chas Brown wrote:
> I think you are confusing two different steps in his development.
>
> In the first step, he starts with a set, call it T_1. He then defines a
> set, T_2, which contains T_1 as a proper subset. He then "succesively"
> continues this process, to define a sequence of sets (T_1, T_2, ...,
> T_n, ...), and proves by induction that they have the property that for
> naturals n < m, T_n < T_m.
>
> In his second step, he generalizes his ideas by defining a /different/
> uncountable collection of sets, S = {S_x}. He defines an ordering on
> these sets which is the same as the usual ordering of the reals; and
> proves that they have certain properties. One of these properties is
> that for real numbers x < y, the set S_x is a proper subset of S_y.
Yep. Stated more simply, H_1 contains all the suprareals based on
eta_1, H_2 contains those based on eta_2 (which includes polynomials
having suprareal coefficients in H_1), and so forth. This is a
hierarchy of sets H_i for all i in N+.
Then I defined the L_i sets as the inverse suprareals, so that L_1
contains 1/h for all h in H_1. And likewise for L_2, L_3, etc.
Then I defined H_0 as equivalent to L_0 and thus equivalent to R,
which implies that eta_0 = 1. (This needs reworking in light of the
problem with inverse suprareal polynomials, however). This gives
us a hierarchy H_i for all i in Z.
The last step was generalizing this hierarchy to H_x for all x in R.
The ordering axiom still applies, so everything defined for the
suprareals in H_i still applies to the suprareals in H_x.
(Chas gives a more detailed response.)
Tony Orlow wrote:
>> Now, if you want to define an uncountable set of eta units, such that
>> each is a real power of some standard eta_1...
>
Chas Brown wrote:
> If he wanted to do that, he wouldn't be using polynomials; because in a
> polynomial, the powers of eta_1 are natural numbers.
>
> Instead, because he /wants/ to use polynomials, he defines an
> uncountable set of eta_'s, ordered as the reals are ordered. If x < y,
> then r*eta_x^n < eta_y for all real numbers r, and all natural numbers n.
I'd like to extend the suprareals to include more than just
polynomials, but first I have to fix the inverse supranumbers.
And it might not be possible in any case, because any addition
to the system must be consistent with what's already there.
Tony Orlow wrote:
>> , then anything above eta_0
>> is beyond the finites, and any real difference in the eta number will
>> result in a value difference beyond any finite number. By introducing a
>> particular formula, you can get a system that works. But, don't listen
>> to me...
>
Chas Brown wrote:
> Instead, I urge you to "listen to" David, so that you can see how one
> defines and examines mathematical objects.
And
"anything above eta_0 is beyond the finites"
is more correctly stated as
any suprareal based on eta_i is outside the set of reals.
You keep thinking of the suprareals in terms of number lines,
but that's only going to confuse you.
David R Tribble
> Yes. I have two choices:
> a) allow suprareals to be polynomials with an infinite number of terms;
> b) accept that H_1 U L_1 is not a field.
>
> Option (b) appears to be the more acceptable at this point, because
> it's simpler and because it allows me to keep the more general
> suprareal construction as being a polynomial with both positive and
> negative integer powers of eta_i.
> h = sum{i= -n to +m} x_i eta_1, x_i in R
> Obviously, I need to rework some of my article.
Oops, that should be:
h = sum{i= -n to +m} x_i eta_1^i, x_i in R
So a given suprareal h in H_1 U L_1 (in its corrected form) is
still a polynomial over integer powers of eta_1:
h = x + x1 eta_1^1 + x2 eta_1^2 + ... + xn eta_1^n,
where x, x1, x2, ..., xn in R and at least one x_i /= 0.
cbr...@cbrownsystems.com
> Chas Brown wrote:
> >> The real number line is /not/ the union of two non-empty disjoint open
> >> sets. Lines are connected; the suprareals are not connected; therefore
> >
>
> Tony Orlow wrote:
> >> And yet, when you introduce the ih-numbers as a non-real halo around 0,
> >> then you have disconnected the positive reals from the negative reals,
> >> as the positive h-numbers are separated by the relatively infinitesimal
> >> expanse of standard reals between them. Where you include all standard
> >> reals, all ih-numbers, and all h-numbers, they form a single line,
> >
>
> Chas Brown wrote:
> > It turns out that not just the positive and negative reals are
> > disconnected in the suprareals; in fact, /every/ two disjoint sets R1
> > and R2 with R = R1 union R2 are disconnected (this is what "totally
> > disconnected" means).
> >
> > I think what you mean to say is that they form a total order (sometimes
> > misleadingly called a linear order). The real line is one /example/ of
> > a total order; but there are many, many other types of total orders,
> > sometimes with quite bizarre properties which make them very much
> > "unlike" the real line.
>
> Right. Introducing the i-suprareals (infinitesimals) does not make the
> existing reals unconnected.
Well, in the sense that we mean "the reals still connected in the
topology of the reals", you're right. But as a subset of the
suprareals, the reals inherit the subspace toplogy of the supra reals;
and in this topology, the reals are completely disconnected.
I'm a bit at a loss to offer a subset of the suprareals which is
connected (in the subspace toplogy); although it seems that there
should be one...
>
> Chas Brown wrote:
> >> (i) "2^eta_1" is not a polynomial in eta_1; so "2^eta_1" is not
> >> meaningful as a suprareal. It follows that "log_2(eta_1)" is not
> >> meaningful for suprareals.
> >
>
> Tony Orlow wrote:
> >> That follows from the h-numbers being restricted to polynomials over
> >> eta1, but that restriction itself seems to be simply declared to
> >> "appear" true, and I don't see why it follows.
> >
>
> Chas Brown wrote:
> > I'm not sure what your objection is.
> >
> > David is simply suggesting a particular mathematical object for our
> > perusal. Of course he "simply declares" that it has certain properties,
> > such as that it contains an element eta_1, that it is closed under
> > addition, multiplication, etc., so that we are sure that both he and we
> > are talking about the "same" mathematical object.
> >
> > It indeed "follows" that the set of polynomials in eta_1 obeys these
> > properties, so it is an example of the kind of mathematical object he
> > is talking about. Since other people have studied polynomials
> > extensively, we can take advantage of their work to figure certain
> > things out (such as that we needed more than simply 1/eta_1 to
> > construct a field).
>
> Yes. I'm trying to work out what is meant by eta_1^p for non-integer
> p, e.g., eta_1^(1/2) or sqrt(eta_1). It follows from the axioms that
> x < eta_1^p for all x in R and p > 0.
>
You could start by considering the algebraic closure of the suprareals
- this would give you at least x^q for suprareals x and all (positive)
rational numbers q. Ensuring that the result of this closure obeys the
total order might be a bit tricky (consider that there is no "usual"
total ordering on the complex numbers).
The problem with option (a) initially is that, in order to have an
inverse, the constant coefficient in the formal power series (technical
name for "polynomials with an infinite number of terms") must not be 0.
If:
x = x_0 + x_1*eta_1 + x_2* eta_1^2 + ...
y = y_0 + y_1*eta_1 + y_2*eta_1^2 + ...
then the product x*y must have constant term x_0*y_0 (assuming we
define multiplication in this case as an extension of the usual
multiplication of polynomials).
Thus if x_0 = 0, there is no y such that x*y = 1.
This problem is mitigated by considering formal power series in two
variables, eta_1 and eps_1, where eta_1*eps_1 = 1; but I'm not sure
that there aren't further problems with the resulting construction.
> Option (b) appears to be the more acceptable at this point, because
> it's simpler and because it allows me to keep the more general
> suprareal construction as being a polynomial with both positive and
> negative integer powers of eta_i.
> h = sum{i= -n to +m} x_i eta_1, x_i in R
> Obviously, I need to rework some of my article.
>
If your goal is to produce a framework in which fractions play a role
(e.g., where the equation f(x)/g(x) = 1 makes sense), you'll want a
field I think.
>
> Tony Orlow wrote:
> >> eta_1 is not a fraction, but a number beyond all reals.
> >
>
> Chas Brown wrote:
> > eta_1 is not a "number beyond all reals", except in a poetic sense.
> > It's a symbol; an abstraction.
>
> You could say that eta_1 is a "real-like number" that exists outside
> the set of reals. I chose "suprareal" to emphasize the fact that they
> are greater than the reals, instead of something like "extrareal" or
> "unreal".
>
The main point I wanted to make is that by "number", we often mean
things that aren't properly considered "numbers" in usual English. To
say that eta_1 is a "number" seems to confuse some people terribly.
And I think it's important to note that the suprareals do not "contain"
the reals - they contain an isomorphic (under field operations) /copy/
of the reals. Just as the reals do not "contain" the finite ordinals -
they contain an isomorphic (under addition) /copy/ of the finite
ordinals.
> Furthermore, eta_1
> a. is not a real
> b. is not a fraction
> c. is not a sum of reals
> d. is not prime
> e. is not composite
>
> Properties (a) and (c) mean that there is no possible decimal or binary
> representation for eta_1.
>
> Properties (d) and (e) apply to eta_1 the same way that they apply to
> 1 or i.
Yep.
There's plenty that has been with polynomials and formal power series
that you might enjoy reading. Any introductory book on Abstract Algebra
would probably be useful to you. From the El Cheapo Dover collection, I
can recommend "Elements of Abstract Algebra" by Allan Clark at $8.
Cheers - Chas
Baldin...@msn.com
David R Tribble wrote:
> Over the last few months I've been noodling around with the concept
> of an extension to the reals, defining real-like numbers that are
> larger than any regular real.
>
> standard arithmetic to extend the reals with what I call, for lack of
> a better term, "h-numbers". Briefly, an h-number has a magnitude
> greater than any real. The first axiom states that n1 (eta_1), a
> primitive h-number constant, exists, and that x < n1 for all x in R.
> >From there, an entire set H of h-numbers is defined as containing other
> h-numbers, being sums and products of reals and n1.
>
> up defining an entire hierarchy of such numbers (n1, n2, etc.) and
> their multiplicative inverses (e1 = 1/n1, e2, etc.).
>
> mathematical HTML characters) is at:
> http://david.tribble.com/text/hnumbers.html
>
> something like this has been done before, or whether it's
> mathematically inconsistent.
I don't see how these numbers relate to the reals. It is all well and
good to build something with a certain property, but your numbers don't
seem to be defined in a way that allows you to actually show they are
larger than the reals -- you just declare them to be and that is that.
For instance, the standard model of the hyperreals is as infinite
sequences of real numbers {r_i} sorted into equivalence classes by a
free ultrafilter on the indices i. Then the reals are (roughly)
sequences whose members agree on an element of the ultrafilter, and the
infinite hyperreals are those sequences which have an unbounded
subsequence indexed by an element of the ultrafilter. With this
construction finite hyperreals (reals) and infinite hyperreals can be
compared, and have the same construction.
What is the common structure of the reals and your h numbers that
allows you to show that the h's are larger?
Baldin Lee Pramer
David R Tribble
>> It turns out that not just the positive and negative reals are
>> disconnected in the suprareals; in fact, /every/ two disjoint sets R1
>> and R2 with R = R1 union R2 are disconnected (this is what "totally
>> disconnected" means).
>>
>> I think what you mean to say is that they form a total order (sometimes
>> misleadingly called a linear order). The real line is one /example/ of
>> a total order; but there are many, many other types of total orders,
>> sometimes with quite bizarre properties which make them very much
>> "unlike" the real line.
>
David R Tribble wrote:
>> Right. Introducing the i-suprareals (infinitesimals) does not make the
>> existing reals unconnected.
>
Chas Brown wrote:
> Well, in the sense that we mean "the reals still connected in the
> topology of the reals", you're right. But as a subset of the
> suprareals, the reals inherit the subspace toplogy of the supra reals;
> and in this topology, the reals are completely disconnected.
Yes, since the inverse suprareals lie "between" or "around"
the reals. Almost forgot about those.
David R Tribble wrote:
>> What is curious (and which I will probably add to the next revision)
>> is the fact that
>> x0 + eta_1
>> x1 eta_1^1
>> x2 eta_1^2
>> ...
>> are unconnected (uncountable) sets. Any member of one set in the
>> list is less than any member of the next set (assuming all positive
>> real x's).
>
Chas Brown wrote:
> I'm a bit at a loss to offer a subset of the suprareals which is
> connected (in the subspace toplogy); although it seems that there
> should be one...
Hmm. If we consider only the reals and the "large" suprareals,
R U H_i, we can say that the reals are still connected. We can
also probably say that at least some of the suprareals in H_1
are connected, since the difference between x+eta_1 and
y+eta_1 is x-y, a real.
But when we talk about the totality of the reals, the large suprareals,
and the inverse suprareals ("infrareals", perhaps?), we lose the
connectedness within the reals, as you said. And since the "small"
suprareals are a never-ending hierarchy (L_1, L_2, etc., or the
equivalent labeling H_{-1}, H_{-2}, etc.), there does not seem to
be any "least separation" between suprareals in any set of the
total hierarchy.
Does that sound right?
jiri...@gmail.com
> Over the last few months I've been noodling around with the concept
> of an extension to the reals, defining real-like numbers that are
> larger than any regular real.
Just glancing over the article it seemed a lot like the ordered ring of
real polynomials in one variable. Just define x to be your n1. Then
making this a field is taking the ordered field of rational functions.
Read the following book
J. Bochnak, M. Coste, and MF. Roy. Real Algebraic Geometry. Springer,
1998.
Jiri
MoeBlee
> Well, at least we both agree with Robinson that there's no smalleest
> infinity in such a system.
There's no smallest infinite non-standard integer and no smallest
infinite non-standard real. There is a smallest infinite ordinal. These
are from different senses of 'smallest' and 'infinite'.
> > But please, this thread is supposed to be about my theory, not
> > yours. I'll be happy to join the thread you start for yours.
> Okay, just noting similarities.
Right, as the movie actor in love himself would say after going on and
on about himself, "Well, enough about me, what do YOU think of these
latest developments in my career?"
MoeBlee
cbr...@cbrownsystems.com
Think of the polynomials a_0 + a_1*eta_1 + a_2*eta_1^2 + ... +
a_n*eta_1^n instead as sequences of real numbers with at most a finite
number of non-zero entries (a_0, a_1, a_2, ..., a_n, 0, 0, 0, ...).
Addition is defined point-wise; and multiplication c = a*b is defined
as:
c_n = sum(i = 0 to n) (a_i*b_(n-i)).
(This gives us multiplication and addition as usually defined for
polynomials).
Then given two such sequences a and b, we can totally order them by: a
>= b iff exists n s.t. a_n >= b_n, and for all m > n, a_m = b_m.
It is no more arbitrary to declare that sequence a > sequence b in this
way than it is to declare that hyperreal a > hyperreal b if a is
greater than b on a "large" set in the ultrafilter.
> For instance, the standard model of the hyperreals is as infinite
> sequences of real numbers {r_i} sorted into equivalence classes by a
> free ultrafilter on the indices i. Then the reals are (roughly)
> sequences whose members agree on an element of the ultrafilter, and the
> infinite hyperreals are those sequences which have an unbounded
> subsequence indexed by an element of the ultrafilter. With this
> construction finite hyperreals (reals) and infinite hyperreals can be
> compared, and have the same construction.
>
... and the reals can be embedded into these polynomials by considering
only those polynomials of degree 0 (these are then the "finite"
numbers).
Note that I am not claiming this is the /best/ extension of the reals
(as I stated it above, it's not even a field); but that it is /an/
extension of the reals.
Cheers - Chas
Virgil
Baldin...@msn.com wrote:
>
> For instance, the standard model of the hyperreals is as infinite
> sequences of real numbers {r_i} sorted into equivalence classes by a
> free ultrafilter on the indices i. Then the reals are (roughly)
> sequences whose members agree on an element of the ultrafilter, and the
> infinite hyperreals are those sequences which have an unbounded
> subsequence indexed by an element of the ultrafilter. With this
> construction finite hyperreals (reals) and infinite hyperreals can be
> compared, and have the same construction.
>
> What is the common structure of the reals and your h numbers that
> allows you to show that the h's are larger?
if {i in N: r_i > s_i} is in the ultrafilter, then {r_i} > {s_i}.
Tony Orlow
Then thesame applies to theh-numbers. If they are not colinear with the
reals, then the reals do not constitute a "gap" within the suprareals.
If you apply the logic of connectedness to the suprareals as you do to
the reals, then they are also gapless. For any two suprareals, positive
or negative, there lies a suprareal between them, Therefore, they are as
"continuous" as the reals.
That was the suggestion I offered: define you H_x as the countable
neighborhood of eta_1^x, whether x is natural or real. Any real
difference in an exponent applied to an infinite value results in an
infinite difference, an uncountably "disconnected" pair of sets.
> Chas Brown wrote:
>>> (i) "2^eta_1" is not a polynomial in eta_1; so "2^eta_1" is not
>>> meaningful as a suprareal. It follows that "log_2(eta_1)" is not
>>> meaningful for suprareals.
>
> Tony Orlow wrote:
>>> That follows from the h-numbers being restricted to polynomials over
>>> eta1, but that restriction itself seems to be simply declared to
>>> "appear" true, and I don't see why it follows.
>
> Chas Brown wrote:
>> I'm not sure what your objection is.
>>
>> David is simply suggesting a particular mathematical object for our
>> perusal. Of course he "simply declares" that it has certain properties,
>> such as that it contains an element eta_1, that it is closed under
>> addition, multiplication, etc., so that we are sure that both he and we
>> are talking about the "same" mathematical object.
>>
>> It indeed "follows" that the set of polynomials in eta_1 obeys these
>> properties, so it is an example of the kind of mathematical object he
>> is talking about. Since other people have studied polynomials
>> extensively, we can take advantage of their work to figure certain
>> things out (such as that we needed more than simply 1/eta_1 to
>> construct a field).
>
> Yes. I'm trying to work out what is meant by eta_1^p for non-integer
> p, e.g., eta_1^(1/2) or sqrt(eta_1). It follows from the axioms that
> x < eta_1^p for all x in R and p > 0.
Right. It further follows from that that, for x,y in R, x<y, then for
all z in R, eta_1^y/eta_1^x>z.
>
> As Chas points out, my current definition of suprareals are polynomials
> (with integer exponents) over eta_i. Perhaps it could be extended to
> include more, but these things require some care. I can't just declare
> it to work without showing that it agrees with the previous axioms
> and theorems.
>
>
Understood. Didn't it fail to work for division, when restricted to
finite polynomials?
> Tony Orlow wrote:
>>> The fact that the inverse
>>> of a finite polynomial is not generally a finite polynomial indicates
>>> that this restriction makes some of the properties David desires in his
>>> system impossible.
>
> Chas Brown wrote:
>> If David "desires" a property to be true, he can assert it so that we
>> know what property he means. Of course, adding such a property may
>> cause a contradiction. So it is wise to proceed step by step to see
>> what the implications of the properties he has already asserted are.
>>
>> For example, he may have "wanted" to preserve connectedness by
>> "inserting" a bunch of infinitesimals; but it turns out that that
>> doesn't work.
>>
>> (And "infinite polynomials", a.k.a, formal power series, have their own
>> problems - particularly when it comes to finding multiplicative
>> inverses.)
>
Sure, that became a problem. I'm not cutting down the process, but
making some suggestions.
> Yes. I have two choices:
> a) allow suprareals to be polynomials with an infinite number of
> terms;
> b) accept that H_1 U L_1 is not a field.
>
> Option (b) appears to be the more acceptable at this point, because
> it's simpler and because it allows me to keep the more general
> suprareal construction as being a polynomial with both positive and
> negative integer powers of eta_i.
If that's the construction, then I don't see how you can form your
"uncountable hierarchy", since there aren't an uncountable number of
integers. That's what I was saying about that part not jibing.
> h = sum{i= -n to +m} x_i eta_1, x_i in R
> Obviously, I need to rework some of my article.
>
>
That's okay. It's not bad as a start at nonstandard approaches.
> Tony Orlow wrote:
>>> eta_1 is not a fraction, but a number beyond all reals.
>
> Chas Brown wrote:
>> eta_1 is not a "number beyond all reals", except in a poetic sense.
>> It's a symbol; an abstraction.
>
> You could say that eta_1 is a "real-like number" that exists outside
> the set of reals. I chose "suprareal" to emphasize the fact that they
> are greater than the reals, instead of something like "extrareal" or
> "unreal".
>
> And the suprareals can be compared to reals, although I'm not sure
> that's what Tony meant by "beyond". Again, the suprareals and the
> reals do not reside within the same "number line" or connected set.
>
>
But, did start by assuming some number beyond the reals, such that
eta_1>x for all x in R, yes?
> Tony Orlow wrote:
>>> While one could
>>> consider a non-integral such number, one would do better to assume eta_1
>>> to be integral, with fractional differences covered by its finite real
>>> "halo".
>
> Chas Brown wrote:
>> One would do better to be specific about what one means, rather than
>> sling around mathematical sounding phrases without specific meaning.
>> "Fractional differences covered by its finite real "halo" " is hardly
>> comprehensible to me.
>>
I don't think that's my problem, and you needn't get snippy about it. A
simple request for clarification will do.
The question is whether eta_1 can be used as a count, or set size. It
cannot, if it's not a whole number. Consider it a whole number, and any
non-integral suprareal as requiring the addition of a finite real
component. The "halo" in NSA is the countable neighborhood of
infinitesimals that can be considered to lie around each real number. In
similar fashion, we can view numbers like the suprareals, which are
separated by an uncountable number of units from each other, to have
countable neighborhoods of unit intervals surrounding them, each exactly
the same as the standard real line. That's what I mean by a "halo". It's
the same thing on a different scale. Questions?
>> We are not considering "numbers" here in the usual sense. We are
>> considering mathematical objects which share many, but not all,
>> properties of what are usually called "numbers". We can define
>> something that acts very much like addition between the usual numbers;
>> so much so that we call it "adding two suprareals"; but it is important
>> to keep track of the fact that they are /not/ numbers as we usually
>> think of them - they are "abstract numbers".
Um, what is the definition of "number", please?
>>
>> As such, eta_1 is not a "quantity" in the sense that "5" is a
>> "quantity". And it's not a "measure" in the sense that we say "the
>> length of the diagonal of the unit square is sqrt(2)".
>>
>> It's a symbol which represents an abstraction.
Yes, well, symbols are all very well and good, but they are not all
there is to math.
>
> Furthermore, eta_1
> a. is not a real
> b. is not a fraction
> c. is not a sum of reals
> d. is not prime
> e. is not composite
I really think you might want to give more thought at some point to
those last two. There are reasons to consider each possibility.
>
> Properties (a) and (c) mean that there is no possible decimal or binary
> representation for eta_1.
That is not correct. Eta_1 CAN be treated like the T-riffic Big'un, but
there's no point arguing that at this point.
>
> Properties (d) and (e) apply to eta_1 the same way that they apply to
> 1 or i.
>
What "way" is that?
Tony Orlow
The T-riffics are infinite values with digital representations.
> Chas Brown wrote:
>> This begs the question of what you mean by "a number of elements".
>>
>> For example, we have already determined that ceiling(log2(eta_1)) is
>> meaningless; because there is no suprareal (and certainly no real!)
>> which is of the form "a^eta_1".
>
> And even if we were to successfully extend the suprareals to include
> consistent definitions for log(eta_1), eta_1^(1/2), x^eta_1, etc., it's
> certain that none of these would be representable as bit strings,
> because they would not be real numbers.
>
>
The T-riffics manage it.
> Tony Orlow wrote:
>>> You can have infinite digital strings to the left,
>>> which represent infinite values, ala p-adics or T-riffics.
>
> You forget that p-adics are not infinite numbers. They are
> "alternative" representations of reals that may employ an
> infinite number of digits, that's all.
p-adics reside to the left of the digital point. You can map them to the
reals in [0,1), but that doesn't make them the same thing.
>
> There are no infinite numbers within the standard reals.
>
>
No, just infinite bit strings to the right.
I'm trying to tighten up the concept. How are you mapping you
polynomials to H_x? Can you tell me the polynomial for H_pi or H_1/2?
Anyway, carry on....
Tony
hagman
cbr...@cbrownsystems.com schrieb:
> hagman wrote:
> > David R Tribble schrieb:
> >
> > > Over the last few months I've been noodling around with the concept
> > > of an extension to the reals, defining real-like numbers that are
> > > larger than any regular real.
> > >
> > > I've written an article exploring the effects of adding a few axioms to
> > > standard arithmetic to extend the reals with what I call, for lack of
> > > a better term, "h-numbers". Briefly, an h-number has a magnitude
> > > greater than any real. The first axiom states that n1 (eta_1), a
> > > primitive h-number constant, exists, and that x < n1 for all x in R.
> > > >From there, an entire set H of h-numbers is defined as containing other
> > > h-numbers, being sums and products of reals and n1.
> > >
> > > The article goes on to explore arithmetic for the h-numbers and ends
> > > up defining an entire hierarchy of such numbers (n1, n2, etc.) and
> > > their multiplicative inverses (e1 = 1/n1, e2, etc.).
> > >
> > > The article (which requires a browser capable of rendering certain
> > > mathematical HTML characters) is at:
> > > http://david.tribble.com/text/hnumbers.html
> > >
> > > Comments and suggestions are welcome. I'm curious to know if
> > > something like this has been done before, or whether it's
> > > mathematically inconsistent.
> > >
> > > -drt
> > >
> > > | Rev 1.0, 2006-12-08
> > > | iQA/AwUBRYQ+eHS9RCOKzj55EQJFrQCeMuZ+B+D9s0iLYs9j80SIjxNzAsoAoJWG
> > > | /I5J051vnLpnmDctzmYe1dy6
> > > | =iGOQ
> >
> > The easiest, most obvious and literally universal method to extend the
> > reals by adding new numbers is to look at R[X].
> > In fact, that is what you did; additionally you defined an order
> > relation on HuR = R[X] by declaring all non-zero-polynomials with
> > positive leading coefficient as positive.
> > It is trivially verified that for this set P we have P*P subset P, P+P
> > subset P and R[X] is the disjoint union of P and -P and {0}.
> >
> > Later you construct HuRuL in a way that essentially boils down to
> > R[X,Y]/(XY-1) and claim that this is a field.
> > However, Theorem 18c. fails to be true (at least with the first version
> > of Axiom 8; the second version is not equivalent):
> > In R[X,Y]/(XY-1), X+1 has no inverse!
>
> Also, for example, 18a claims that u + v in H u R u L; but given h in
> H, h + 1/h is not in any of H, R, or L.
>
> >
> > Instead, You should have used R(X), the field of rational functions in
> > one variable.
>
> This is actually what I /thought/ the OP had defined (so much for my
> careful reading!).
>
> > The order on R[X] induces an order on R(X), i.e. f/g is positive if f
> > and g are both positive or both negative.
> > (Again, it is trivially verified that for this set P we have P*P subset
> > P, P+P subset P and R[X] is the disjoint union of P and -P and {0}).
> >
> > To obtain nested hierarchies, repeat the step, i.e. use more variables:
> > Let S be a totally ordered set (of variables, i.e. viewed as disjoint
> > with R etc.; if you want to use S=R or the like, use some standard
> > trick to obtain a disjoint copy)
> > Then R[S] is a ring and R(S) is a field.
> > To define an order relation on R[S] (and thus on R(S)), declare a
> > non-zero element of R[S] as positive if the leading coefficient is a
> > positive real.
> > Note that each element of R[S] is in fact an element of R[X1,...,Xn]
> > for a finite number of variables X1<...<Xn in S. Since
> > R[X1,..,Xn]=R[X1,...,X{n-1}][Xn], the leading coefficient method can be
> > applied step by step.
> >
> > This works for any totally ordered set S, be it finite, countable,
> > continuum-sized or whatever.
>
> Which leads us back to the question of what topology to use for these
> numbers, and how/whether limits are to be defined.
>
> Using open intervals seems the most "natural" basis for a topology for
> this space; the sub-space topology on R is then the usual topology.
> This gives us a Hausdorff space (I think, regular Hausdorff?).
>
> Then, for example, the set of "finite" suprareals is an open set as is
> the set of "infinite" suprareals, so we get the lack of connectedness
> implied in the drawings the OP made. (Many other
> examples of the disconnectedness of this system can be found.) Thus,
> this space is not normal (the above two sets are also closed and
> disjoint, but are not separated).
>
> Let us denote the suprareals as H^. We can define lim (x->c) {f(x)} = L
> as:
>
> L in H^ is the limit of f(x) as x approaches c if, for every 0 <
> epsilon in H^, there exists a 0 < delta in H^ such that for all x such
> that |x-c| < delta, |f(x) -L| < epsilon.
>
> It doesn't seem like this yields a complete space; but I can't
> construct any counterexamples off the top of my head.
One needs to distinguish R[[X]] from R(X), i.e.
one need a power series that is not a rational function,
for example the exponential (contrary to my former post, I view X as
the infinitesimal unit here, not the infinite unit; but that's just to
simplify notation).
Let S be the subset of R(X) consisting of all f(X)/(1+X*g(X)).
Since 1+X*g(X) is invertible in R[[X]], we can view S as (part of) the
intersection of R(X) and R[[X]]. ("Fortunately", the topology on R[[X]]
is also defined by the fact that powers of X are small).
One can show that the series (a_n) with a_n = sum_{k=0,...,n} X^n/n!
is a Cauchy sequence.
Assume it converges against an element f(X)/g(X) of R(X) where f(X) and
g(X) have no irreducible factors in common.
Obviously, the limit has to be finite (namely, very close to a_0 = 1
but obviuously different from a_0).
Hence we can assume that g(X) = 1 + X*h(X).
But then f(X)/g(X) is in S and we must obtain the same limit in R[[X]].
But there we have exp(X) as limit.
Hence f(X) = g(X)*exp(X) as formal power series.
The equality must still hold when we insert a complex number for X as
long as we have convergence. Since exp converges everywhere and has no
zeroes, we see that f(X) and g(X) have the same zeroes (in same
multiplicity) in C, hence have the same irreducible factors.
Since they have no irreducible factors in common, it follows that they
f(X) and g(X) are constant, i.e. the limit is indeed real.
But as the limit is infinitesimal close to but different from 1, we
have a contradiction.
> Cheers - Chas
Virgil
Tony Orlow <to...@lightlink.com> wrote:
> If you apply the logic of connectedness to the suprareals as you do to
> the reals, then they are also gapless. For any two suprareals, positive
> or negative, there lies a suprareal between them, Therefore, they are as
> "continuous" as the reals.
Then, by that same argument, the density of the rationals makes the
rationals are as "continuous" as the reals.
cbr...@cbrownsystems.com
For the third time in this thread, "dense", "connected", and
"continuous" have different meanings. And "gapless" has no particular
meaning to me at all; unless it is supposed to mean "dense".
(i) The fact that the suprareals are a dense set does not mean that the
suprareals are connected.
(ii) The fact that the suprareals are a dense set does not mean that
there is a continuous function from the reals onto the suprareals.
These two statements follow from the definitions of "dense",
"connected" and "continuous".
http://en.wikipedia.org/wiki/Dense_set
http://en.wikipedia.org/wiki/Connected_set
http://en.wikipedia.org/wiki/Continuous_function
Or better yet, read a book on topology: Hocking and Young's "Topology"
is about $13, and they probably have it at Barnes and Nobles.
http://www.amazon.com/Topology-John-G-Hocking/dp/0486656764
(i) What is a "countable neighborhood"? A neighborhood is usually
equivalent to "an open set", which in this topology means "an open
interval in the suprareals". But what is a "countable neighborhood"?
And how do you define an open interval prior to defining a total order
on the elements you aim to define?
(ii) Why do you say "/the/ countable neighborhood"? Why is there only
one unique such object associated with eta_1^x?
(iii) How does your definition of "countable neighborhood" /define/
H_x; by which I mean provide a /definition/ of addition, subtraction
and multiplication on the elements of H_x?
> Any real
> difference in an exponent applied to an infinite value results in an
> infinite difference, an uncountably "disconnected" pair of sets.
>
(i) How does a real difference in the exponents x, y of eta_1 get
"applied" to an "infinite value" to "result" in an "infinite
difference"?
(ii) What is the difference between a "disconnected" set and an
"uncountably disconnected set"?
(iii) What is an "uncountably "disconnected" pair of sets"? What two
sets of suprareals are you referring to?
(iv) Why does it follow from these definitions that "an infinite
difference" is therefore "an uncountably "disconnected" pair of sets"?
> > Yes. I have two choices:
> > a) allow suprareals to be polynomials with an infinite number of
> > terms;
> > b) accept that H_1 U L_1 is not a field.
> >
> > Option (b) appears to be the more acceptable at this point, because
> > it's simpler and because it allows me to keep the more general
> > suprareal construction as being a polynomial with both positive and
> > negative integer powers of eta_i.
>
> If that's the construction, then I don't see how you can form your
> "uncountable hierarchy", since there aren't an uncountable number of
> integers. That's what I was saying about that part not jibing.
>
Let Eta = {eta_x : x in R+}. Let A = {a, b, c, ..., m} be a finite set
of reals. Let E be the set of eta_'s defined by E = {eta_u : u in A}.
Let h = p(E) be a polynomial with real coefficients over the set
{eta_a, eta_b, eta_c, ..., eta_m}.
Then if y is a real number greater than any real number in the (finite)
set A, then we /define/ eta_y > h.
Given that definition, can you work out whether
1 + (eta_2)^100*(eta_3)^2 > 3 - 2*eta_e + 17*eta_pi
is true or false?
If not, can you instead first work out whether
eta_pi < 1/17*(-2 + (eta_2)^100*(eta_3)^2 + 2*eta_e)
is true or false?
> > And the suprareals can be compared to reals, although I'm not sure
> > that's what Tony meant by "beyond". Again, the suprareals and the
> > reals do not reside within the same "number line" or connected set.
> >
> >
>
> But, did start by assuming some number beyond the reals, such that
> eta_1>x for all x in R, yes?
>
If by "eta_1 is beyond the reals", you mean the assertion "in the
suprareals, r < eta_1 for all r in H_1 such that r is of the form (r_0,
0, 0, ...)", then yes. If you mean something else, then I can't opine
at this point.
> > Tony Orlow wrote:
> >>> While one could
> >>> consider a non-integral such number, one would do better to assume eta_1
> >>> to be integral, with fractional differences covered by its finite real
> >>> "halo".
> >
> > Chas Brown wrote:
> >> One would do better to be specific about what one means, rather than
> >> sling around mathematical sounding phrases without specific meaning.
> >> "Fractional differences covered by its finite real "halo" " is hardly
> >> comprehensible to me.
> >>
>
> I don't think that's my problem, and you needn't get snippy about it. A
> simple request for clarification will do.
>
OK. Given /David's definition/ of H_1:
(i) What does it mean for eta_1 to be "non-integral" in the suprareals?
What does it mean for eta_1 to be "integral" in the suprareals?
(ii) What are "fractional differences", in the context of the
suprareals? Are the "fractional differences" of eta_1 the same as, or
different from, the "fractional differences" of any other suprareal?
(iii) What is the "finite real halo" of eta_1 in the suprareals?
(iv) What does it mean to say "the fractional differences of eta_1 are
covered by its finite real halo" in the suprareals?
(v) Assuming that we can now make sense of your assertion, why would it
be /better/ to "assume" eta_1 to be integral, with fractional
differences covered by its finite real "halo", instead of "merely"
assuming that eta_1 is "non-integral"? What, exactly, is "better" about
it?
> The question is whether eta_1 can be used as a count, or set size. It
> cannot, if it's not a whole number.
As I use the term "the number of elements in set X = card(X)", it
cannot, if it is not a /cardinal number/.
If eta_1 is "the number of elements in the set X", it means to me that
there is a bijective function f : eta_1 -> X.
But the /suprareal/ 2*eta_1 is /not a set/; except in the trivial sense
that the sequence (0, 2, 0, 0, 0,...) is a set.
So to say "the set Y has 2*eta_1 elements" no longer makes sense to me;
unless you mean that there is a bijective function between the set (0,
2, 0, ...) and set Y (which would be an odd thing to mean, since then
there also exists a bijection between the suprareal 0 = (0, 0, ...) and
Y).
I have no real idea of what you mean by "eta_1 is the number of
elements in set X", if you /don't/ mean "the cardinality of X".
> Consider it a whole number
What is a "whole number", when we are not talking about the reals? In
particular, what is a "whole number" in David's H_1?
> and any
> non-integral suprareal as requiring the addition of a finite real
> component. The "halo" in NSA is the countable neighborhood of
> infinitesimals that can be considered to lie around each real number.
> In
> similar fashion, we can view numbers like the suprareals, which are
> separated by an uncountable number of units from each other, to have
> countable neighborhoods of unit intervals surrounding them, each exactly
> the same as the standard real line. That's what I mean by a "halo". It's
> the same thing on a different scale. Questions?
>
(i) What does it mean to say "suprareals x and y are separated by an
uncountable number of units"?
(ii) What does it mean for a suprareal y to "be considered to lie
around" the suprareal x = 1 + eta_1^2?
(iii) From these definitions, why does it follow that there is only a
countable number of such infinitesimal y's? Why isn't there an
uncountable number of such infinitesimal y's?
(iv) What does it mean for a neighborhood in the suprareals to be "a
countable neighborhood of a unit interval"?
(v) How does a collection of "countable neighborhoods of unit
intervals" then "surround" a suprareal x? When do they /not/ "surround"
some other suprareal, y?
(v) What does it mean for a countable neighborhood of a unit interval
in the suprareals to be "exactly the same as" the real line?
(vi) When you say "that's what (you) mean by a halo", what is "that"?
Is "that" a "way of viewing" the suprareals? Is "that" some particular
set of suprareals? Is "that" a property of certain sets of suprareals?
Etc.
If I asked you what you meant every time you made an assertion, these
threads would get even longer. It's usually better to ignore your
statements unless I can make /some/ sense of them.
That's why you should /learn/ some of the terminology. Then we can
understand each other much better!
> >> We are not considering "numbers" here in the usual sense. We are
> >> considering mathematical objects which share many, but not all,
> >> properties of what are usually called "numbers". We can define
> >> something that acts very much like addition between the usual numbers;
> >> so much so that we call it "adding two suprareals"; but it is important
> >> to keep track of the fact that they are /not/ numbers as we usually
> >> think of them - they are "abstract numbers".
>
> Um, what is the definition of "number", please?
>
There is no "one definition"; it depends on /context/.
For the H_1, a "number" means a sequence of real numbers x = (a_0, a_1,
..., a_n, 0, 0, 0, ...) where for some natural n, m > n imples a_m = 0.
"+" is a commutative group operation defined as pointwise addition;
multiplication is a little more complicated to describe. The result is
a (commutative) ordered ring.
For the hyperreals, a "number" means an /equivalence class/ of
sequences of reals, where sequences a and b are considered equivalent
if they agree on some element of the chosen ultrafilter. Both addition
and multiplication are defined point-wise; and the result is an ordered
field.
For cardinal numbers, "numbers" is a particular a subset of the
ordinals. Adiition and multiplication are defined, but the result is
not a ring (neither + nor * are groups; they are monoids) (although it
is totally ordered).
For the complex numbers, "numbers" are expressions of the form "a +
b*i", where and b are real numbers. The complex numbers form a field;
but /not/ an ordered field: it is not the case that, for any two
complex numbers u, v, that exactly one of u<v, u>v, or u=v must hold
For algebraic numbers, "numbers" are a subset of the complex numbers a
in A that satisfy P(a) = 0 for some polynomial P in Z[x]. + and * are
defined so that the result is a field.
For a prime number p, the field F_p^n consists of p^n "numbers" which
form a field. It is not ordered.
Etc.
In general, "numbers" is usually shorthand for "the elements of some
particular set X, with operations + and * defined so that (X, +, *)
forms a commutative monoid with multiplication"; but even that is
probably best considered just a guideline.
> >>
> >> As such, eta_1 is not a "quantity" in the sense that "5" is a
> >> "quantity". And it's not a "measure" in the sense that we say "the
> >> length of the diagonal of the unit square is sqrt(2)".
> >>
> >> It's a symbol which represents an abstraction.
>
> Yes, well, symbols are all very well and good, but they are not all
> there is to math.
>
And yet in this case, it is all that is being claimed.
> >
> > Furthermore, eta_1
> > a. is not a real
> > b. is not a fraction
> > c. is not a sum of reals
> > d. is not prime
> > e. is not composite
>
> I really think you might want to give more thought at some point to
> those last two. There are reasons to consider each possibility.
>
> >
> > Properties (a) and (c) mean that there is no possible decimal or binary
> > representation for eta_1.
>
> That is not correct. Eta_1 CAN be treated like the T-riffic Big'un, but
> there's no point arguing that at this point.
>
> >
> > Properties (d) and (e) apply to eta_1 the same way that they apply to
> > 1 or i.
> >
>
> What "way" is that?
By the definition of the complex numbers, every complex number divides
i; just as every complex number divides every other complex number.
"Primeness" and "compositeness" make no sense (except perhaps
trivially) in the complex numbers.
On the other hand, in the Guassian integers (ring of numbers of the
form a + b*i where a and b are integers), there /are/ "prime numbers"
and "composite numbers". But there are also "units", which are
/neither/ prime nor composite; and i is neither prime nor composite in
the Guassian integers.
Primeness and compositeness are not properties of the /symbol/ "i".
They are properties of the "number" i /in a particular context/. Change
the context, and the meaning of these properties may change, and even
become meaningless /in that context/.
Cheers - Chas
Ross A. Finlayson
>
> For the third time in this thread, "dense", "connected", and
> "continuous" have different meanings. And "gapless" has no particular
> meaning to me at all; unless it is supposed to mean "dense".
>
So is it a complete ordered field?
Otherwise there's not the arithmetic on it, defying its stated
properties.
Sets are defined by their elements and N in the generic extension
bijects to R. Transfinite cardinals is finitism.
The real numbers are _definitely_ useful, and _widely_ used. They're
just not standard yet.
Ross
cbr...@cbrownsystems.com
> cbr...@cbrownsystems.com schrieb:
>
<snippage>
> > Let us denote the suprareals as H^. We can define lim (x->c) {f(x)} = L
> > as:
> >
> > L in H^ is the limit of f(x) as x approaches c if, for every 0 <
> > epsilon in H^, there exists a 0 < delta in H^ such that for all x such
> > that |x-c| < delta, |f(x) -L| < epsilon.
> >
> > It doesn't seem like this yields a complete space; but I can't
> > construct any counterexamples off the top of my head.
>
> One needs to distinguish R[[X]] from R(X), i.e.
> one need a power series that is not a rational function,
> for example the exponential (contrary to my former post, I view X as
> the infinitesimal unit here, not the infinite unit; but that's just to
> simplify notation).
Yes; I understand: we are simply considering R[[X]] and R(X) as
polynomials with some total ordering.
I should note that the only exposure I've gotten in any depth regarding
formal power series was from a quick introduction in Stanley's
"Enumerative Combinatorics", where he just needs to describe enough of
their properties to make them useful for describing generating
functions; and to establish a calculus such that F(X) + G(X),
F(X)*G(X), and F(G(X)) all have meaning.
So just to make sure I remember this correctly (since you refer to it
below):
In Stanley's description, the topology is one where a sequence of
elements of R[[X]] {Fi(X)} = (F1(X), F2(X), ..., Fn(X), ...) converges
to the element F(X) according to the following rules:
Let a_(i, n) be the coeffcient of X^n in Fi(X); and a_n be the similar
coefficient in F(X).
Then for every n, there is a natural j such that for all k > j, a_(k,
j) = a_j.
This is different than requiring that the limit k->oo {a_(k, j)} = a_j;
and I presume this is equivalent to your statement below "the fact that
powers of X are small"? Or do you use a different notion of convergence
than Stanley defines?
> Let S be the subset of R(X) consisting of all f(X)/(1+X*g(X)).
> Since 1+X*g(X) is invertible in R[[X]], ...
... because an element of R[[X]] is invertible iff its "constant
coefficient" is non-zero...
> ...we can view S as (part of) the
> intersection of R(X) and R[[X]]...
... because surely (1 + X*g(X) designates a particular element in R(X).
> ("Fortunately", the topology on R[[X]]
> is also defined by the fact that powers of X are small).
See above. In any case, I assume you imply that the open sets of S =
R[[X]] intersect R(X) are the open sets of R(X) intersect S, which are
the same as the open sets of R[[X]] intersect S; and so if {Fi(X)}
converges to F in R[[X]], it also converges to the equivalent member of
R(X) in R(X).
> One can show that the series (a_n) with a_n = sum_{k=0,...,n} X^n/n!
> is a Cauchy sequence.
> Assume it converges against an element f(X)/g(X) of R(X) where f(X) and
> g(X) have no irreducible factors in common.
> Obviously, the limit has to be finite (namely, very close to a_0 = 1
> but obviuously different from a_0).
> Hence we can assume that g(X) = 1 + X*h(X).
> But then f(X)/g(X) is in S and we must obtain the same limit in R[[X]].
> But there we have exp(X) as limit.
> Hence f(X) = g(X)*exp(X) as formal power series.
With you so far...
> The equality must still hold when we insert a complex number for X as
> long as we have convergence.
I am definitely missing a theorem in your toolbox. Can you amplify on
this assertion?
Are you saying that f(X) and g(X) /must/ belong to the subset of R[[X]]
where every series converges in C when X is replaced by any complex
number?
cbr...@cbrownsystems.com
Ross A. Finlayson wrote:
> cbr...@cbrownsystems.com wrote:
> >
> > For the third time in this thread, "dense", "connected", and
> > "continuous" have different meanings. And "gapless" has no particular
> > meaning to me at all; unless it is supposed to mean "dense".
> >
>
> So is it a complete ordered field?
>
> Otherwise there's not the arithmetic on it, defying its stated
> properties.
>
> Sets are defined by their elements and N in the generic extension
> bijects to R. Transfinite cardinals is finitism.
Or how about this:
Fractal functions form the fjords of finitism.
(Well actually, fractal functions form the fjords of /computability/.
But "computability" doesn't start with "f".)
Cheers - Chas
Tony Orlow
True, I guess I meant dense more than continuous, but "gapless" could
mean either, I suppose.
>
> (i) The fact that the suprareals are a dense set does not mean that the
> suprareals are connected.
>
> (ii) The fact that the suprareals are a dense set does not mean that
> there is a continuous function from the reals onto the suprareals.
>
> These two statements follow from the definitions of "dense",
> "connected" and "continuous".
>
> http://en.wikipedia.org/wiki/Dense_set
> http://en.wikipedia.org/wiki/Connected_set
> http://en.wikipedia.org/wiki/Continuous_function
Okay, my bad. Still, if the ih-numbers are included, and lie between the
reals and 0, then the reals consist of three disjoint sets,
(-eta_1,0),0, and (0,eta_1). Since the hierarchy continues forever, each
such H_x would be disjoint from 0. 0 is really eta_-oo. So, in order to
maintain connectedness on each level, it seems to me that each level of
polynomial eta_x on eta_1 needs to regress infinitely, having the form
union(n=-eta_1->x: H_n). Then each level is included in the last, and no
gap occurs. Of course, you mentioned there was a problem with infinite
polynomials with regard to 0 not being allowed for the 0th term or there
can be no multiplicative inverse, or something? Perhaps there's a way
around that by including eta_-oo=0?
>
> Or better yet, read a book on topology: Hocking and Young's "Topology"
> is about $13, and they probably have it at Barnes and Nobles.
>
> http://www.amazon.com/Topology-John-G-Hocking/dp/0486656764
>
That's a good idea, but I have a few other books I need to finish first,
I think.
A countable neighborhood is one where each point is finitely distant
from each other point, given some unit of measure. The nonstandard halo
is a countable neighborhood of infinitesimals in the vicinity of each
real, between it and any other standard real. Similarly, the countable
neighborhood of eta_x+y, where yeR, surrounds eta_x, and is between it
and any other distinct eta_n. Indeed, this could be considered to be an
open set, I suppose, as (eta_n - eta_n-1, eta_n + eta_n-1). It's also an
external set, in essence. I'm not sure I consider it quite the same as
an open interval. I'll have to think about that.
I think I misspoke above. Where you have successive etas, you can define
a countable neighborhood around eta_x as being within a finite number of
eta_x-1's. Where you attempt to have an uncountable hierarchy, there is
no base unit with which to define a countable neighborhood. You would
still have a total order, however, for the set of numbers you produce,
since ordering two elements is simply a matter of finding the term with
the highest power in which they agree. (Where one does not have a term
of that power, the coefficient is 0). How off am I?
> (ii) Why do you say "/the/ countable neighborhood"? Why is there only
> one unique such object associated with eta_1^x?
>
I just corrected myself there. That's only true with successive,
integral etas. An uncountable continuous set of etas would not lend
itself to the particular relatively infinitesimal unit required to
define "the" countable neighborhood.
> (iii) How does your definition of "countable neighborhood" /define/
> H_x; by which I mean provide a /definition/ of addition, subtraction
> and multiplication on the elements of H_x?
>
I don't think it does. But, if you have A=a*eta_x and B=b*eta_y, and you
define eta_x as eta_1^x, then the product A*B is a*b*eta_(x+y), and A/B
is a/b * eta(x-y). In adding and subtracting, only like terms are
combined, and where one has a term with an eta that the other is
missing, the other's coefficient is assumed to be 0. Adding or
subtracting multiple terms is not a problem, and neither is multiplying
multiple terms. Dividing by multiple terms may be a problem here, or
seem to be, but I think a digital interpretation such as the T-riffics
may help resolve that. Or, maybe not....I'll have to mull that over.
>> Any real
>> difference in an exponent applied to an infinite value results in an
>> infinite difference, an uncountably "disconnected" pair of sets.
>>
>
> (i) How does a real difference in the exponents x, y of eta_1 get
> "applied" to an "infinite value" to "result" in an "infinite
> difference"?
For real x and y such that x<y, lim(n->oo: n^y-n^x)=oo. :)
>
> (ii) What is the difference between a "disconnected" set and an
> "uncountably disconnected set"?
In this context, not much. I wouldn't say the set of standard nonzero
reals was "uncountably" disconnected, in the context of the standard
reals. Even in terms of infinitesimals it would be only countably
disconnected. But, all reals not in [0,1], for instance, would be
uncountably disconnected, at that infinitesimal level, even if only
finitely disconnected at the finite level.
When I say "uncountably disconnected", I mean that the difference
between eta_1^x and eta_1^y is greater than any finite number, if y>x
and eta_1 is greater than any finite number.
>
> (iii) What is an "uncountably "disconnected" pair of sets"? What two
> sets of suprareals are you referring to?
Those within a countable distance of eta_1^x and those within a
countable distance of eta_1^y.
>
> (iv) Why does it follow from these definitions that "an infinite
> difference" is therefore "an uncountably "disconnected" pair of sets"?
>
Because one cannot count from a member of one set to a member of the
other set.
>>> Yes. I have two choices:
>>> a) allow suprareals to be polynomials with an infinite number of
>>> terms;
>>> b) accept that H_1 U L_1 is not a field.
>>>
>>> Option (b) appears to be the more acceptable at this point, because
>>> it's simpler and because it allows me to keep the more general
>>> suprareal construction as being a polynomial with both positive and
>>> negative integer powers of eta_i.
>> If that's the construction, then I don't see how you can form your
>> "uncountable hierarchy", since there aren't an uncountable number of
>> integers. That's what I was saying about that part not jibing.
>>
>
> Let Eta = {eta_x : x in R+}. Let A = {a, b, c, ..., m} be a finite set
> of reals. Let E be the set of eta_'s defined by E = {eta_u : u in A}.
>
> Let h = p(E) be a polynomial with real coefficients over the set
> {eta_a, eta_b, eta_c, ..., eta_m}.
>
> Then if y is a real number greater than any real number in the (finite)
> set A, then we /define/ eta_y > h.
>
> Given that definition, can you work out whether
>
> 1 + (eta_2)^100*(eta_3)^2 > 3 - 2*eta_e + 17*eta_pi
>
> is true or false?
I don't think so, without knowing what raising an eta to a given power
does to its number. If the number is the power of eta_1, then we can say
1+(eta_2)^100*eta_3^2=1+eta_200*eta_6=1+eta_206.
3-3*eta_e + 17*eta_pi is in its lowest form, with a greatest eta of
eta_pi, far less than eta_206, so the statement would be true. That's
the only way I see the continuous hierarchy working properly.
>
> If not, can you instead first work out whether
>
> eta_pi < 1/17*(-2 + (eta_2)^100*(eta_3)^2 + 2*eta_e)
>
> is true or false?
>
Again, not without knowing how exponentiation works on etas, but given
my assumption, that's clearly true.
>>> And the suprareals can be compared to reals, although I'm not sure
>>> that's what Tony meant by "beyond". Again, the suprareals and the
>>> reals do not reside within the same "number line" or connected set.
>>>
>>>
>> But, did start by assuming some number beyond the reals, such that
>> eta_1>x for all x in R, yes?
>>
>
> If by "eta_1 is beyond the reals", you mean the assertion "in the
> suprareals, r < eta_1 for all r in H_1 such that r is of the form (r_0,
> 0, 0, ...)", then yes. If you mean something else, then I can't opine
> at this point.
>
>>> Tony Orlow wrote:
>>>>> While one could
>>>>> consider a non-integral such number, one would do better to assume eta_1
>>>>> to be integral, with fractional differences covered by its finite real
>>>>> "halo".
>>> Chas Brown wrote:
>>>> One would do better to be specific about what one means, rather than
>>>> sling around mathematical sounding phrases without specific meaning.
>>>> "Fractional differences covered by its finite real "halo" " is hardly
>>>> comprehensible to me.
>>>>
>> I don't think that's my problem, and you needn't get snippy about it. A
>> simple request for clarification will do.
>>
>
> OK. Given /David's definition/ of H_1:
>
> (i) What does it mean for eta_1 to be "non-integral" in the suprareals?
> What does it mean for eta_1 to be "integral" in the suprareals?
What does it mean? I guess that is can be considered a count of whole
entities, albeit larger than any finite count. As a digital number, all
digits right of the digital point are 0.
>
> (ii) What are "fractional differences", in the context of the
> suprareals? Are the "fractional differences" of eta_1 the same as, or
> different from, the "fractional differences" of any other suprareal?
>
I mean that eta_1 should be considered integral, and eta_1 + a
non-intergral value should be considered non-integral. This really isn't
particularly central at this point anyway. It was a response, I think,
to an objection to eta_1 as a set size...
> (iii) What is the "finite real halo" of eta_1 in the suprareals?
The set of all numbers eta_1+x for xeR.
>
> (iv) What does it mean to say "the fractional differences of eta_1 are
> covered by its finite real halo" in the suprareals?
That any fractional component would be handled by the finite eta_0 term.
>
> (v) Assuming that we can now make sense of your assertion, why would it
> be /better/ to "assume" eta_1 to be integral, with fractional
> differences covered by its finite real "halo", instead of "merely"
> assuming that eta_1 is "non-integral"? What, exactly, is "better" about
> it?
>
That it serves as a count and an infinite unit. Finite fractions are
lost at that level. Even finite units are lost to an extent, which is
why you're objecting, I think, but it pays to view numbers like eta_1 or
aleph_0 as whole numbers, without fractional part. Then again, it may be
interesting the other way..
>> The question is whether eta_1 can be used as a count, or set size. It
>> cannot, if it's not a whole number.
>
> As I use the term "the number of elements in set X = card(X)", it
> cannot, if it is not a /cardinal number/.
Cardinality is ONE approach to set size. Did I use the term? No? Then
please don't inject it. This has nothing to do with cardinality.
>
> If eta_1 is "the number of elements in the set X", it means to me that
> there is a bijective function f : eta_1 -> X.
That's what it means to you, because you've been trained to think about
set size that way.
>
> But the /suprareal/ 2*eta_1 is /not a set/; except in the trivial sense
> that the sequence (0, 2, 0, 0, 0,...) is a set.
Hand waving doesn't prove anything. Every number represents the set of
finite units between it and zero.
>
> So to say "the set Y has 2*eta_1 elements" no longer makes sense to me;
> unless you mean that there is a bijective function between the set (0,
> 2, 0, ...) and set Y (which would be an odd thing to mean, since then
> there also exists a bijection between the suprareal 0 = (0, 0, ...) and
> Y).
>
> I have no real idea of what you mean by "eta_1 is the number of
> elements in set X", if you /don't/ mean "the cardinality of X".
>
You should know by now I don't mean "gardenality".
>> Consider it a whole number
>
> What is a "whole number", when we are not talking about the reals? In
> particular, what is a "whole number" in David's H_1?
One in which there are no terms of the eta_1^x where x is negative, and
if there is a term with eta_0, it is integral. Anything above eta_0
should be considered integral.
>
>> and any
>> non-integral suprareal as requiring the addition of a finite real
>> component. The "halo" in NSA is the countable neighborhood of
>> infinitesimals that can be considered to lie around each real number.
>
>> In
>> similar fashion, we can view numbers like the suprareals, which are
>> separated by an uncountable number of units from each other, to have
>> countable neighborhoods of unit intervals surrounding them, each exactly
>> the same as the standard real line. That's what I mean by a "halo". It's
>> the same thing on a different scale. Questions?
>>
>
> (i) What does it mean to say "suprareals x and y are separated by an
> uncountable number of units"?
It means you cannot count from one to the other, assuming there is a
difference in terms at a level above eta_0.
>
> (ii) What does it mean for a suprareal y to "be considered to lie
> around" the suprareal x = 1 + eta_1^2?
To be on one side or the other, between it and any other suprareal which
differs in a term higher than eta_0.
>
> (iii) From these definitions, why does it follow that there is only a
> countable number of such infinitesimal y's? Why isn't there an
> uncountable number of such infinitesimal y's?
That would produce a measurable real interval.
>
> (iv) What does it mean for a neighborhood in the suprareals to be "a
> countable neighborhood of a unit interval"?
That all elements in the neighborhood are within a finite number of
finite units from each other.
>
> (v) How does a collection of "countable neighborhoods of unit
> intervals" then "surround" a suprareal x? When do they /not/ "surround"
> some other suprareal, y?
Because, being finitely distant from x, they are all infinitely distant
from y.
>
> (v) What does it mean for a countable neighborhood of a unit interval
> in the suprareals to be "exactly the same as" the real line?
You misread. The countable neighborhood of unit intervals which
surrounds each suprareal is merely a copy of the real line, translated
so that the origin is that suprareal.
>
> (vi) When you say "that's what (you) mean by a halo", what is "that"?
> Is "that" a "way of viewing" the suprareals? Is "that" some particular
> set of suprareals? Is "that" a property of certain sets of suprareals?
I was explaining what I meant, when I extended the nonstandard notion of
the halo to larger scales, such that there is a finite halo around 0,
and eta_x halos around those.
>
> Etc.
>
> If I asked you what you meant every time you made an assertion, these
> threads would get even longer. It's usually better to ignore your
> statements unless I can make /some/ sense of them.
>
> That's why you should /learn/ some of the terminology. Then we can
> understand each other much better!
>
Oh, well, then ignore them.
So, number in itself means almost nothing....
And how does that apply to eta_1, particularly?
Have a nice day.
Tony
hagman
cbr...@cbrownsystems.com schrieb:
> hagman wrote:
> > One needs to distinguish R[[X]] from R(X), i.e.
> > one need a power series that is not a rational function,
> > for example the exponential (contrary to my former post, I view X as
> > the infinitesimal unit here, not the infinite unit; but that's just to
> > simplify notation).
>
> Yes; I understand: we are simply considering R[[X]] and R(X) as
> polynomials with some total ordering.
>
> I should note that the only exposure I've gotten in any depth regarding
> formal power series was from a quick introduction in Stanley's
> "Enumerative Combinatorics", where he just needs to describe enough of
> their properties to make them useful for describing generating
> functions; and to establish a calculus such that F(X) + G(X),
> F(X)*G(X), and F(G(X)) all have meaning.
>
> So just to make sure I remember this correctly (since you refer to it
> below):
>
> In Stanley's description, the topology is one where a sequence of
> elements of R[[X]] {Fi(X)} = (F1(X), F2(X), ..., Fn(X), ...) converges
> to the element F(X) according to the following rules:
>
> Let a_(i, n) be the coeffcient of X^n in Fi(X); and a_n be the similar
> coefficient in F(X).
>
> Then for every n, there is a natural j such that for all k > j, a_(k,
> j) = a_j.
This should probably read:
Then for every n, there is a natural j such that for all k > j, a_(k,n)
= a_n.
>
> This is different than requiring that the limit k->oo {a_(k, j)} = a_j;
... at least if one takes the standard topology on R.
If you take discrete topology on R, both notions coincide.
In other words, for power series one uses pointwise convergence of
coefficients with respect
to discrete topology on the base ring.
> and I presume this is equivalent to your statement below "the fact that
> powers of X are small"?
Exactly.
For all n there is a j_n such that for all k>j_n, a_(k,n) = a_n
<=>
For all n there is a m (namely m=max{j_0, j_1, ..., j_n}) such that
for all k>m, Fk(X)-F(X) is a multiple of X^n.
Usually one would want the latter to read
For all eps>0 there is a m such that for all k>m
|Fk(X)-F(X)| < eps.
Therefore one may say "something is mindbogglingly small"
instead of "something is divisible by a mindbogglingly high power of
X".
> Or do you use a different notion of convergence
> than Stanley defines?
Fortunately not.
When I started reading your post I got shocked that I might have
erroneously ignored the topology of R in R[[X]].
Well, partly I have as we do use the topology of R in R(X).
Interestingly, this poses no problem - see below.
>
> > Let S be the subset of R(X) consisting of all f(X)/(1+X*g(X)).
> > Since 1+X*g(X) is invertible in R[[X]], ...
>
> ... because an element of R[[X]] is invertible iff its "constant
> coefficient" is non-zero...
exactly. And of course we can hide any non-zero constant in f if it
should be
different from 1.
>
> > ...we can view S as (part of) the
> > intersection of R(X) and R[[X]]...
>
> ... because surely (1 + X*g(X) designates a particular element in R(X).
>
> > ("Fortunately", the topology on R[[X]]
> > is also defined by the fact that powers of X are small).
>
> See above. In any case, I assume you imply that the open sets of S =
> R[[X]] intersect R(X) are the open sets of R(X) intersect S, which are
> the same as the open sets of R[[X]] intersect S; and so if {Fi(X)}
> converges to F in R[[X]], it also converges to the equivalent member of
> R(X) in R(X).
Well, at least I thought so.
But since R(X) respects the standard topology of R and R[[X]]
implicitly uses discrete topology on R, a sequence converging in R(X)
need not necessarily converge in R[[X]], or does it?
Well, a zero sequence in R(X) will have to have almost all members
smaller than eps for any given positive eps.
But every eps>0 can be written as
r*X^m*(1+X*f(X))/(1+X*g(X)) with m in Z, r>0, f(X), g(X) polynomials
(check that!).
But being <eps in absolute value
- implies being divisible by X^m and
- is implied by being divisible by X^(m+1)
Thus respecting or ignoring the standard topology of R made no
difference.
Lucky me.
Note that this implies that 1/n does not converge to 0.
Indeed, 1/n will never be less than X.
But that does not show incompleteness, as 1/n is no Cauchy sequence...
>
> > One can show that the series (a_n) with a_n = sum_{k=0,...,n} X^n/n!
> > is a Cauchy sequence.
... because differences between "late" elements of the sequence are
divisible
by high powers of X.
> > Assume it converges against an element f(X)/g(X) of R(X) where f(X) and
> > g(X) have no irreducible factors in common.
> > Obviously, the limit has to be finite (namely, very close to a_0 = 1
> > but obviuously different from a_0).
> > Hence we can assume that g(X) = 1 + X*h(X).
> > But then f(X)/g(X) is in S and we must obtain the same limit in R[[X]].
> > But there we have exp(X) as limit.
> > Hence f(X) = g(X)*exp(X) as formal power series.
>
> With you so far...
>
> > The equality must still hold when we insert a complex number for X as
> > long as we have convergence.
>
> I am definitely missing a theorem in your toolbox. Can you amplify on
> this assertion?
Maybe a less invasive tool would have sufficed if it had come to my
mind in time:.
The (or at least almost all) coefficients of the formal power series of
f(X)/g(X) fulfill a linear recursion (in a similar fashion as the
decimal expansion of fractions is eventually periodic).
This is clear from writing down the condition that the coefficient of
X^n in g(X) * (f(X)/g(X)) must be 0 if n exceeds deg(f).
Thus automatically f(X)/g(X) cannot equal exp(X), where the
coefficients do /not/ follow a linear recursion (needs to be checked).
This much simpler trick works with any formal power series instead of
exp(X),
as long as there is no linear recursion for the coefficients.
It works even for those series that do not converge when evaluated at
any nonzero number.
Now that I've thought of it I'm much happier with this proof method
than my previous one...
A much simpler power series that /obviously/ doesn't allow a linear
recursion for its coefficients would be sum_(n>=0) X^(n^2), which has
arbitrary long blocks of 0's.
If I had that used that series in the proof, the proof would end a
couple of lines earlier
and without using properties of R or C.
>
> Are you saying that f(X) and g(X) /must/ belong to the subset of R[[X]]
> where every series converges in C when X is replaced by any complex
> number?
I admit that I magically (but legally) went from R[[X]] to C[[X]].
What we do have is:
If z is in C, then in C[[X]] we have the subring of series converging
at z.
Then setting X=z is a homomorphism of that subring to C.
Fortunately, f(X) and g(X) are polynomials and exp is a very happy
converger.
But according to my remarks above you may ignore this part anyway.
cbr...@cbrownsystems.com
> cbr...@cbrownsystems.com wrote:
> > Tony Orlow wrote:
> >> David R Tribble wrote:
> >>> Chas Brown wrote:
> >>> Which means that it's not proper to think of the union of the reals
> >>> and the suprareals as a single "number line", but more as separate
> >>> disconnected number lines. More properly, they are disconnected
> >>> sets.
> >>>
> >>>
> >> Then thesame applies to theh-numbers. If they are not colinear with the
> >> reals, then the reals do not constitute a "gap" within the suprareals.
> >> If you apply the logic of connectedness to the suprareals as you do to
> >> the reals, then they are also gapless. For any two suprareals, positive
> >> or negative, there lies a suprareal between them, Therefore, they are as
> >> "continuous" as the reals.
> >>
> >
> > For the third time in this thread, "dense", "connected", and
> > "continuous" have different meanings. And "gapless" has no particular
> > meaning to me at all; unless it is supposed to mean "dense".
>
>
> True, I guess I meant dense more than continuous, but "gapless" could
> mean either, I suppose.
>
The main point being: a set of suprareals can "have no gaps" (in the
sense that between any two elements there is another element); while at
the same time it is "has gaps" (e.g., between the finites and
non-finites in the sense that they are not connected topologically).
These latter "gaps" prevent the existence of a /continuous function/
from the reals onto the suprareals, and are part of why the suprareals
are different than the reals.
Just try to remember that when you say "X is continuous" on sci.math,
people are going to think you are saying that X is the image of a
continuous function from R to (whatever topological space X belongs
to).
> >
> > (i) The fact that the suprareals are a dense set does not mean that the
> > suprareals are connected.
> >
> > (ii) The fact that the suprareals are a dense set does not mean that
> > there is a continuous function from the reals onto the suprareals.
> >
> > These two statements follow from the definitions of "dense",
> > "connected" and "continuous".
> >
> > http://en.wikipedia.org/wiki/Dense_set
> > http://en.wikipedia.org/wiki/Connected_set
> > http://en.wikipedia.org/wiki/Continuous_function
>
> Okay, my bad. Still, if the ih-numbers are included, and lie between the
> reals and 0, then the reals consist of three disjoint sets,
> (-eta_1,0),0, and (0,eta_1).
If by (0, eta_1), you mean the set of all suprareals h such that 0 < h
< eta_1, then you are implying that you feel that "eta_1 - 1" should be
considered a real number.
Is that your intent? The sensibility of the remainder of your argument
seems to rely on interpreting this.
> Since the hierarchy continues forever, each
> such H_x would be disjoint from 0. 0 is really eta_-oo.
> So, in order to
> maintain connectedness on each level, it seems to me that each level of
> polynomial eta_x on eta_1 needs to regress infinitely, having the form
> union(n=-eta_1->x: H_n).
Not sure what
union(n=-eta_1->x: H_n)
is supposed to mean.
Is n supposed to be a real number? If so, -eta_1 is not a real number.
On the other hand, if n is supposed to be a suprareal, then "H_n" is
not defined (so far) in David's system except for cases where n is a
real number.
> Then each level is included in the last, and no
> gap occurs.
Again, I can't tell what you mean by "gap" here, because you use it in
two different ways.
Yes, there are no "gaps", because the resulting set is dense.
But no, there /is/ a "gap", because each "level" is the disjoint union
of open sets which then "don't touch": each "level" is disconnected
topologically from each of the other "levels".
> Of course, you mentioned there was a problem with infinite
> polynomials with regard to 0 not being allowed for the 0th term or there
> can be no multiplicative inverse, or something? Perhaps there's a way
> around that by including eta_-oo=0?
>
> >
> > Or better yet, read a book on topology: Hocking and Young's "Topology"
> > is about $13, and they probably have it at Barnes and Nobles.
> >
> > http://www.amazon.com/Topology-John-G-Hocking/dp/0486656764
> >
>
> That's a good idea, but I have a few other books I need to finish first,
> I think.
>
Next time you're in B&N or the like, check out the mathematics section.
There's a ton of cheap Dover books that cover a lot of stuff. That's
how I learned most of the mathematics I know on a tight budget.
> >>> What is curious (and which I will probably add to the next revision)
> >>> is the fact that
> >>> x0 + eta_1
> >>> x1 eta_1^1
> >>> x2 eta_1^2
> >>> ...
> >>> are unconnected (uncountable) sets. Any member of one set in the
> >>> list is less than any member of the next set (assuming all positive
> >>> real x's).
> >>>
> >>>
> >> That was the suggestion I offered: define you H_x as the countable
> >> neighborhood of eta_1^x, whether x is natural or real.
> >
> > (i) What is a "countable neighborhood"? A neighborhood is usually
> > equivalent to "an open set", which in this topology means "an open
> > interval in the suprareals". But what is a "countable neighborhood"?
> > And how do you define an open interval prior to defining a total order
> > on the elements you aim to define?
> >
>
> A countable neighborhood is one...
... one what? One neighborhood? What do you consider a neighborhood, in
your definition of H_x?
> ... where each point is finitely distant
> from each other point, given some unit of measure.
What does "finitely distant" mean in this context? What "unit of
measure" are you proposing?
And why do you call the neighborhood "countable"? This usually implies
that the neighborhood contains a countable number of elements; which
will never be a neighborhood in the suprareals.
Why is it not simply a neighborhood; i.e., a union of open intervals?
You never even addressed this part of the question...
> The nonstandard halo
> is a countable neighborhood of infinitesimals in the vicinity of each
> real, between it and any other standard real.
You still haven't /defined/ a countable neighborhood, nor a
neighborhood; so I don't know what you mean here.
> Similarly, the countable
> neighborhood of eta_x+y, where yeR, surrounds eta_x, and is between it
> and any other distinct eta_n. Indeed, this could be considered to be an
> open set, I suppose, as (eta_n - eta_n-1, eta_n + eta_n-1). It's also an
> external set, in essence. I'm not sure I consider it quite the same as
> an open interval. I'll have to think about that.
I'm not being prickly here, I'm merely pointing out that, from your
comments, not even you appear to know exactly what you mean by a
"countable neighborhood". Maybe it's an open interval; maybe it's not.
Maybe it's the union of open intervals. Who knows, if you don't?
So it's hard to expect me, or other people reading this thread, to make
sense of your comments when you yourself have trouble making sense from
them.
>
> I think I misspoke above. Where you have successive etas, you can define
> a countable neighborhood around eta_x as being within a finite number of
> eta_x-1's. Where you attempt to have an uncountable hierarchy, there is
> no base unit with which to define a countable neighborhood.
This part makes no sense to me.
> You would
> still have a total order, however, for the set of numbers you produce,
> since ordering two elements is simply a matter of finding the term with
> the highest power in which they agree. (Where one does not have a term
> of that power, the coefficient is 0). How off am I?
>
Your last statement is almost correct: we can impose a total order on
the numbers we produce by ordering the pair of suprareals x and y by
the coefficient of the highest power of the /largest/ eta_ in their
difference (x - y).
If that coefficient is < 0 then we /define/ that y > x. If that
coefficient is > 0, then we /define/ that x > y.
And we determine the "largest" eta_ in x - y by /defining/ that eta_a <
eta_b iff a < b as real numbers.
For example:
1 + (eta_2)^100*(eta_3)^2 > 3 - 2*eta_e + 17*eta_pi
is false; because
1 + (eta_2)^100*(eta_3)^2 - (3 - 2*eta_e + 17*eta_pi)
= -2 + (eta_2)^100*(eta_3)^2 + 2*eta_e - 17*eta_pi
The "largest" eta_ by /definition/ is eta_pi; because pi > 0, and pi >
2 and pi > 3 and pi > e.
The highest power of eta_pi is 1.
The coefficient of the highest power of eta_pi is -17.
But -17 < 0, so
-2 + (eta_2)^100*(eta_3)^2 - + 2*eta_e - 17*eta_pi < 0
which implies that actually
1 + (eta_2)^100*(eta_3)^2 < 3 - 2*eta_e + 17*eta_pi
Do you understand now?
>
>
> > (ii) Why do you say "/the/ countable neighborhood"? Why is there only
> > one unique such object associated with eta_1^x?
> >
>
> I just corrected myself there. That's only true with successive,
> integral etas. An uncountable continuous...
Please! /Stop/ using "continuous" in this way! You mean "dense set of",
or perhaps "divisible set of".
> ... set of etas would not lend
> itself to the particular relatively infinitesimal unit required to
> define "the" countable neighborhood.
>
> > (iii) How does your definition of "countable neighborhood" /define/
> > H_x; by which I mean provide a /definition/ of addition, subtraction
> > and multiplication on the elements of H_x?
> >
>
> I don't think it does.
Then why did you say "That was the suggestion I offered: define your
H_x as the countable neighborhood of eta_1^x, whether x is natural or
real."? What good is such a definition, if it does not also tell us
what we mean by a + b or a*b?
> But, if you have A=a*eta_x and B=b*eta_y, and you
> define eta_x as eta_1^x, then the product A*B is a*b*eta_(x+y), and A/B
> is a/b * eta(x-y).
Probably. That would, of course be a an entirely different system than
the one David proposes. And the topic of this thread is David's
proposal, not some other proposal.
> In adding and subtracting, only like terms are
> combined, and where one has a term with an eta that the other is
> missing, the other's coefficient is assumed to be 0. Adding or
> subtracting multiple terms is not a problem, and neither is multiplying
> multiple terms. Dividing by multiple terms may be a problem here, or
> seem to be, but I think a digital interpretation such as the T-riffics
> may help resolve that. Or, maybe not....I'll have to mull that over.
>
> >> Any real
> >> difference in an exponent applied to an infinite value results in an
> >> infinite difference, an uncountably "disconnected" pair of sets.
> >>
> >
> > (i) How does a real difference in the exponents x, y of eta_1 get
> > "applied" to an "infinite value" to "result" in an "infinite
> > difference"?
>
> For real x and y such that x<y, lim(n->oo: n^y-n^x)=oo. :)
>
What is "n"? Is it a natural number, a real number, or a suprareal?
When you say "=oo", oo is neither a natural, a real, nor a suprareal.
Do you mean that the limit increases without bound /in the suprareals/,
i.e., that the limit is not defined?
> >
> > (ii) What is the difference between a "disconnected" set and an
> > "uncountably disconnected set"?
>
> In this context, not much. I wouldn't say the set of standard nonzero
> reals was "uncountably" disconnected, in the context of the standard
> reals. Even in terms of infinitesimals it would be only countably
> disconnected. But, all reals not in [0,1], for instance, would be
> uncountably disconnected, at that infinitesimal level, even if only
> finitely disconnected at the finite level.
>
> When I say "uncountably disconnected", I mean that the difference
> between eta_1^x and eta_1^y is greater than any finite number, if y>x
> and eta_1 is greater than any finite number.
>
Then it appears that "uncountably disconnected" has nothing to do with
something being either uncountable or disconnected. It is simply a way
of saying that x > y implies that eta_1^x > eta_1^y.
> >
> > (iii) What is an "uncountably "disconnected" pair of sets"? What two
> > sets of suprareals are you referring to?
>
> Those within a countable distance of eta_1^x and those within a
> countable distance of eta_1^y.
>
What is a "countable distance"? How is it different from "a finite
distance"? If you mean "finite distance", why don't you just say
"finite distance"?
> >
> > (iv) Why does it follow from these definitions that "an infinite
> > difference" is therefore "an uncountably "disconnected" pair of sets"?
> >
>
> Because one cannot count from a member of one set to a member of the
> other set.
What does it mean to say that you /can/ count from "a member of one
set" to "a member of another set? Can you "count" from a member of the
set (0,1/pi) to a member of the set (1/2,1/4)?
>
> >>> Yes. I have two choices:
> >>> a) allow suprareals to be polynomials with an infinite number of
> >>> terms;
> >>> b) accept that H_1 U L_1 is not a field.
> >>>
> >>> Option (b) appears to be the more acceptable at this point, because
> >>> it's simpler and because it allows me to keep the more general
> >>> suprareal construction as being a polynomial with both positive and
> >>> negative integer powers of eta_i.
> >> If that's the construction, then I don't see how you can form your
> >> "uncountable hierarchy", since there aren't an uncountable number of
> >> integers. That's what I was saying about that part not jibing.
> >>
> >
> > Let Eta = {eta_x : x in R+}. Let A = {a, b, c, ..., m} be a finite set
> > of reals. Let E be the set of eta_'s defined by E = {eta_u : u in A}.
> >
> > Let h = p(E) be a polynomial with real coefficients over the set
> > {eta_a, eta_b, eta_c, ..., eta_m}.
> >
> > Then if y is a real number greater than any real number in the (finite)
> > set A, then we /define/ eta_y > h.
> >
> > Given that definition, can you work out whether
> >
> > 1 + (eta_2)^100*(eta_3)^2 > 3 - 2*eta_e + 17*eta_pi
> >
> > is true or false?
>
> I don't think so, without knowing what raising an eta to a given power
> does to its number. If the number is the power of eta_1, then we can say
> 1+(eta_2)^100*eta_3^2=1+eta_200*eta_6=1+eta_206.
We are working in /David's system/. So no, it is not true that
eta_2^100 = eta_200. Each eta_x is /distinct/; the product eta_x*eta_y
is NOT THE SAME AS eta_(x+y) or eta_(x*y):
(eta_1)^2 IS NOT EQUAL TO eta_2.
In fact eta_2 > r*(eta_1)^m for /all/ reals r and /all/ naturals m.
(eta_2)^2 IS NOT EQUAL TO eta_4.
In fact eta_4 > r*(eta_2)^m for /all/ reals r and /all/ naturals m.
eta_2*eta_3 IS NOT EQUAL TO eta_5.
In fact eta_5 > r*(eta_2)^j*(eta_3)^k for /all/ reals r and /all/
naturals j,k.
eta_2 is not "numerically" associated with the number 2. Instead, "2"
is just the "name" we give to a particular member of the set of eta_'s.
> 3-3*eta_e + 17*eta_pi is in its lowest form, with a greatest eta of
> eta_pi, far less than eta_206, so the statement would be true. That's
> the only way I see the continuous hierarchy working properly.
>
Then you are still misunderstanding David's construction.
> >
> > If not, can you instead first work out whether
> >
> > eta_pi < 1/17*(-2 + (eta_2)^100*(eta_3)^2 + 2*eta_e)
> >
> > is true or false?
> >
>
> Again, not without knowing how exponentiation works on etas, but given
> my assumption, that's clearly true.
You still misunderstand even David's most basic construction, H_1.
We have /defined/ how exponentiation works on eta_'s. It's very simple.
For any eta_x, eta_x = (eta_x)^1. And for any natural number n,
eta_x*(eta_x)^n = (eta_x)^(n+1).
So for example, (eta_x)*(eta_y)*(eta_x)^2 = (eta_y)*(eta_x)^3, yes?
In David's system, by definition, there is /no/ eta_z such that eta_1 +
eta_2 = eta_z. Do you agree?
The thing it appears you are missing is that also:
IT IS NOT THE CASE THAT there exists a z such that (eta_x)*(eta_y) =
eta_z.
Every eta_ is "independent" of every other eta_. This is just like:
(1 + y)*(1 + x)^2 = 1 + y + 2*x*y + 2*x + y*x^2 + x^2
There is no way in an equation like this to somehow "reduce" it to just
an expression involving one variable x. It's an "equation in two
variables".
Similarly, we use the same /formal/ rules to say:
(1 + eta_y)*(1 + eta_x)^2
= 1 + eta_y + 2*eta_x*eta_y + 2*eta_x + eta_y*(eta_x)^2 +
(eta_x)^2
There is (by David's definition) no way to "reduce" this equation to be
an expression involving just eta_x (or any other eta_). It is an
"equation in two eta_'s".
>
> >>> And the suprareals can be compared to reals, although I'm not sure
> >>> that's what Tony meant by "beyond". Again, the suprareals and the
> >>> reals do not reside within the same "number line" or connected set.
> >>>
> >>>
> >> But, did start by assuming some number beyond the reals, such that
> >> eta_1>x for all x in R, yes?
> >>
> >
> > If by "eta_1 is beyond the reals", you mean the assertion "in the
> > suprareals, r < eta_1 for all r in H_1 such that r is of the form (r_0,
> > 0, 0, ...)", then yes. If you mean something else, then I can't opine
> > at this point.
> >
> >>> Tony Orlow wrote:
> >>>>> While one could
> >>>>> consider a non-integral such number, one would do better to assume eta_1
> >>>>> to be integral, with fractional differences covered by its finite real
> >>>>> "halo".
> >>> Chas Brown wrote:
> >>>> One would do better to be specific about what one means, rather than
> >>>> sling around mathematical sounding phrases without specific meaning.
> >>>> "Fractional differences covered by its finite real "halo" " is hardly
> >>>> comprehensible to me.
> >>>>
> >> I don't think that's my problem, and you needn't get snippy about it. A
> >> simple request for clarification will do.
> >>
> >
> > OK. Given /David's definition/ of H_1:
> >
> > (i) What does it mean for eta_1 to be "non-integral" in the suprareals?
> > What does it mean for eta_1 to be "integral" in the suprareals?
>
> What does it mean? I guess that...
Don't guess! If you don't know what you mean, how can we be expected to
know what you mean?
> ... is can be considered a count of whole
> entities, albeit larger than any finite count. As a digital number, all
> digits right of the digital point are 0.
eta_1 is then a "count" of whole entities: it is 1 of the whole entity,
eta_1.
eta_1 + sqrt(2) is also a "count" of whole entities: it is 1 of the
whole entity, eta_1 + sqrt(2).
So I think you mean something else.
>
> >
> > (ii) What are "fractional differences", in the context of the
> > suprareals? Are the "fractional differences" of eta_1 the same as, or
> > different from, the "fractional differences" of any other suprareal?
> >
>
> I mean that eta_1 should be considered integral, and eta_1 + a
> non-intergral value should be considered non-integral.
A definition is not an example; and an example is not a definition.
What is a /definition/ of "x is a fractional difference"?
> This really isn't
> particularly central at this point anyway. It was a response, I think,
> to an objection to eta_1 as a set size...
>
> > (iii) What is the "finite real halo" of eta_1 in the suprareals?
>
> The set of all numbers eta_1+x for xeR.
>
Yippeeee! Okay! That /is/ a definition! We can even make it clearer:
For /any/ h in the suprareals, the finite real halo of h is the set
Halo(h) = {g : g = h + r, r in R}.
Or, in words rather than symbols, for any suprareal h, by "Halo(h)" we
mean the set of all suprareals g of the form "h + r", where r is a real
number (which is exactly what the above mathematical symbols mean).
Since all reals are finite by definition, I would recommend following
the standard espoused by the Department of Redundancy Department: Don't
call it "the finite real halo of h", just call it "the real halo of h".
> >
> > (iv) What does it mean to say "the fractional differences of eta_1 are
> > covered by its finite real halo" in the suprareals?
>
> That any fractional component would be handled by the finite eta_0 term.
What is "a fractional component"? How is it "handled" by the finite
term?
Be /specific/, and use terms that we /both/ understand.
For example, if by "a fractional difference of eta_1", you mean in your
mind "a suprareal x such that eta_1 - x is a real number", then don't
define it by calling it "a fractional component"; because that really
doesn't tell me anything, and forces me to ask "what is a fractional
component?".
Instead, just call it "a suprareal x such that eta_1 - x is a real
number". We don't need any fancier name for it than that.
If by "handled by the finite eta_0 term", you mean "eta_1 - x is a real
number; so x is a member of Halo(eta_1)", then don't say "it is handled
by the finite eta_0 term"; because then I have to ask "what do you mean
by 'handled by'?".
Just say "eta_1 - x is a real number; so x is a member of Halo(eta_1)".
>
> >
> > (v) Assuming that we can now make sense of your assertion, why would it
> > be /better/ to "assume" eta_1 to be integral, with fractional
> > differences covered by its finite real "halo", instead of "merely"
> > assuming that eta_1 is "non-integral"? What, exactly, is "better" about
> > it?
> >
>
> That it serves as a count and an infinite unit. Finite fractions are
> lost at that level.
"Lost"? Where did they go? What "level" are we "at"?
> Even finite units are lost to an extent, which is
> why you're objecting, I think...
No I'm objecting (as usual) that, with each successive request for
clarification, instead of just saying what you mean, you use words
which you don't define to define the words that you are asked to
define.
For example, what do you mean by "even finite units are lost to an
extent"? What finite units are you talking about? The natural numbers?
How can they be "lost" at all, let alone be lost "to an extent"?
>, but it pays to view numbers like eta_1 or
> aleph_0 as whole numbers, without fractional part. Then again, it may be
> interesting the other way..
>
If I knew what you meant, I could respond.
But I think you are forgetting that, for example, for any two real
numbers r and s, Halo(r*eta_1) intersect Halo(s*eta_1) is empty - so
even just considering eta_1 (and not even looking at, say, (eta_1)^2),
r*eta_1 has no "fractional part" as you have defined it.
So, is (1/sqrt(2))*eta_1 a "whole number"? It has no "fractional
component", correct?
> >> The question is whether eta_1 can be used as a count, or set size. It
> >> cannot, if it's not a whole number.
> >
> > As I use the term "the number of elements in set X = card(X)", it
> > cannot, if it is not a /cardinal number/.
>
> Cardinality is ONE approach to set size. Did I use the term? No? Then
> please don't inject it. This has nothing to do with cardinality.
>
Well, you are claiming that, by assuming that eta_1 is "integral", you
can somehow define the "number of elements" in some set. I only know
one meaning for that term: cardinality. If you have another, it is your
job to /define/ it.
Otherwise, I have no idea what you're talking about; and as we have
seen above, with all due respect, it may be the case that even /you/
have no idea really what you're talking about either.
> >
> > If eta_1 is "the number of elements in the set X", it means to me that
> > there is a bijective function f : eta_1 -> X.
>
> That's what it means to you, because you've been trained to think about
> set size that way.
>
Yup. It turns out that it's quite handy; which helps explain its
popularity.
> >
> > But the /suprareal/ 2*eta_1 is /not a set/; except in the trivial sense
> > that the sequence (0, 2, 0, 0, 0,...) is a set.
>
> Hand waving doesn't prove anything. Every number represents the set of
> finite units between it and zero.
>
What is "the set of finite units between 2*eta_1 and 0"? What are the
elements of this set?
What is "the set of finite units between (eta_1)^2 and 0"? What are the
elements of this set?
What is the difference between "the set of finite units between eta_1
and 0" and "the set of finite units between (1/sqrt(2))*eta_1 and 0"?
How does the number eta_1 + (eta_1)^2 "represent" the set of finite
units between it and zero?
> >
> > So to say "the set Y has 2*eta_1 elements" no longer makes sense to me;
> > unless you mean that there is a bijective function between the set (0,
> > 2, 0, ...) and set Y (which would be an odd thing to mean, since then
> > there also exists a bijection between the suprareal 0 = (0, 0, ...) and
> > Y).
> >
> > I have no real idea of what you mean by "eta_1 is the number of
> > elements in set X", if you /don't/ mean "the cardinality of X".
> >
>
> You should know by now I don't mean "gardenality".
>
Yes; but I don't know what you /do/ mean by it.
I can tell you /exactly/ what I mean by "the number of suprareals =
|R|".
But you /cannot/ tell me exactly what you mean by "eta_1^2 is the
number of finite units between eta_1^2 and 0", because you haven't
/defined/ what you /mean/ by that.
I don't think what you have in mind is actually a /definition/ yet.
It's a /metaphor/; a mental image.
It turns out that most of the things you are talking about, like
"Halo", "integral", etc., can already be expressed in David's system;
but they don't yet add up to a coherent meaning of what you are trying
to imply by calling "eta_1" the "number of elements in some set".
> >> Consider it a whole number
> >
> > What is a "whole number", when we are not talking about the reals? In
> > particular, what is a "whole number" in David's H_1?
>
> One in which there are no terms of the eta_1^x where x is negative, and
> if there is a term with eta_0, it is integral. Anything above eta_0
> should be considered integral.
So (1/pi)*eta_1 + 2 is integral, because 2 is a whole number and
(1/pi)*eta_1 is "above" eta_0? That seems to contradict your above
statements.
Here's what I /think/ you're trying to say:
Let the integer suprareals be those polynomials of the form:
a_0 + a_1*eta_1 + a_2*eta_1^2 + ... + a_n*eta_1^n
where each a_i is an integer.
Now, if you like, given an integer suprareal x >= 0, you can say "by x
is the number of integer suprareals < x, I mean that x represents the
set {y : y an integer suprareal and 0 <= y < x}".
I can still then observe: for any integer suprareal x, the set x
represents is countable.
And yet this does not "destroy" your assertion "if y < x, then the set
represented by y is a proper subset of the set represented by x", or
whatever else you might like to say.
For example, you can say that "by the set of elements represented by x
plus the set of the elements represented by y, I mean the set of
elements repesented by (x+y)".
/Then/ if you like, you can say "eta_^2 + 1 is the number of elements
in the set eta_1^2, added to the number of elements in the set 1". Or,
once we have established that that's what you /mean/, we can use the
shorthand "if we add 1 to the set eta_1^2, we get the set eta_1^2 + 1"
Go nuts! But /define/ what you mean when you use english words like
"number of", "more" and "less"; so that others will be able to
understand what you say.
> >
> >> and any
> >> non-integral suprareal as requiring the addition of a finite real
> >> component. The "halo" in NSA is the countable neighborhood of
> >> infinitesimals that can be considered to lie around each real number.
> >
> >> In
> >> similar fashion, we can view numbers like the suprareals, which are
> >> separated by an uncountable number of units from each other, to have
> >> countable neighborhoods of unit intervals surrounding them, each exactly
> >> the same as the standard real line. That's what I mean by a "halo". It's
> >> the same thing on a different scale. Questions?
> >>
> >
> > (i) What does it mean to say "suprareals x and y are separated by an
> > uncountable number of units"?
>
> It means you cannot count from one to the other, assuming there is a
> difference in terms at a level above eta_0.
What does it mean to say that you can "count from" one suprareal "to"
the other? Can I count from "eta_1" to "eta_1 + 1/sqrt(2)"? How would I
determine whether or not this is possible?
>
> >
> > (ii) What does it mean for a suprareal y to "be considered to lie
> > around" the suprareal x = 1 + eta_1^2?
>
> To be on one side or the other,
well, all suprareals are on "one side or the other" of x; either y < x
or y > x.
> ... between it and any other suprareal which
> differs in a term higher than eta_0.
>
How about this:
"A suprareal y is considered to lie around the suprareal x if, and only
if, y - x in R".
In other words, iff y in Halo(x).
> >
> > (iii) From these definitions, why does it follow that there is only a
> > countable number of such infinitesimal y's? Why isn't there an
> > uncountable number of such infinitesimal y's?
>
> That would produce a measurable real interval.
>
Huh?
Neve rmind. Halo(x) is an uncountable set in NSA. (Both the set of
elements in the ultrafilter and it's complement are uncountable).
> >
> > (iv) What does it mean for a neighborhood in the suprareals to be "a
> > countable neighborhood of a unit interval"?
>
> That all elements in the neighborhood are within a finite number of
> finite units from each other.
If by a "finite unit", you mean "1", then (in the suprareals) certainly
every element y of Halo(x) satisfies that x - y is a real number. Every
real number is less than some finite natural. Therefore, every element
of Halo(x) is "within a finite number of finite units" of x.
I'm still baffled as to why you insist on calling Halo(x) "countable",
when it is clearly uncountable.
>
> >
> > (v) How does a collection of "countable neighborhoods of unit
> > intervals" then "surround" a suprareal x? When do they /not/ "surround"
> > some other suprareal, y?
>
> Because, being finitely distant from x, they are all infinitely distant
> from y.
... assuming you mean "x - y not in R", sure.
>
> >
> > (v) What does it mean for a countable neighborhood of a unit interval
> > in the suprareals to be "exactly the same as" the real line?
>
> You misread. The countable neighborhood of unit intervals which
> surrounds each suprareal is merely a copy of the real line, translated
> so that the origin is that suprareal.
>
Just remember that when you say "is a copy of the real line", not every
property of the reals is automatically "copied" to Halo(x).
For example, if u and v are members of Halo(x), then it does /not/
follow that u + v is also a member of Halo(x). But if u and v are real
numbers, u + v are also real numbers; so we need to be a bit careful
about what we say we are "copying".
> >
> > (vi) When you say "that's what (you) mean by a halo", what is "that"?
> > Is "that" a "way of viewing" the suprareals? Is "that" some particular
> > set of suprareals? Is "that" a property of certain sets of suprareals?
>
> I was explaining what I meant, when I extended the nonstandard notion of
> the halo to larger scales, such that there is a finite halo around 0,
> and eta_x halos around those.
>
And with a bit of teeth pulling, we now have a nice simple definition
(at least for H_1): Halo(h).
* The real halo of a suprareal h is defined as the set Halo(x) = {y : y
= x + r, r in R}.
>From this definition, you could have easily answered all my
otherquestions:
* By "a fractional difference of a suprareal h", you mean any member of
Halo(h).
* By "a fractional component of a suprareal h", you mean any member of
Halo(h).
* By "the/a countable neighborhood of unit intervals which surrounds
the suprareal h", you mean the set Halo(h).
* By "a suprareal y which is considered to lie around the suprareal x",
you mean that y is in Halo(x)
* By "the suprareal y is finitely distant from the suprareal x" you
mean that y is in Halo(x).
* By "all elements in the neighborhood of suprareal h are within a
finite number of finite units from each other", you mean that for all y
in Halo(h), h - y = r in R is finite.
I'm going to guess that:
* By "it is possible to count from suprareal x to suprareal y", you
mean y is in Halo(x).
And so on.
<snip>
> > In general, "numbers" is usually shorthand for "the elements of some
> > particular set X, with operations + and * defined so that (X, +, *)
> > forms a commutative monoid with multiplication"; but even that is
> > probably best considered just a guideline.
> >
>
> So, number in itself means almost nothing....
>
Pretty much, yes; at least from a mathematical viewpoint. It provides
us with no really useful information about the thing we are calling a
"number".
It's a bit like asking a musician "what is Jazz"? You'll get about as
many definitions as there are musicians; despite the fact that almost
all musicians would consider Motzart "not Jazz" and Duke Ellington
"Jazz".
Similarly, almost all mathematicians would call the naturals "numbers".
And they would call a continuous differentiable function R->R "not a
number".
Other than that, calling something "a number" is mostly a matter of
taste and interests.
> > Primeness and compositeness are not properties of the /symbol/ "i".
> > They are properties of the "number" i /in a particular context/. Change
> > the context, and the meaning of these properties may change, and even
> > become meaningless /in that context/.
> >
> > Cheers - Chas
> >
>
> And how does that apply to eta_1, particularly?
We've got about four different definitions of the suprareals right now;
and it depends on exactly which definition we are applying.
For example, if all we're considering is the integer suprareals we
defined above, then eta_1 is "prime"; which is to say that there are no
integer suprareals x and y such that neither is equal to 1 or -1; and
yet x*y = eta_1. eta_1 has no "factors" other than itself and 1 (and
-1, and -eta).
This is as opposed to the integer suprareal "1 - (eta_1)^2"; because it
is the product of (1 - eta_1)*(1 + eta_1). So "1 - eta_1^2" is
"composite": it is the product of the two "primes" (1 + eta_1) and (1-
eta_1).
On the other hand, when we introduce infinitesimals (e.g., the inverse
of eta_1), there is then a suprareal x such that x*eta_1 = 1. In that
case, we call eta_1 a "unit" (because it has an inverse); and units are
neither prime nor composite.
It doesn't make much difference to say that
1 - eta_1^2 = (1 - eta_1)*(1 + eta_1)
is "really" a distinct set of factors from
1 - eta_1^2 = (1 - eta_1)*(1 + eta_1)
= (1 - eta_1)*(1 + eta_1)*(eta_1 * 1/eta_1)
= ((1 - eta_1)*eta_1) *((1 + eta_1)* 1/eta_1)
= (eta_1 - eta_1^2) * ((1 + eta_1) /eta_1)
In some sense (eta_1 - eta_1^2) could be called a "factor" of (1 -
eta_1^2) in this context; but we usually define primes and composite as
being understood as equivalence classes "up to" multiplication by the
"units".
In the "full glory" of suprareals, /every/ suprareal has an inverse,
and so every suprareal is a "unit", and there are no "primes" nor
"composites".
Cheers - Chas
cbr...@cbrownsystems.com
> cbr...@cbrownsystems.com schrieb:
>
> > hagman wrote:
As a preface, I /think/ this describes our discussion:
We want to show that R(X) in not complete under the order topology.
To show this, we have an ordered field monomorphism T : R(X) -> R[[X]]
allowing us to embed R(X) in R[[X]]; and where the order topology on
R(X) is the subspace topology of R[[X]].
We prove that given any element F(X) in R[[X]], there is a sequence of
elements of R(X) (F0(X), F1(X), ..., Fn(X), ...) which is Cauchy in
R(X), and such that {T(Fi(X))} converges to F(X) in R[[X]] (this being
the "tricky" part of the proof).
Therefore, if there is some F(X) in R[[X]] which is not in the image of
T, then we can construct a sequence {Fi} which is Cauchy in R(X), but
does not converge to an element of R(X).
Is that about right?
Yup. That's a typo, not a "think-o" :).
> >
> > This is different than requiring that the limit k->oo {a_(k, j)} = a_j;
>
> ... at least if one takes the standard topology on R.
> If you take discrete topology on R, both notions coincide.
> In other words, for power series one uses pointwise convergence of
> coefficients with respect
> to discrete topology on the base ring.
>
Yes, that is a helpful way of describing the state of affairs.
> > and I presume this is equivalent to your statement below "the fact that
> > powers of X are small"?
>
> Exactly.
> For all n there is a j_n such that for all k>j_n, a_(k,n) = a_n
> <=>
> For all n there is a m (namely m=max{j_0, j_1, ..., j_n}) such that
> for all k>m, Fk(X)-F(X) is a multiple of X^n.
... i.e., a multiple of X^n in R[[X]] (isomorphically, to a multiple of
X^n in R(X)), because we are working in S. And all the coeffcients (in
R[[X]]) up to n) have been "fixed", so the coefficient of their
difference must be 0.
I'm with you.
>
> Usually one would want the latter to read
> For all eps>0 there is a m such that for all k>m
> |Fk(X)-F(X)| < eps.
>
> Therefore one may say "something is mindbogglingly small"
> instead of "something is divisible by a mindbogglingly high power of
> X".
>
LOL.
This brings to mind:
http://dailyvideo.googlepages.com/hubble-deep-field
> Or do you use a different notion of convergence
> > than Stanley defines?
>
> Fortunately not.
> When I started reading your post I got shocked that I might have
> erroneously ignored the topology of R in R[[X]].
> Well, partly I have as we do use the topology of R in R(X).
> Interestingly, this poses no problem - see below.
>
After reading through the article, I wonder whether we are actually
using the /topology/ of R, or we are using the /ordering/ of R to
/define/ a topology on R(X).
> >
> > > Let S be the subset of R(X) consisting of all f(X)/(1+X*g(X)).
> > > Since 1+X*g(X) is invertible in R[[X]], ...
> >
> > ... because an element of R[[X]] is invertible iff its "constant
> > coefficient" is non-zero...
>
> exactly. And of course we can hide any non-zero constant in f if it
> should be
> different from 1.
>
I agree that R is a field :).
> >
> > > ...we can view S as (part of) the
> > > intersection of R(X) and R[[X]]...
> >
> > ... because surely (1 + X*g(X) designates a particular element in R(X).
> >
> > > ("Fortunately", the topology on R[[X]]
> > > is also defined by the fact that powers of X are small).
> >
> > See above. In any case, I assume you imply that the open sets of S =
> > R[[X]] intersect R(X) are the open sets of R(X) intersect S, which are
> > the same as the open sets of R[[X]] intersect S; and so if {Fi(X)}
> > converges to F in R[[X]], it also converges to the equivalent member of
> > R(X) in R(X).
>
> Well, at least I thought so.
> But since R(X) respects the standard topology of R and R[[X]]
> implicitly uses discrete topology on R, a sequence converging in R(X)
> need not necessarily converge in R[[X]], or...
... Bum - Bum - buhhhhhh....
> ... does it?
(Rising Evil Viola Crescendo!)
> Well, a zero sequence in R(X) will have to have almost all members
> smaller than eps for any given positive eps.
Where eps is taken here, first and foremost, to be some element of
R(X). So our topology is the ordering topology (open sets are unions of
open intervals), with the ordering of elements of R(X) is being
performed in the "obvious" way hinted at in this thread.
> But every eps>0 can be written as
> r*X^m*(1+X*f(X))/(1+X*g(X)) with m in Z, r>0, f(X), g(X) polynomials
> (check that!).
Check!
> But being <eps in absolute value
> - implies being divisible by X^m and
> - is implied by being divisible by X^(m+1)
... assuming that we are talking about an element of R(X) < eps, this
follows; and of course, if there exists such an element in R(X), there
is then a corresponding element in R[[X]].
> Thus respecting or ignoring the standard topology of R made no
> difference.
> Lucky me.
>
> Note that this implies that 1/n does not converge to 0.
> Indeed, 1/n will never be less than X.
> But that does not show incompleteness, as 1/n is no Cauchy sequence...
>
Right. This is part of the massive "disconnectedness" of R(X) (which is
not the same as incompleteness).
I note that if we use the ordering topology on R(X), then the
/subspace/ topology on R yields the discrete topology (proven elsewhere
in this thread). And this may eliminate any "luck" that you had in your
exposition :).
> >
> > > One can show that the series (a_n) with a_n = sum_{k=0,...,n} X^n/n!
> > > is a Cauchy sequence.
>
> ... because differences between "late" elements of the sequence are
> divisible
> by high powers of X.
>
I see now what you meant by : Divisibility by high powers of X in R(X)
is equivalent to the "fixed-ness" of a "high" number of of the initial
coefficients in R[[X]].
> > > Assume it converges against an element f(X)/g(X) of R(X) where f(X) and
> > > g(X) have no irreducible factors in common.
> > > Obviously, the limit has to be finite (namely, very close to a_0 = 1
> > > but obviuously different from a_0).
> > > Hence we can assume that g(X) = 1 + X*h(X).
> > > But then f(X)/g(X) is in S and we must obtain the same limit in R[[X]].
> > > But there we have exp(X) as limit.
> > > Hence f(X) = g(X)*exp(X) as formal power series.
> >
> > With you so far...
> >
> > > The equality must still hold when we insert a complex number for X as
> > > long as we have convergence.
> >
> > I am definitely missing a theorem in your toolbox. Can you amplify on
> > this assertion?
>
> Maybe a less invasive tool ...
... Ouch!!! ...
> ... would have sufficed if it had come to my
> mind in time:.
> The (or at least almost all) coefficients of the formal power series of
> f(X)/g(X) fulfill a linear recursion (in a similar fashion as the
> decimal expansion of fractions is eventually periodic).
This already makes /way/ more sense.
> This is clear from writing down the condition that the coefficient of
> X^n in g(X) * (f(X)/g(X)) must be 0 if n exceeds deg(f).
> Thus automatically f(X)/g(X) cannot equal exp(X), where the
> coefficients do /not/ follow a linear recursion (needs to be checked).
> This much simpler trick works with any formal power series instead of
> exp(X),
> as long as there is no linear recursion for the coefficients.
> It works even for those series that do not converge when evaluated at
> any nonzero number.
> Now that I've thought of it I'm much happier with this proof method
> than my previous one...
>
Yes, this is /much/ clearer as a proof that R(X) < R[[X]]. Phrased in
this way, we needn't actually provide some /particular/ element of
R[[X]] such exp(X); because it is pretty obvious from its definition
that there exist elements of R[[X]] whose coefficients do not follow a
linear recursion (just as it is clear there exist elements of R whose
decimal expansions are non-repeating).
Cheers - Chas
hagman
cbr...@cbrownsystems.com schrieb:
> hagman wrote:
> > cbr...@cbrownsystems.com schrieb:
> >
> > > hagman wrote:
>
> As a preface, I /think/ this describes our discussion:
>
> We want to show that R(X) in not complete under the order topology.
>
> To show this, we have an ordered field monomorphism T : R(X) -> R[[X]]
> allowing us to embed R(X) in R[[X]]; and where the order topology on
> R(X) is the subspace topology of R[[X]].
No, the homomorphism is only "partial" as T(1/X) cannot be defined
(R[[X]] is not a field).
Now that you mention it, is R[[X]] the ring of all
a_0 + a_1*X + a_2*X^2 + ...
or of field of all all
a_(-k)*X^(-k) + ... ?
I thought it was the ring, but the latter forces me to withdraw my "no"
and see that the prove can be simplified accordingly. :)
>
> We prove that given any element F(X) in R[[X]], there is a sequence of
> elements of R(X) (F0(X), F1(X), ..., Fn(X), ...) which is Cauchy in
> R(X), and such that {T(Fi(X))} converges to F(X) in R[[X]] (this being
> the "tricky" part of the proof).
>
> Therefore, if there is some F(X) in R[[X]] which is not in the image of
> T, then we can construct a sequence {Fi} which is Cauchy in R(X), but
> does not converge to an element of R(X).
>
> Is that about right?
Sounds familiar.
Yes, exp(X) would have been really too R-centric.
Thank you for following through it all. :)
> Cheers - Chas
--hagman
David R Tribble
>> Which means that it's not proper to think of the union of the reals
>> and the suprareals as a single "number line", but more as separate
>> disconnected number lines. More properly, they are disconnected
>> sets.
>
Tony Orlow wrote:
>> Then the same applies to the h-numbers. If they are not colinear with the
>> reals, then the reals do not constitute a "gap" within the suprareals.
>> If you apply the logic of connectedness to the suprareals as you do to
>> the reals, then they are also gapless. For any two suprareals, positive
>> or negative, there lies a suprareal between them, Therefore, they are as
>> "continuous" as the reals.
>
Chas Bbrown wrote:
>> For the third time in this thread, "dense", "connected", and
>> "continuous" have different meanings. And "gapless" has no particular
>> meaning to me at all; unless it is supposed to mean "dense".
>
Tony Orlow wrote:
> True, I guess I meant dense more than continuous, but "gapless" could
> mean either, I suppose.
>
Chas Bbrown wrote:
>> (i) The fact that the suprareals are a dense set does not mean that the
>> suprareals are connected.
>> (ii) The fact that the suprareals are a dense set does not mean that
>> there is a continuous function from the reals onto the suprareals.
>
Tony Orlow wrote:
> Okay, my bad. Still, if the ih-numbers are included, and lie between the
> reals and 0, then the reals consist of three disjoint sets,
> (-eta_1,0),0, and (0,eta_1). Since the hierarchy continues forever, each
> such H_x would be disjoint from 0.
You're still thinking of the suprareals as somehow being colinear
(on the same "line", within the same set, whatever) as the reals, and
they aren't. All of the reals are in H_0 (equivalent to R). All of
the suprareals based on eta_1 are in H_1. No member of H_1 is
a member of H_0, and vice versa. Any given suprareal h, which
is a polynomial with a largest term x*eta_j, is a member of H_j,
and not a member of any other suprareal (or real) set.
All of the suprareal sets H_i are disconnected, meaning that
no two sets share any elements. That's what we mean by
"unconnected sets".
However, you'll notice that my definition of H_0 (which is R)
includes 0 and 1. So yes, each H_i (other than H_0) is disjoint
from 0, i.e., does not include 0 as a member.
Since this is a hierarchy of unconnected/disjoint sets, it makes
no sense to talk about the interval (0, eta_1), because 0 is a
member of H_0 but eta_1 is a member of H_1, two disjoint
sets. You could say that they line on separate (non-touching)
number lines.
> 0 is really eta_-oo.
No, 0 is just plain 0, the additive identity for the suprareals and a
member of H_0.
> So, in order to
> maintain connectedness on each level, it seems to me that each level of
> polynomial eta_x on eta_1 needs to regress infinitely, having the form
> union(n=-eta_1->x: H_n). Then each level is included in the last, and no
> gap occurs.
There is no way to connect the disjoint H_i sets without breaking
the order relation. "Gap" is a misleading term (mea culpa) -
perhaps I should have made it more clear how the sets are
"not connected" instead.
> Of course, you mentioned there was a problem with infinite
> polynomials with regard to 0 not being allowed for the 0th term or there
> can be no multiplicative inverse, or something? Perhaps there's a way
> around that by including eta_-oo=0?
One solution is to allow suprareal power series. The problem is
then how to define a meaning for them:
h = sum{i = 0 to oo} x_i eta_1^i
Assuming the coefficients x_i "converge" in some sense, what does
the sum h converge to, exactly? I'm not saying it can't be done,
but I need to learn more about power series and R[X] to do so.
For the time being, the suprareals are still only defined as
finite polynomials over eta_i. (By "finite", I mean having a finite
number of terms).
David R Tribble wrote:
>> What is curious (and which I will probably add to the next revision)
>> is the fact that
>> x0 + eta_1
>> x1 eta_1^1
>> x2 eta_1^2
>> ...
>> are unconnected (uncountable) sets. Any member of one set in the
>> list is less than any member of the next set (assuming all positive
>> real x's).
>
Tony Orlow wrote:
>> That was the suggestion I offered: define you H_x as the countable
>> neighborhood of eta_1^x, whether x is natural or real.
>
Chas Bbrown wrote:
>> (i) What is a "countable neighborhood"? A neighborhood is usually
>> equivalent to "an open set", which in this topology means "an open
>> interval in the suprareals". But what is a "countable neighborhood"?
>> And how do you define an open interval prior to defining a total order
>> on the elements you aim to define?
>
Tony Orlow wrote:
> A countable neighborhood is one where each point is finitely distant
> from each other point, given some unit of measure. The nonstandard halo
> is a countable neighborhood of infinitesimals in the vicinity of each
> real, between it and any other standard real. Similarly, the countable
> neighborhood of eta_x+y, where yeR, surrounds eta_x, and is between it
> and any other distinct eta_n. Indeed, this could be considered to be an
> open set, I suppose, as (eta_n - eta_n-1, eta_n + eta_n-1). It's also an
> external set, in essence. I'm not sure I consider it quite the same as
> an open interval. I'll have to think about that.
Since I have not defined a metric for the suprareals (I've only defined
an order relation), it's not fitting to talk about the "distance"
between two given suprareals. At best we can only really talk
about the difference between suprareals. For example, the
difference between
x1 + y1 eta_1 + z1 eta_1^2 and
x2 + y2 eta_1 + z2 eta_1^2
is (x2-x1) + (y2-y1)eta_1 + (z2-z1)eta_1^2, for a simple case.
Any notion of "distance" for the suprareals will only make sense for
members of the same (connected) H_i set. Otherwise it's like
asking what the distance is between the members of {1, 2, 3}
and {red, blue, green}.
David R Tribble
>> (ii) What is the difference between a "disconnected" set and an
>> "uncountably disconnected set"?
>
Tony Orlow wrote:
> In this context, not much. I wouldn't say the set of standard nonzero
> reals was "uncountably" disconnected, in the context of the standard
> reals. Even in terms of infinitesimals it would be only countably
> disconnected. But, all reals not in [0,1], for instance, would be
> uncountably disconnected, at that infinitesimal level, even if only
> finitely disconnected at the finite level.
>
> When I say "uncountably disconnected", I mean that the difference
> between eta_1^x and eta_1^y is greater than any finite number, if y>x
> and eta_1 is greater than any finite number.
It does not make much sense to use the term "finite" when
talking about the suprareals. All suprareals are, in effect, "finite"
numbers. Some just happen to be larger than others, e.g., any
suprareal h in H_1 is larger than any real in H_0 (which is R).
In almost all other respects, the suprareals act like regular
"finite" reals; they just have a special order topology.
Being larger than all reals (specifically, as defined by the
"<" and ">" order relation) is not enough to make a number
"infinite". In fact I make it clear than h < oo for all h in H_i.
Chas Brown wrote:
>> (iii) What is an "uncountably "disconnected" pair of sets"? What two
>> sets of suprareals are you referring to?
>
Tony Orlow wrote:
> Those within a countable distance of eta_1^x and those within a
> countable distance of eta_1^y.
How do you count the distance (points?) between two suprareals?
Chas Brown wrote:
>> Given that definition, can you work out whether
>> 1 + (eta_2)^100*(eta_3)^2 > 3 - 2*eta_e + 17*eta_pi
>> is true or false?
>
Tony Orlow wrote:
> I don't think so, without knowing what raising an eta to a given power
> does to its number. If the number is the power of eta_1, then we can say
> 1+(eta_2)^100*eta_3^2=1+eta_200*eta_6=1+eta_206.
No, eta_2^100 is already expressed in its lowest form, and eta_200
is a different and much larger number. Likewise, eta_1*eta_2 is
in its lowest form and is less than eta_3. So
1 + eta_2^100*eta_3^2
cannot be reduced to simpler terms.
Tony Orlow wrote:
> 3-3*eta_e + 17*eta_pi is in its lowest form, with a greatest eta of
> eta_pi, far less than eta_206, so the statement would be true. That's
> the only way I see the continuous hierarchy working properly.
eta_pi is less than eta_206 because pi < 206.
The continuous hierarchy of suprareals is a logical extension
of the countable hierarchy of sets H_i for all i in Z. I simply
stated that there exist sets H_x for all x in R, and they obey the
same order topology.
Thus 1 + 17 eta_e + 3/2 eta_pi^2 is expressed in its lowest form,
and is less than 1/2 eta_4.
cbr...@cbrownsystems.com
> cbr...@cbrownsystems.com schrieb:
>
> > hagman wrote:
> > > cbr...@cbrownsystems.com schrieb:
> > >
> > > > hagman wrote:
> >
> > As a preface, I /think/ this describes our discussion:
> >
> > We want to show that R(X) in not complete under the order topology.
> >
> > To show this, we have an ordered field monomorphism T : R(X) -> R[[X]]
> > allowing us to embed R(X) in R[[X]]; and where the order topology on
> > R(X) is the subspace topology of R[[X]].
>
> No, the homomorphism is only "partial" as T(1/X) cannot be defined
> (R[[X]] is not a field).
Yes, you are correct. Obviously, all these different spaces are
beginning to confuse me!
> Now that you mention it, is R[[X]] the ring of all
> a_0 + a_1*X + a_2*X^2 + ...
> or of field of all all
> a_(-k)*X^(-k) + ... ?
Well, we need a total order which is (somehow) compatible with the
order topology of R(X). But the "obvious" ordering in "positive" powers
of X doesn't even seem to be a total order.
Consider the formal power series {-1^n}. If X^(n+1) > X^n, should it be
greater than, or less than, 0? (Unless it's greater than 0 on the
indices of an element of an ultrafilter... whoops! Wrong number
system!)
> I thought it was the ring, but the latter forces me to withdraw my "no"
> and see that the prove can be simplified accordingly. :)
I think the proof would clearly carry through if we thought of R'[[X]]
as the "semi-formal" power series being defined by elements of the
form:
sum (-oo < n <= d) a_n*x^n
where n in Z, a_n in R; and d in Z is the "degree" of the "polynomial".
R'[[X]] is a field with an easy total ordering, which also contains
R(X) with /its/ desired ordering. The relationship between R'[[X]] and
R(X) is parallel to the relationship between R and Q.
And this allows us to use the "pointwise" convergence of R'[[X]] to
show that there are Cauchy sequences in R(X) which don't converge in
R(X) by your previous logic.
Which proves that R(X) is /not/ complete. Which I /think/ was the
original question we were considering; but I was learning so much I
lost track :).
<snip>
>
> Yes, exp(X) would have been really too R-centric.
> Thank you for following through it all. :)
>
Thank you for first understanding the question, and second going to the
effort of explaining the answer. It really improves my understanding.
Cheers - Chas
Tony Orlow
If that is the case, then there is no gap in H_1, since the reals that
consitute that gap are not in the set. Between any two members of H_1
lies another, no?
>
> All of the suprareal sets H_i are disconnected, meaning that
> no two sets share any elements. That's what we mean by
> "unconnected sets".
Only if you consider all H_x for x<i to be part of any H_i. But, you say
H_0 is not part of H_1, and so this is not the case. None of your H_1
have a gap if they are not colinear with some H_x "inside" of it which
is not included.
>
> However, you'll notice that my definition of H_0 (which is R)
> includes 0 and 1. So yes, each H_i (other than H_0) is disjoint
> from 0, i.e., does not include 0 as a member.
And what about your sets of inverses of eta_1? Does that include 0? It
would seem to me that 0 would be equal to H_-oo, so it's either included
in all higher H_i, or in none.
>
> Since this is a hierarchy of unconnected/disjoint sets, it makes
> no sense to talk about the interval (0, eta_1), because 0 is a
> member of H_0 but eta_1 is a member of H_1, two disjoint
> sets. You could say that they line on separate (non-touching)
> number lines.
I would say they reside on uncountable distant countable intervals of
the real line.
>
>
>
>> 0 is really eta_-oo.
>
> No, 0 is just plain 0, the additive identity for the suprareals and a
> member of H_0.
Did you even think for a second about what eta_-oo would be?
>
>
>> So, in order to
>> maintain connectedness on each level, it seems to me that each level of
>> polynomial eta_x on eta_1 needs to regress infinitely, having the form
>> union(n=-eta_1->x: H_n). Then each level is included in the last, and no
>> gap occurs.
>
> There is no way to connect the disjoint H_i sets without breaking
> the order relation. "Gap" is a misleading term (mea culpa) -
> perhaps I should have made it more clear how the sets are
> "not connected" instead.
How does it break the order relation? A reR eta_1>r and -eta_1<r. The
only problem is identifying the boundary between R and H_1/R, or rather,
in coming to grips with the fact there can be no such boundary.
>
>
>> Of course, you mentioned there was a problem with infinite
>> polynomials with regard to 0 not being allowed for the 0th term or there
>> can be no multiplicative inverse, or something? Perhaps there's a way
>> around that by including eta_-oo=0?
>
> One solution is to allow suprareal power series. The problem is
> then how to define a meaning for them:
> h = sum{i = 0 to oo} x_i eta_1^i
> Assuming the coefficients x_i "converge" in some sense, what does
> the sum h converge to, exactly? I'm not saying it can't be done,
> but I need to learn more about power series and R[X] to do so.
Yeah, I don't know what it would mean for the coefficients to converge.
The highest power of eta_1 would be the most significant by "far".
Basically, you're describing a digital number system. You could include
negative powers of eta_1 to get your inverse values. It would be an
eta_1-base digital system, a subset of the T-riffics. :)
>
> For the time being, the suprareals are still only defined as
> finite polynomials over eta_i. (By "finite", I mean having a finite
> number of terms).
Okay. Then they're not closed under division?
I see that as an expression of the distance, but you call it difference
if you like.
>
> Any notion of "distance" for the suprareals will only make sense for
> members of the same (connected) H_i set. Otherwise it's like
> asking what the distance is between the members of {1, 2, 3}
> and {red, blue, green}.
>
If you insist, but I don't understand why you do. What, again, does it
break for these numbers to all be colinear?
Tony Orlow
> Chas Brown wrote:
>>> (ii) What is the difference between a "disconnected" set and an
>>> "uncountably disconnected set"?
>
> Tony Orlow wrote:
>> In this context, not much. I wouldn't say the set of standard nonzero
>> reals was "uncountably" disconnected, in the context of the standard
>> reals. Even in terms of infinitesimals it would be only countably
>> disconnected. But, all reals not in [0,1], for instance, would be
>> uncountably disconnected, at that infinitesimal level, even if only
>> finitely disconnected at the finite level.
>>
>> When I say "uncountably disconnected", I mean that the difference
>> between eta_1^x and eta_1^y is greater than any finite number, if y>x
>> and eta_1 is greater than any finite number.
>
> It does not make much sense to use the term "finite" when
> talking about the suprareals. All suprareals are, in effect, "finite"
> numbers. Some just happen to be larger than others, e.g., any
> suprareal h in H_1 is larger than any real in H_0 (which is R).
> In almost all other respects, the suprareals act like regular
> "finite" reals; they just have a special order topology.
If "finite numbers" means R, which it usually does when talking about
raw quantity, then you just said any element of H_1 is larger than any
finite. Where y>x, n^y-n^x diverges as n->oo. So, for all n in H_1,
n^y-n^x would be larger than any r in R.
>
> Being larger than all reals (specifically, as defined by the
> "<" and ">" order relation) is not enough to make a number
> "infinite". In fact I make it clear than h < oo for all h in H_i.
There is a difference between a specific infinite number, larger than
any finite, and absolute oo, larger than any number.
>
>
> Chas Brown wrote:
>>> (iii) What is an "uncountably "disconnected" pair of sets"? What two
>>> sets of suprareals are you referring to?
>
> Tony Orlow wrote:
>> Those within a countable distance of eta_1^x and those within a
>> countable distance of eta_1^y.
>
> How do you count the distance (points?) between two suprareals?
>
>
Using an expression for the difference as you showed above. You cannot
count uncountable quantities in the sequential sense. You deal with
them, as you are trying, formulaically.
> Chas Brown wrote:
>>> Given that definition, can you work out whether
>>> 1 + (eta_2)^100*(eta_3)^2 > 3 - 2*eta_e + 17*eta_pi
>>> is true or false?
>
> Tony Orlow wrote:
>> I don't think so, without knowing what raising an eta to a given power
>> does to its number. If the number is the power of eta_1, then we can say
>> 1+(eta_2)^100*eta_3^2=1+eta_200*eta_6=1+eta_206.
>
> No, eta_2^100 is already expressed in its lowest form, and eta_200
> is a different and much larger number. Likewise, eta_1*eta_2 is
> in its lowest form and is less than eta_3. So
> 1 + eta_2^100*eta_3^2
> cannot be reduced to simpler terms.
>
>
No, not without an interpretation of the index on eta. You don't seem to
have one. So, I wonder how you can justify the statement that
eta_200>eta_2^100? If you took my suggestion of making the eta_1 on any
level equal to eta_1^i, then they'd be equal. If you don't like that
idea, what's your alternative?
> Tony Orlow wrote:
>> 3-3*eta_e + 17*eta_pi is in its lowest form, with a greatest eta of
>> eta_pi, far less than eta_206, so the statement would be true. That's
>> the only way I see the continuous hierarchy working properly.
>
> eta_pi is less than eta_206 because pi < 206.
Right.
>
> The continuous hierarchy of suprareals is a logical extension
> of the countable hierarchy of sets H_i for all i in Z. I simply
> stated that there exist sets H_x for all x in R, and they obey the
> same order topology.
Yes, but that loses the relationship between the etas, which started out
as sequential. So that relationship needs to be established otherwise.
>
> Thus 1 + 17 eta_e + 3/2 eta_pi^2 is expressed in its lowest form,
> and is less than 1/2 eta_4.
>
Understood.
cbr...@cbrownsystems.com
> David R Tribble wrote:
> > You're still thinking of the suprareals as somehow being colinear
> > (on the same "line", within the same set, whatever) as the reals, and
> > they aren't. All of the reals are in H_0 (equivalent to R). All of
> > the suprareals based on eta_1 are in H_1. No member of H_1 is
> > a member of H_0, and vice versa. Any given suprareal h, which
> > is a polynomial with a largest term x*eta_j, is a member of H_j,
> > and not a member of any other suprareal (or real) set.
>
> If that is the case, then there is no gap in H_1, since the reals that
> consitute that gap are not in the set. Between any two members of H_1
> lies another, no?
>
Yes. Yes, H_1 is dense, yes, yes, yes. Dense is /not the same/ as
connected. Dense is /not the same/ as connected. Dense is /not the
same/ as connected. Dense...
Consider the set A = [0,1) union (1,2] (i.e., the interval [0,2] with
the point "1" removed).
Given any distinct a, b in A, do you agree that there is a c in A which
is between a and b?
But is A a connected line? It sure looks like two /disconnected/ pieces
to me!
Now remove from A the points 1/4, 1/2, 3/4, 5/4, 3/2, and 7/4.
Again, for any a,b in A, there is a c in A which is between a and b;
but A now has 8 disconnected "pieces".
Now remove /all/ points of the form p/2^n, p and n naturals, from A.
You will /still/ find that for any a, b in A, there is a c in A which
is between a and b; but now instead of A being in two "pieces", or
eight "pieces", A has an /uncountably infinite/ number of disconnected
"pieces"; and no individual "piece" will have any real length
associated with it.
That is hardly what I would call a "line"; it's more like sawdust. And
that's much more what {H_1} is "like" than a line (and even that
example does not capture the ordering of {H_1}).
> > All of the suprareal sets H_i are disconnected, meaning that
> > no two sets share any elements. That's what we mean by
> > "unconnected sets".
>
> Only if you consider all H_x for x<i to be part of any H_i. But, you say
> H_0 is not part of H_1, and so this is not the case. None of your H_1
> have a gap if they are not colinear with some H_x "inside" of it which
> is not included.
>
I think you are having a terminology breakdown here; and I propose a
new naming.
Given a non-negaive real number x, we want to talk about two different
things:
* H_x is the set of all suprareals which are polynomials over some
(finite) set of eta_'s, where each eta_ is less than or equal to eta_x;
and where the coefficient of /some/ power of eta_x is non-zero.
* {H_x} is the set of all suprareals which are polynomials over a
(finite) set of eta_'s, where each eta_ is less than or equal to eta_x;
so {H_x} is the union of all H_y where y <= x.
H_x is a subset of {H_x}; and in fact for any 0 <= y <= x, both H_y and
{H_y} are subsets of {H_x}.
But given x <> y, H_x intersect H_y is empty.
H_x is closed under addition and multiplication; but not under
subtraction: (eta_x + 1) - eta_x = 1, which is not in H_x.
{H_x} is closed under *, +, and -; and is in fact a ring (but not a
field).
> > Since this is a hierarchy of unconnected/disjoint sets, it makes
> > no sense to talk about the interval (0, eta_1), because 0 is a
> > member of H_0 but eta_1 is a member of H_1, two disjoint
> > sets. You could say that they line on separate (non-touching)
> > number lines.
>
Let us restrict our selves to just polynomials in eta_0 = 1 and eta_1.
It certainly /does/ make perfect sense to talk about the interval (0,
eta_1): it is simply the set of all h in {H_1} such that 0 < h < eta_1.
1 is in this set, as is eta_1 - 1, eta_1 - 2, (eta_1)/2, etc.; and -5,
eta_1, 2*eta_1 - 1, -(eta_1)^2 etc. are not in this set.
But it also makes sense to talk about the two sets H_0 and H_1. H_0
intersect H_1 is empty; and {H_1} = H_0 union H_1.
So {H_1} is the disjoint union of two sets.
So what, one may ask? The reals numbers are /also/ the disjoint union
of two sets: the set of reals < 40, and the set of reals >= 40.
The difference is that the set of reals < 40 is an open set; but the
set of reals >= 40 is a closed set. It's not possible (under the usual
topology) to make R be the disjoint union of /two/ open sets. If one
set is open, then the other is /always/ not open.
We can think of 40 as being "the boundary" between these two sets:
there is no number less than 40 which is in the set of all reals >= 40
(obviously!).
But /both/ H_0 and H_1 are /open/ sets. So there is no h in {H_1} which
is "the boundary" between H_0 and H_1; in the way that "40" was the
boundary: H_1 has no "least element" which can act as this boundary,
and H_0 has no "greatest element" which can act as this boundary.
But it's far worse than that (as David alludes to below)!
Given any real r, let S(r) be the set of all polynomials of the form s
+ r*eta_1 (where s in R). TO might call this the "real halo of r*eta_1,
Halo(r*eta_1)".
Then both S(r) and {H_1}\S(r) are again open sets, and so for example
all suprareals of the form s + 1.5*eta_1 form a component (disconnected
subset) of {H_1}. There is no "smallest" or "largest" element of S(1.5)
or {H_1}\S(1.5) which can act as a "boundary".
And just as there is no "next" real after 1.5, there is also no set
S(x) which is the "next" disconnected component of {H_1} "after"
S(1.5).
> I would say they reside on uncountable distant countable intervals of
> the real line.
>
I will respond as usual:
I know what the real line is.
I know what an interval of the real line is.
I know what countable means.
I know what uncountable means (it's an adjective BTW).
Does "countable intervals of the real line" mean a countable /number/
of intervals of the real line? Or does "countable intervals of the real
line" have some special personal meaning to you, which is not actually
related to the words "countable" and "interval of the real line"?
Does "uncountable distant (somethings)" refer to a set consisting of an
uncountable /number/ of "distant" (somethings) (which I /might/
understand if you explained it)? Or do you have some special personal
meaning to "uncountably distant (somethings)" (note use of adverb),
which is not actually related to the words "uncountable" and "distant"?
If what you're /trying/ to say is that the suprareals can be embedded
in the real line and maintain their ordering, then you're wrong, it
cannot.
The real line is not the union of an uncountable number of disjoint
intervals, while the suprareals are.
How do I know this? Well, every interval of the reals contains a
rational number which is distinct from any rational number in any other
disjoint interval of R. Thus, there cannot be an uncountable number of
disjoint intervals in R; because that would imply that the rationals
have an uncountable subset (one distinct rational for each one of the
intervals).
This, by the way, is one reason why the cardinalities "countable" and
"uncountable" are useful as a measure of set "size". There simply
aren't "enough" intervals in R to support the intervals in {H_1}.
> >
> >
> >
> >> 0 is really eta_-oo.
> >
> > No, 0 is just plain 0, the additive identity for the suprareals and a
> > member of H_0.
>
> Did you even think for a second about what eta_-oo would be?
>
Why should he? -oo is not a real number. Why should he think for a
second about what "eta_triangleABC" might be? A triangle is not a real
number.
> >
> >
> >> So, in order to
> >> maintain connectedness on each level, it seems to me that each level of
> >> polynomial eta_x on eta_1 needs to regress infinitely, having the form
> >> union(n=-eta_1->x: H_n). Then each level is included in the last, and no
> >> gap occurs.
> >
> > There is no way to connect the disjoint H_i sets without breaking
> > the order relation. "Gap" is a misleading term (mea culpa) -
> > perhaps I should have made it more clear how the sets are
> > "not connected" instead.
>
> How does it break the order relation? A reR eta_1>r and -eta_1<r. The
> only problem is identifying the boundary between R and H_1/R, or rather,
> in coming to grips with the fact there can be no such boundary.
>
In order for there to be a "boundary" between H_0 and H_1, we need to
have a suprareal h such that x <= h for all x in H_0 = R, and h <= y
for all y in H_1.
But in this ordering, there can be no such h; because if h in H_1, then
h - 1 < h, and h - 1 is in H_1; so h cannot be in H_1. And if h in H_0
= R, then h+1 > h, and h + 1 in H_0; so h cannot be in H_0.
That's the "gap" in the connectedness of {H_1}: {H_1} lacks an element
h which "straddles" both H_0 and H_1. Compare this with the reals - you
can't say the same thing!
> > Since I have not defined a metric for the suprareals (I've only defined
> > an order relation), it's not fitting to talk about the "distance"
> > between two given suprareals. At best we can only really talk
> > about the difference between suprareals. For example, the
> > difference between
> > x1 + y1 eta_1 + z1 eta_1^2 and
> > x2 + y2 eta_1 + z2 eta_1^2
> > is (x2-x1) + (y2-y1)eta_1 + (z2-z1)eta_1^2, for a simple case.
>
> I see that as an expression of the distance, but you call it difference
> if you like.
>
> >
> > Any notion of "distance" for the suprareals will only make sense for
> > members of the same (connected) H_i set.
I'd have to disagree. If we define distance as the absolute value of
the difference |g - h| between suprareals, then it satisfies:
|g - h| = 0 iff h = g.
|g - h| = |h - g|
|g - h| + |h - k| >= |g - k|
which is all we usually require of a metric (well, we also require that
the range of || be the reals). And using this definition of "distance",
we can produce the same topology as the order topology: one that is
/based on/ open intervals.
But remember that in the order topology, open sets are the union of
/any/ number of open intervals. So even though H_0 is not an interval
(it is not a set of the form {h : x < h < y}), it is still an open set
(it is the union of all intervals (x, x+1) for x in H_0).
At the same time, there is no meaning to referring to the "distance"
from "one end" of H_0 to the "other end", because H_0 is /not/ an
interval - it has no "ends".
> Otherwise it's like
> > asking what the distance is between the members of {1, 2, 3}
> > and {red, blue, green}.
> >
>
> If you insist, but I don't understand why you do. What, again, does it
> break for these numbers to all be colinear?
As I have said many times before: because a line is connected, and the
suprareals are very much not connected, it doesn't make much sense to
call the suprareals "linear" = "like a line"; except in the simplistic
sense that both a line and the suprareals have (very different) total
orders.
Cheers - Chas
David R Tribble
>> Okay, my bad. Still, if the ih-numbers are included, and lie between the
>> reals and 0, then the reals consist of three disjoint sets,
>> (-eta_1,0),0, and (0,eta_1). Since the hierarchy continues forever, each
>> such H_x would be disjoint from 0.
>
David R Tribble wrote:
>> You're still thinking of the suprareals as somehow being colinear
>> (on the same "line", within the same set, whatever) as the reals, and
>> they aren't. All of the reals are in H_0 (equivalent to R). All of
>> the suprareals based on eta_1 are in H_1. No member of H_1 is
>> a member of H_0, and vice versa. Any given suprareal h, which
>> is a polynomial with a largest term x*eta_j, is a member of H_j,
>> and not a member of any other suprareal (or real) set.
>
Tony Orlow wrote:
> If that is the case, then there is no gap in H_1, since the reals that
> consitute that gap are not in the set. Between any two members of H_1
> lies another, no?
Except that the negative suprareals in H_1 (in H_1-) are not connected
to the positive suprareals in H_1 (in H_1+). There is no least
positive suprareal in H_1, nor is there a least negative in H_1.
Thus the negative and positive half-sets are not connected.
You can't get from say, -eta_1, to +eta_1 by successive incrementing
(or multiplying) by a positive number.
And in fact, H_1 is even more disconnected than that. The following
subsets of H_1 are not connected with each other:
x0 + eta_1
x0 + x1 eta_1
x0 + x1 eta_1^p1
x0 + x1 eta_1^p1 + x2 eta_1^p2
etc.
David R Tribble wrote:
>> All of the suprareal sets H_i are disconnected, meaning that
>> no two sets share any elements. That's what we mean by
>> "unconnected sets".
>
Tony Orlow wrote:
> Only if you consider all H_x for x<i to be part of any H_i. But, you say
> H_0 is not part of H_1, and so this is not the case. None of your H_1
> have a gap if they are not colinear with some H_x "inside" of it which
> is not included.
All the reals x are members of R, which is H_0.
All supranumbers based on eta_1, i.e., those of the form
x0 + sum{i = -n to +m} x_i eta_1^i, x_i in R
are members of H_1. Likewise, all suprarerals of the form
h0 + sum{i = -n to +m} h_i eta_1^i, h_i in H_1 U R
are members of H_2, and so forth. There is no subsetting
of H_i in H_j when i<j. All H_i are disjoint sets.
David R Tribble wrote:
>> However, you'll notice that my definition of H_0 (which is R)
>> includes 0 and 1. So yes, each H_i (other than H_0) is disjoint
>> from 0, i.e., does not include 0 as a member.
>
Tony Orlow wrote:
> And what about your sets of inverses of eta_1? Does that include 0? It
> would seem to me that 0 would be equal to H_-oo, so it's either included
> in all higher H_i, or in none.
There is no suprareal h in any H_i such that 1/h = 0.
Just like there is no real x in R where 1/x = 0.
But 0 is in R, so it's in H_0, and it is the additive identity for the
suprareals, just like it is for the reals.
There is no set H_oo or H_-oo. There is a set H_x for every x in R,
but oo and -oo are not members of R.
David R Tribble wrote:
>> Since this is a hierarchy of unconnected/disjoint sets, it makes
>> no sense to talk about the interval (0, eta_1), because 0 is a
>> member of H_0 but eta_1 is a member of H_1, two disjoint
>> sets. You could say that they line on separate (non-touching)
>> number lines.
>
Tony Orlow wrote:
> I would say they reside on uncountable distant countable intervals of
> the real line.
Then you would be wrong. The term "line" implies a connected set
of points. H_i is a collection of unconnected sets.
Tony Orlow wrote:
>> 0 is really eta_-oo.
>
David R Tribble wrote:
>> No, 0 is just plain 0, the additive identity for the suprareals and a
>> member of H_0.
>
Tony Orlow wrote:
> Did you even think for a second about what eta_-oo would be?
No, because -oo is not a member of R, so there is no H_-oo.
I think having an uncountable number of H_x sets is enough to be
interesting for the time being.
David R Tribble wrote:
>> Any notion of "distance" for the suprareals will only make sense for
>> members of the same (connected) H_i set. Otherwise it's like
>> asking what the distance is between the members of {1, 2, 3}
>> and {red, blue, green}.
>
Tony Orlow wrote:
> If you insist, but I don't understand why you do. What, again, does it
> break for these numbers to all be colinear?
They can't be "colinear" because they are unconnected sets.
There is no way to start at, say, 2-eta_1 and start counting
(incrementing, adding, whatever you choose to call it) and end
up at, say, 3+eta_1^2 without somehow jumping from eta_1 to
eta_1^2. Which can't be done under the axioms I provide.
cbr...@cbrownsystems.com
> David R Tribble wrote:
> > Chas Brown wrote:
> >>> (ii) What is the difference between a "disconnected" set and an
> >>> "uncountably disconnected set"?
> >
> > Tony Orlow wrote:
> >> In this context, not much. I wouldn't say the set of standard nonzero
> >> reals was "uncountably" disconnected, in the context of the standard
> >> reals. Even in terms of infinitesimals it would be only countably
> >> disconnected. But, all reals not in [0,1], for instance, would be
> >> uncountably disconnected, at that infinitesimal level, even if only
> >> finitely disconnected at the finite level.
> >>
> >> When I say "uncountably disconnected", I mean that the difference
> >> between eta_1^x and eta_1^y is greater than any finite number, if y>x
> >> and eta_1 is greater than any finite number.
> >
> > It does not make much sense to use the term "finite" when
> > talking about the suprareals. All suprareals are, in effect, "finite"
> > numbers. Some just happen to be larger than others, e.g., any
> > suprareal h in H_1 is larger than any real in H_0 (which is R).
> > In almost all other respects, the suprareals act like regular
> > "finite" reals; they just have a special order topology.
>
> If "finite numbers" means R, which it usually does when talking about
> raw quantity, then you just said any element of H_1 is larger than any
> finite. Where y>x, n^y-n^x diverges as n->oo.
What on earth, in this context, do you mean by x, y, n, and "diverges
as n->oo"? As it stands, your comment literally makes as much sense as
claiming that: because the triangle n, x, y approaches the origin as x
= Area(y), therefore any element of H_1 is larger than any finite.
> So, for all n in H_1,
> n^y-n^x would be larger than any r in R.
>
There's no need or sense to use (frankly, silly-sounding) undefined
expressions and poetic language such as "n^y - n^x diverges as n->oo"
to assert:
If n in H_1, and x, y in N with not x = y , it is a fact that /from the
definitions/, n^y - n^x is in H_1.
Do you want a proof?
Alternatively, do you not even /care/ what can or cannot be proven from
David's definition of {H_x}, because your hobby is just to babble
vaguely mathematical sounding phrases claiming that "the T-rrifics" do
this or that, and "the H-riffics" do this, that, and the other thing?
Assuming you want to do /more/ than that, you should observe how we go
about saying things like "In David's system, n in H_1 and x <> y in N,
implies that n^x - n^y is also in H_1"; so that you can then learn how
to say "In the T-riffics, (whatever it is you want to say)".
C'mon! Even as a troll, you have to learn how to engage your subjects;
otherwise your most interesting victims will become desensitized to
your rhetoric and cease to respond!
> >
> > Being larger than all reals (specifically, as defined by the
> > "<" and ">" order relation) is not enough to make a number
> > "infinite". In fact I make it clear than h < oo for all h in H_i.
>
> There is a difference between a specific infinite number, larger than
> any finite, and absolute oo, larger than any number.
That is one way of saying, in English (and thus, imprecisely), the
"meaning" of the mathematical statement:
(h in H_1) implies that:
For all x in R, h > x; and
h < oo
But English is /not/ mathematics; it is only an /interpretation/ of
that mathematics in English. Your response does not clarify David's
statement; instead it blurs it.
>
> >
> >
> > Chas Brown wrote:
> >>> (iii) What is an "uncountably "disconnected" pair of sets"? What two
> >>> sets of suprareals are you referring to?
> >
> > Tony Orlow wrote:
> >> Those within a countable distance of eta_1^x and those within a
> >> countable distance of eta_1^y.
> >
> > How do you count the distance (points?) between two suprareals?
> >
> >
>
> Using an expression for the difference as you showed above. You cannot
> count uncountable quantities in the sequential sense. You deal with
> them, as you are trying, formulaically.
What is "uncountable" about the "quantity" "eta_1 -1" that is not
equally "uncountable" about the "quantity" "sqrt(17)", where by
"quantity" I assume you mean "suprareal"?
I can only imagine that, in general, by "countable quantity" you mean
"any real number smaller than some integer" which is identical to
saying "any real number"; and by "uncountable quantity" you mean, in
this context only, "a suprareal larger than any real, or smaller than
any real"; i.e., "any element of H_1".
But it is difficult to understand why you insist on referring to some
"finite" suprareal h as being "countable"; and a "non-finite" suprareal
h as being "uncountable".
So far as I have been given definitions of these words: "Uncountable"
means "cannot be injected to a subset of the naturals". And "countable"
means "can be bijected with the naturals". And finite means "cannot be
injected to a proper subset of itself".
>
> > Chas Brown wrote:
> >>> Given that definition, can you work out whether
> >>> 1 + (eta_2)^100*(eta_3)^2 > 3 - 2*eta_e + 17*eta_pi
> >>> is true or false?
> >
> > Tony Orlow wrote:
> >> I don't think so, without knowing what raising an eta to a given power
> >> does to its number. If the number is the power of eta_1, then we can say
> >> 1+(eta_2)^100*eta_3^2=1+eta_200*eta_6=1+eta_206.
> >
> > No, eta_2^100 is already expressed in its lowest form, and eta_200
> > is a different and much larger number. Likewise, eta_1*eta_2 is
> > in its lowest form and is less than eta_3. So
> > 1 + eta_2^100*eta_3^2
> > cannot be reduced to simpler terms.
> >
> >
>
> No, not without an interpretation of the index on eta. You don't seem to
> have one.
What is an "interpretation" of the index on eta? Is it anything like a
/definition/ of the index on eta? That has been provided to you several
times.
> So, I wonder how you can justify the statement that
> eta_200>eta_2^100?
One "justifies" it by referring to the /definition/: eta_x > eta_y iff
x > y, where x and y are non-negative real numbers. It also follows
from the definition that if eta_x > eta_y, that eta_x > r*eta_y^n for
all real numbers r, and naturals n.
What other kind of "justification" do you require before something is
valid as an object of mathematical interest?
> If you took my suggestion of making the eta_1 on any
> level equal to eta_1^i, then they'd be equal. If you don't like that
> idea, what's your alternative?
>
Well, how about the one he has been using all along? What exactly is
your difficulty in understanding it?
> > Tony Orlow wrote:
> >> 3-3*eta_e + 17*eta_pi is in its lowest form, with a greatest eta of
> >> eta_pi, far less than eta_206, so the statement would be true. That's
> >> the only way I see the continuous hierarchy working properly.
> >
> > eta_pi is less than eta_206 because pi < 206.
>
> Right.
>
So you /do/ understand it? You just fail to see how it relates to your
own personal obsessions, then?
> >
> > The continuous hierarchy of suprareals is a logical extension
> > of the countable hierarchy of sets H_i for all i in Z. I simply
> > stated that there exist sets H_x for all x in R, and they obey the
> > same order topology.
>
> Yes, but that loses the relationship between the etas, which started out
> as sequential. So that relationship needs to be established otherwise.
>
Do you see how the natural numbers 0, 1, 2, and so on can also be
considered the real numbers 0, 1, 2, and so on?
Do you claim that we "lose the relationship" between the natural
numbers 0, 1, 2, and so on when we consider them as real numbers? Does
pi somehow "invalidate" the real numbers 0, 1, 2, and so on; so that we
can no longer say "2 > 1"?
Cheers - Chas
David R Tribble
>> Since I have not defined a metric for the suprareals (I've only defined
>> an order relation), it's not fitting to talk about the "distance"
>> between two given suprareals. At best we can only really talk
>> about the difference between suprareals. For example, the
>> difference between
>> x1 + y1 eta_1 + z1 eta_1^2 and
>> x2 + y2 eta_1 + z2 eta_1^2
>> is (x2-x1) + (y2-y1)eta_1 + (z2-z1)eta_1^2, for a simple case.
>>
>> members of the same (connected) H_i set.
>
Chas Brown wrote:
> I'd have to disagree. If we define distance as the absolute value of
> the difference |g - h| between suprareals, then it satisfies:
>
> |g - h| = 0 iff h = g.
> |g - h| = |h - g|
> |g - h| + |h - k| >= |g - k|
>
> which is all we usually require of a metric (well, we also require that
> the range of || be the reals). And using this definition of "distance",
> we can produce the same topology as the order topology: one that is
> /based on/ open intervals.
>
> But remember that in the order topology, open sets are the union of
> /any/ number of open intervals. So even though H_0 is not an interval
> (it is not a set of the form {h : x < h < y}), it is still an open set
> (it is the union of all intervals (x, x+1) for x in H_0).
That sounds good. I kind of suspected that difference alone could
be used as a metric, but I was concerned about the meaning of
"distance" between unconnected sets. Since differences are
suprareals themselves (obviously), which are ordered, I see that
this works.
> At the same time, there is no meaning to referring to the "distance"
> from "one end" of H_0 to the "other end", because H_0 is /not/ an
> interval - it has no "ends".
Exactly.
David R Tribble
>> Of course, you mentioned there was a problem with infinite
>> polynomials with regard to 0 not being allowed for the 0th term or there
>> can be no multiplicative inverse, or something?
>
David R Tribble wrote:
>> One solution is to allow suprareal power series. The problem is
>> then how to define a meaning for them:
>> h = sum{i = 0 to oo} x_i eta_1^i
>> Assuming the coefficients x_i "converge" in some sense, what does
>> the sum h converge to, exactly? I'm not saying it can't be done,
>> but I need to learn more about power series and R[X] to do so.
>
Tony Orlow wrote:
> Yeah, I don't know what it would mean for the coefficients to converge.
> The highest power of eta_1 would be the most significant by "far".
There is no highest power in a power series.
> Basically, you're describing a digital number system.
Basically, I'm describing a power series.
> You could include negative powers of eta_1 to get your inverse values.
Only if it could be proved that those negative powers are actually
inverses of suprareals in H_1. And it's been shown that they aren't
(at least not within the suprareals as they are currently defined).
> It would be an eta_1-base digital system, a subset of the T-riffics. :)
Do you have a proof?
David R Tribble wrote:
>> For the time being, the suprareals are still only defined as
>> finite polynomials over eta_i.
>
Tony Orlow wrote:
> Okay. Then they're not closed under division?
That's right. So polynomials over negative powers of eta_1
cannot be inverses of polynomials over positive powers of
eta_1.
cbr...@cbrownsystems.com
You are right to be cautious in assuming that, simply because
"distance" obeys the three axioms regarding identity, reflexivity and
the triangle inequality, that therefore {H_1} is a metric space under
the order topology - because {H_1} /isn't/ a metric space under the
ordering topology.
> Since differences are
> suprareals themselves (obviously), which are ordered, I see that
> this works.
>
It "works" in a topological space which is "almost" but not quite a
metric space. Still, it follows that many things that are true in a
metric space are also true in this topological space.
As an example of the kinds of things that are /not/ true in {H_1}, but
are true in metric spaces, consider "compactness".
Every set which is the union of open intervals in a metric space is
also the union of at most a countable number of those intervals (1
unique rational per interval in the union, please!).
On the other hand, H_1 is the union of an uncountable number of
intervals, for which no countable subset is sufficient.
Cheers - Chas
David R Tribble
>> Of course, you mentioned there was a problem with infinite
>> polynomials with regard to 0 not being allowed for the 0th term or there
>> can be no multiplicative inverse, or something?
>
David R Tribble wrote:
>> One solution is to allow suprareal power series. The problem is
>> then how to define a meaning for them:
>> h = sum{i = 0 to oo} x_i eta_1^i
>> Assuming the coefficients x_i "converge" in some sense, what does
>> the sum h converge to, exactly? I'm not saying it can't be done,
>> but I need to learn more about power series and R[X] to do so.
>
Assuming I extend the definition of suprareals to include power series
(not just polynomials), what meaning could we ascribe to them?
h = sum{i=0 to oo} x_i eta_1^i
e = sum{i=-oo to 0} x_i eta_1^i
Specifically, could we establish criteria as to whether a given series
converges or diverges?
cbr...@cbrownsystems.com
Suppose we admit both h and e for some set of coefficients. How would
you define h * e in this scenario? For example what is the coefficient
of eta_1^0 in this product?
Cheers - Chas
cbr...@cbrownsystems.com
Let me expand a little on my previous terse response.
In general, I don't think of the formal power series 1 + x + x^2 + ...
+ x^n + ... as "converging" (or not) in the usual sense (although this
series does converge for |x| < 1, and doesn't for |x| >= 1).
I find it easier to think of this in a very abstract way: the
conceptualization that I sometimes use is is that each formal power
series p in R[[x]] is really an element of R^N (equivalently, p : N ->
R). So p = (p_0, p_1, ..., p_n, ...) = {p_n}.
Then the ring operations are:
{p_n} + {q_n} = {p_n + q_n}
{p_n}*{q_n} = {sum (i = 0 to n) (p_i*q_(n-i))}
If we think of polynomials as being formal power series that at some
point end up being "all zeros", then this is isomorphic to the ring of
polynomials R[x].
(I'll leave open for now the question of how to define when a
/sequence/ of formal power series converges).
Under this definition, each coefficient in the product p * q is the sum
of a finite number of terms, and so * is well defined for all of R^N.
But your comments above seem to indicate that instead elements of R^N,
you would like to consider elements of R^Z.
A consistent way of defining * might then be
{p_n}*{q_n} = {sum (i in Z) (p_i*q_(n-i))}
(In some sense this is a bit more like convolution.)
But in that case, only given "suitable" p, q can we be sure that for
all n, sum (i in Z) (p_i*q_(n-i)) actually exists; and it's not clear
to me what the exact conditions are, or even whether a unique maximal
set of elements forming a ring exists.
As a guess, it "feels" necessary that sum (i in Z) (p_i)^2 exist for p
to be a friendly member of the club.
Cheers - Chas
cbr...@cbrownsystems.com
> As a guess, it "feels" necessary that sum (i in Z) (p_i)^2 exist for p
> to be a friendly member of the club.
>
Scratch that - there is no unique maximal ring in this system. We can
generate a ring from your original e; and one from your original h; but
we cannot generate one from e + h.
Cheers - Chas
Keith Ramsay
You might be amused by the system in the paper,
"Analysis without Actual Infinity", The Journal of
Symbolic Logic, volume 46, page 625. From the
title you might get the impression it's completely
opposite from what you're doing, but it's not.
Keith Ramsay
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