"Surreal Numbers" Filled With Gaps?
mike3
I read this:
http://valdostamuseum.org/hamsmith/surreal.html
"But
the far greater scope of the surreals also spans enormous
numbers of "holes", which riddle the realm of the surreals. One,
for example, pops up at the frontier between the finite and infinite
numbers, while another divides the finite from the infinitesimally
small numbers. Unless mathematicians can find some way of
leaping across these gaps, integration is always going to be a
problem for surreal numbers."
And at another site:
http://discovermagazine.com/1995/dec/infinityplusonea599
"For one thing, unlike the real number line, which has no holes in it,
the surreal number line is riddled with gaps. There’s a gaping hole
between the finite numbers and the positive infinites, for example.
(That particular hole is what people often mean when they talk about
infinity.) There’s another gap between the infinitesimals and the
positive real numbers. There’s another that separates zero from
every positive number. (That one’s not quite a gap; I call it a cut,
says
Kruskal. It acts a lot like zero, but it isn’t really a number.) These
gaps and cuts can be represented by arrow sequences of length
greater than any ordinal number, and they’re everywhere. You can
find one between any pair of numbers."
But what *are* these "gaps" or "holes", anyway? And just as
perplexing is the statement that one particular hole "is the one
often meant by "infinity"". What does this mean?
(P.S. I'd like to pick a nit on that 2nd website. They say "The
ordinal
numbers of finite sets are simply the natural numbers; the ordinal
numbers
of infinite sets are called transfinite ordinals. The ordinal that
describes the
size and order of the set of natural numbers is called omega. So far,
no
problem. But if you’re planning to do much arithmetic with these
babies,
you’ll find they’re a washout. Adding finite ordinals works fine--
you’re just
adding ordinary integers. But what if you want to add a transfinite
ordinal
(call it n) to a finite one (call it m)? The transfinite ordinal is so
much bigger
than the finite one that it swallows it whole, leaving no trace; m
+n=n.
Cantor’s ordinals are arithmetically limited, like the natural
numbers, says
Kruskal ruefully. We’re back to square one!" Ordinals don't do that,
cardinals
do. omega+1 is NOT equal to omega!)
David C. Ullrich
wrote:
>Hi.
>[...]
>
>But what *are* these "gaps" or "holes", anyway?
Well, that's an informal way of putting it.
The real numbers have the following property:
(*) If A and B are nonempty sets of reals, x < y for
every x in A and y in B, and A union B = R, then
either A has a largest element or B has a smallest
element.
When people talk about this or that ordered set
having "gaps" they mean that (*) does not hold.
You have two sets A and B as in (*) but A does
not have a largest element and B does not have
a smallest element. So conceptually there's this
gap between A and B, a place where there might
be a number but no number's there.
>And just as
>perplexing is the statement that one particular hole "is the one
>often meant by "infinity"". What does this mean?
If A is the set of finite surreals and B the set of infinite
surreals then (*) fails.
>(P.S. I'd like to pick a nit on that 2nd website. They say "The
>ordinal
>numbers of finite sets are simply the natural numbers; the ordinal
>numbers
>of infinite sets are called transfinite ordinals. The ordinal that
>describes the
>size and order of the set of natural numbers is called omega. So far,
>no
>problem. But if you’re planning to do much arithmetic with these
>babies,
>you’ll find they’re a washout. Adding finite ordinals works fine--
>you’re just
>adding ordinary integers. But what if you want to add a transfinite
>ordinal
>(call it n) to a finite one (call it m)? The transfinite ordinal is so
>much bigger
>than the finite one that it swallows it whole, leaving no trace; m
>+n=n.
>Cantor’s ordinals are arithmetically limited, like the natural
>numbers, says
>Kruskal ruefully. We’re back to square one!" Ordinals don't do that,
>cardinals
>do. omega+1 is NOT equal to omega!)
Well, the bit about swallowing whole is a little silly, but
it's hard to see what your complaint is. It's true that
omega + 1 does not equal omega, but the quote above
doesn't say anything about omega + 1. The quote says
that, for example, 1 + omega = omega, which is true.
(Not that I see how this means the ordinals are
"arithmetically limited" or that "we're back to
square one" - ordinal arithmetic is a perfectly
repectable thing.)
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
paulde...@att.net
> On Mon, 23 Mar 2009 01:55:23 -0700 (PDT), mike3 <mike4...@yahoo.com>
> wrote:
>
> >Hi.
> >[...]
>
> >But what *are* these "gaps" or "holes", anyway?
>
> Well, that's an informal way of putting it.
>
> The real numbers have the following property:
>
> (*) If A and B are nonempty sets of reals, x < y for
> every x in A and y in B, and A union B = R, then
> either A has a largest element or B has a smallest
> element.
>
> When people talk about this or that ordered set
> having "gaps" they mean that (*) does not hold.
> You have two sets A and B as in (*) but A does
> not have a largest element and B does not have
> a smallest element. So conceptually there's this
> gap between A and B, a place where there might
> be a number but no number's there.
>
> >And just as
> >perplexing is the statement that one particular hole "is the one
> >often meant by "infinity"". What does this mean?
>
> If A is the set of finite surreals and B the set of infinite
> surreals then (*) fails.
I don't like this last sentence because the class of infinite surreals
is not a set. When discussing foundations, the fact that surreal
numbers form a class rather than a set is quite important to note.
Also, if by finite surreals, you mean surreals x such that -n < x <
n for some positive integer n, then the class of finite surreals is
not a set either.
Paul Epstein
paulde...@att.net
> I don't like this last sentence because the class of infinite surreals
> is not a set. When discussing foundations, the fact that surreal
> numbers form a class rather than a set is quite important to note.
> Also, if by finite surreals, you mean surreals x such that -n < x <
> n for some positive integer n, then the class of finite surreals is
> not a set either.
>
> Paul Epstein
To clarify, the non-setness of the above classes explains why you
can't "fill the gap" by putting the positive finite surreals on the
left and the positive infinite surreals on the right -- a surreal
number has a left _set_ and a right _set_.
Paul Epstein
Dave Seaman
> On Mar 23, 10:55?am, pauldepst...@att.net wrote:
>> I don't like this last sentence because the class of infinite surreals
>> is not a set. ?When discussing foundations, the fact that surreal
>> numbers form a class rather than a set is quite important to note.
>> Also, if by finite surreals, you mean surreals x such that -n < x <
>> n ? for some positive integer n, then the class of finite surreals is
>> not a set either.
>>
>> Paul Epstein
> To clarify, the non-setness of the above classes explains why you
> can't "fill the gap" by putting the positive finite surreals on the
> left and the positive infinite surreals on the right -- a surreal
> number has a left _set_ and a right _set_.
Then how do you explain the fact that the hyperreals have a similar gap,
even though they do form a set?
Specifically, the hyperreals do not satisfy the least upper bound
principle. Putting it another way, you start with a complete ordered
field (the reals), add some more stuff, and end up with an ordered field
that is no longer complete. By adding more stuff, you get gaps.
--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php>
G. A. Edgar
> Then how do you explain the fact that the hyperreals have a similar gap,
> even though they do form a set?
>
> Specifically, the hyperreals do not satisfy the least upper bound
> principle. Putting it another way, you start with a complete ordered
> field (the reals), add some more stuff, and end up with an ordered field
> that is no longer complete. By adding more stuff, you get gaps.
Not surprising at all. There is a gap "just above" the finite stuff.
In fact, there are gaps in ANY ordered field that is not archimedean,
even in countable fields.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Aatu Koskensilta
> Then how do you explain the fact that the hyperreals have a similar
> gap, even though they do form a set?
What's the point of this question? It is difficult to see what it has
to do with Paul's explanation of why there is no surreal number having
the finite surreals as its left component and the infinite surreals as
its right component.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Dave Seaman
> Dave Seaman <dse...@no.such.host> writes:
>> Then how do you explain the fact that the hyperreals have a similar
>> gap, even though they do form a set?
> What's the point of this question? It is difficult to see what it has
> to do with Paul's explanation of why there is no surreal number having
> the finite surreals as its left component and the infinite surreals as
> its right component.
My point is that Paul's attempted explanation, which you snipped, was
entirely off the mark:
>> To clarify, the non-setness of the above classes explains why you
>> can't "fill the gap" by putting the positive finite surreals on the
>> left and the positive infinite surreals on the right -- a surreal
>> number has a left _set_ and a right _set_.
In short, I was explaining why the non-setness of those classes has
absolutely nothing whatever to do with the existence of gaps. As G. A.
Edgar has pointed out, you always get such gaps in any non-Archimedean
ordered field, and that is precisely the point I was trying to make.
Aatu Koskensilta
> My point is that Paul's attempted explanation, which you snipped,
> was entirely off the mark:
>
>>> To clarify, the non-setness of the above classes explains why you
>>> can't "fill the gap" by putting the positive finite surreals on
>>> the left and the positive infinite surreals on the right -- a
>>> surreal number has a left _set_ and a right _set_.
>
> In short, I was explaining why the non-setness of those classes has
> absolutely nothing whatever to do with the existence of gaps.
Not with the existence of gaps in general, but with the existence (and
the impossibility of filling) this particular gap.
Dave Seaman
> Dave Seaman <dse...@no.such.host> writes:
>> My point is that Paul's attempted explanation, which you snipped,
>> was entirely off the mark:
>>
>>>> To clarify, the non-setness of the above classes explains why you
>>>> can't "fill the gap" by putting the positive finite surreals on
>>>> the left and the positive infinite surreals on the right -- a
>>>> surreal number has a left _set_ and a right _set_.
>>
>> In short, I was explaining why the non-setness of those classes has
>> absolutely nothing whatever to do with the existence of gaps.
> Not with the existence of gaps in general, but with the existence (and
> the impossibility of filling) this particular gap.
No, you misunderstand. *Every* non-Archimedean ordered field, whether it
is a set or not, has gaps. It is impossible to fill those gaps, because
you cannot find an extension field of any non-Archimedean ordered field
that is order-complete. The existence of gaps is essential.
Aatu Koskensilta
> No, you misunderstand. *Every* non-Archimedean ordered field,
> whether it is a set or not, has gaps.
Sure, that is not the issue, but rather the specific gap in the
surreals mentioned. Its existence is indeed explained by the fact that
two particular collections of surreals happen to be proper
classes. The necessary existence of gaps in a non-Archimedean ordered
field doesn't tell us anything about the gap in question.
Dave Seaman
> Dave Seaman <dse...@no.such.host> writes:
>> No, you misunderstand. *Every* non-Archimedean ordered field,
>> whether it is a set or not, has gaps.
> Sure, that is not the issue, but rather the specific gap in the
> surreals mentioned. Its existence is indeed explained by the fact that
> two particular collections of surreals happen to be proper
> classes. The necessary existence of gaps in a non-Archimedean ordered
> field doesn't tell us anything about the gap in question.
Sure, and the reason my car can't move the Empire State Building is that
my car doesn't have a trailer hitch.
You asked what my point was earlier. Is it clear now? Did you actually
have a point?
Aatu Koskensilta
> You asked what my point was earlier. Is it clear now? Did you
> actually have a point?
Well, apparently your point was that all non-Archimedean ordered
fields have gaps. Paul's original observation concerning a specific
gap was
To clarify, the non-setness of the above classes explains why you
can't "fill the gap" by putting the positive finite surreals on the
left and the positive infinite surreals on the right -- a surreal
number has a left _set_ and a right _set_.
To this you replied with the question
Then how do you explain the fact that the hyperreals have a similar
gap, even though they do form a set?
the point of which, as you later explained, was that you were
explaining why the non-setness of those classes has absolutely
nothing whatever to do with the existence of gaps.
I was just wondering what your question has to do with Paul's
observation, which, despite your assertions, is a perfectly fine
explanation of the existence of the gap in question. In order to
answer questions about the hyperreals and particular gaps therein we
must examine the structure of the hyperreals. We may of course also
shed light on these questions with general order-theoretic
considerations, but merely observing that non-Archimedean ordered
fields always have gaps is in itself not particularly illuminating.
Dave Seaman
If you think that explanation is perfectly fine, then you evidently think
there must be a way to fix it by relaxing that restriction and allowing
left-and-right proper classes in place of left-and-right sets, and thus
obtaining an extension field that is order complete.
Is this, in fact, your claim? Can you explain how to do that?
Isn't it true that the surreals have the universal property, meaning that
every ordered field can be embedded in the surreals?
Aatu Koskensilta
> If you think that explanation is perfectly fine, then you evidently
> think there must be a way to fix it by relaxing that restriction and
> allowing left-and-right proper classes in place of left-and-right
> sets, and thus obtaining an extension field that is order complete.
Why should I think that? Paul's explanation is simply an answer to
someone wondering why, given how a surreal number is defined, a
particular construction that might seem to fill the gap in question
fails.
> Is this, in fact, your claim?
No. The obvious extensions, by means of some proper class machinery
formalised in e.g. some Ackermann style set theory, also have a gap,
explained by the fact that the relevant non-set collections fail to be
classes.
> Isn't it true that the surreals have the universal property, meaning
> that every ordered field can be embedded in the surreals?
Sure.
Denis Feldmann
Denis Feldmann
> Dave Seaman <dse...@no.such.host> writes:
>
>> No, you misunderstand. *Every* non-Archimedean ordered field,
>> whether it is a set or not, has gaps.
>
> Sure, that is not the issue, but rather the specific gap in the
> surreals mentioned. Its existence is indeed explained by the fact that
> two particular collections of surreals happen to be proper
> classes. The necessary existence of gaps in a non-Archimedean ordered
> field doesn't tell us anything about the gap in question.
It doesn't? I thought the idea was there was necessarily a gap between
infinitesimals and appreciable numbers
Denis Feldmann
> Dave Seaman <dse...@no.such.host> writes:
>
>> You asked what my point was earlier. Is it clear now? Did you
>> actually have a point?
>
> Well, apparently your point was that all non-Archimedean ordered
> fields have gaps. Paul's original observation concerning a specific
> gap was
>
> To clarify, the non-setness of the above classes explains why you
> can't "fill the gap" by putting the positive finite surreals on the
> left and the positive infinite surreals on the right -- a surreal
> number has a left _set_ and a right _set_.
>
> To this you replied with the question
>
> Then how do you explain the fact that the hyperreals have a similar
> gap, even though they do form a set?
>
> the point of which, as you later explained, was that you were
>
> explaining why the non-setness of those classes has absolutely
> nothing whatever to do with the existence of gaps.
>
> I was just wondering what your question has to do with Paul's
> observation, which, despite your assertions, is a perfectly fine
> explanation of the existence of the gap in question. In order to
> answer questions about the hyperreals and particular gaps therein we
> must examine the structure of the hyperreals. We may of course also
> shed light on these questions with general order-theoretic
> considerations, but merely observing that non-Archimedean ordered
> fields always have gaps is in itself not particularly illuminating.
>
But the fact that they have all a gap there (between finite and
infinite) could be...
Aatu Koskensilta
> But the fact that they have all a gap there (between finite and
> infinite) could be...
Sure, that fact is indeed illuminating -- and indeed what I had in
mind by alluding to "general order theoretic considerations". Paul's
original observation was in effect an explanation of why there is no
surreal number that fills the gap, which may appear puzzling given the
definition of a surreal number as a pair of collections satisfying
certain criteria, and was prompted by David's use of the term "set" in
this context. I'm afraid there's nothing very exciting to this whole
discussion.
Denis Feldmann
Please buy ONAG ; you will find all the answers there. Anyway, gaps are
Dedekind cuts between *classes* (and not sets) of surreal numbers, and
they shar"e some (but not all) properties of true surreal numbers
>
> (P.S. I'd like to pick a nit on that 2nd website. They say "The
> ordinal
> numbers of finite sets are simply the natural numbers; the ordinal
> numbers
> of infinite sets are called transfinite ordinals. The ordinal that
> describes the
> size and order of the set of natural numbers is called omega. So far,
> no
> problem. But if you’re planning to do much arithmetic with these
> babies,
> you’ll find they’re a washout. Adding finite ordinals works fine--
> you’re just
> adding ordinary integers. But what if you want to add a transfinite
> ordinal
> (call it n) to a finite one (call it m)? The transfinite ordinal is so
> much bigger
> than the finite one that it swallows it whole, leaving no trace; m
> +n=n.
> Cantor’s ordinals are arithmetically limited, like the natural
> numbers, says
> Kruskal ruefully. We’re back to square one!" Ordinals don't do that,
> cardinals
> do. omega+1 is NOT equal to omega!)
No, read more carefully : what is said is that 1+w = w, which is
perfectly true...
Denis Feldmann
> On Mon, 23 Mar 2009 01:55:23 -0700 (PDT), mike3 <mike...@yahoo.com>
> wrote:
>
>> Hi.
>> [...]
>>
>> But what *are* these "gaps" or "holes", anyway?
>
> Well, that's an informal way of putting it.
>
> The real numbers have the following property:
>
> (*) If A and B are nonempty sets of reals, x < y for
> every x in A and y in B, and A union B = R, then
> either A has a largest element or B has a smallest
> element.
>
> When people talk about this or that ordered set
> having "gaps" they mean that (*) does not hold.
> You have two sets A and B as in (*) but A does
> not have a largest element and B does not have
> a smallest element. So conceptually there's this
> gap between A and B, a place where there might
> be a number but no number's there.
>
>> And just as
>> perplexing is the statement that one particular hole "is the one
>> often meant by "infinity"". What does this mean?
>
> If A is the set of finite surreals and B the set of infinite
> surreals then (*) fails.
>
(for surreals) that there is a hole there
Aatu Koskensilta
> It doesn't? I thought the idea was there was necessarily a gap
> between infinitesimals and appreciable numbers
Well, I perhaps expressed myself poorly. David's original explanation
of what it means for there to be no gaps, generalised to an arbitrary
possibly proper class field F, was:
(*) If A and B are nonempty collections of members of F, x < y for
every x in A and y in B, and A union B = F, then either A has a
largest element or B has a smallest element.
Merely noting that any non-Archimedean field has gaps in the sense
that (*) fails isn't in itself not particularly illuminating when
considering the gap between finite and infinite surreals. Recall the
context of Paul's observation: the original formulation of (*) was in
terms of sets. Paul then noted that in case of the gap in question in
the surreals the relevant collections A and B are proper classes, and
that this explains why there is no surreal {A | B}.
mike3
> Dave Seaman <dsea...@no.such.host> writes:
> > No, you misunderstand. *Every* non-Archimedean ordered field,
> > whether it is a set or not, has gaps.
>
> Sure, that is not the issue, but rather the specific gap in the
> surreals mentioned. Its existence is indeed explained by the fact that
> two particular collections of surreals happen to be proper
> classes. The necessary existence of gaps in a non-Archimedean ordered
> field doesn't tell us anything about the gap in question.
>
Hmm.
It seems that the surreal numbers "max out" any given set theory.
You mention in a later post that "the obvious extensions, by means of
some proper class machinery formalised in e.g. some Ackermann style
set
theory, also have a gap, explained by the fact that the relevant non-
set
collections fail to be classes.". So what happens if you just keep
going higher and higher up the "ladder"? Although if no non-
Archimedean
ordered field can be free of gaps, it would seem that there would be
absolutely no way to resolve the general gap problem without
sacrificing
either the order (in which case we can't talk about gaps anymore) or
the
field properties.
I'm wondering: is there an even stronger result than all of this,
namely
that a continuum ("ordered field without gaps") cannot be larger than
some
fixed amount?
Aatu Koskensilta
> It seems that the surreal numbers "max out" any given set theory.
> You mention in a later post that "the obvious extensions, by means
> of some proper class machinery formalised in e.g. some Ackermann
> style set theory, also have a gap, explained by the fact that the
> relevant non- set collections fail to be classes.". So what happens
> if you just keep going higher and higher up the "ladder"?
Just more of the same, essentially.
> I'm wondering: is there an even stronger result than all of this,
> namely that a continuum ("ordered field without gaps") cannot be
> larger than some fixed amount?
The reals are the only complete Archimedean field, up to isomorphism.
mike3
> mike3 <mike4...@yahoo.com> writes:
> > It seems that the surreal numbers "max out" any given set theory.
> > You mention in a later post that "the obvious extensions, by means
> > of some proper class machinery formalised in e.g. some Ackermann
> > style set theory, also have a gap, explained by the fact that the
> > relevant non- set collections fail to be classes.". So what happens
> > if you just keep going higher and higher up the "ladder"?
>
> Just more of the same, essentially.
>
So then there's no way to construct a theory that has gapless
surreals...
> > I'm wondering: is there an even stronger result than all of this,
> > namely that a continuum ("ordered field without gaps") cannot be
> > larger than some fixed amount?
>
> The reals are the only complete Archimedean field, up to isomorphism.
>
So indeed, referring to the reals as "*THE* continuum" (as opposed to
merely "a continuum") makes a lot of sense. As there aren't any
*complete*
(in the sense of having no gaps of the type under discussion) non-
Archimedean ordered fields.
lwa...@lausd.net
> But what *are* these "gaps" or "holes", anyway? And just as
> perplexing is the statement that one particular hole "is the one
> often meant by "infinity"". What does this mean?
As the other posters have already pointed out, the "hole at
infinity" refers to the fact that there is no largest
finite surreal number, nor a smallest infinite surreal. So
there is always a "hole" separating them.
Many so-called "cranks" have discovered this when trying to
come up with their new forms of analysis. Indeed, I believe
that one of the best arguments the standard analysts have
against the so-called "cranks" is how the set R of standard
reals is the only complete ordered field. This is why
standard analysis is considered to be superior to both
finitism and infinitesimalism -- neither theory can produce
the complete ordered field.
Any infinitesimal (i.e., non-Archimedean) field fails to be
complete, since the set of all infinitesimals can have no
maximum, nor can the set of all positive non-infinitesimals
ever have a minimum. There is always a hole separating the
non-Archimedean numbers from the Archimedean numbers.
One so-called "crank," AP, has already noticed this. So he
tried to rectify this by declaring the largest number that
has any use in physics to be finite, and any larger number
to be infinite. What AP did is take the number of nucleons
in a plutonium atom -- I believe he chose 242, though I
forget exactly which isotope he used -- and then found its
factorial, 242!, which is about googol^5. This AP declared
to be the largest finite number, and so 242!+1 would be
considered to be the smallest infinite AP-adic.
(Notice that this is _not_ the same as ultrafinitism. An
ultrafinitist would declare 242! to be the largest number
and that 242!+1 doesn't exist. AP states that 242!+1 does
exist, only that it's infinite.)
Has AP really removed the "hole" between the finite and
infinite by declaring 242!+1 to be infinite?
amy666
SIGH.
tell me something , how do you rime
AP - adic 354131874....34513854351,3548713...134835
and finitism ??
its amazing that AP wants to EXTEND THE NATURALS WHILE SUPPORTING FINITISM ???!!!???!!!???
further i dont find your comments very relevant.
mike3 is really getting there.
his eyes are almost open.
although " holes " are perhaps not the best arguments , its getting clear to him that ordinals and surreal numbers arent the BIG answers.
i have been objecting to ordinals and surreals since the beginning of time ...
calculus is just fine , even better.
give me back my epsilons !
newton was RIGHT.
regards
tommy1729
Aatu Koskensilta
> although " holes " are perhaps not the best arguments , its getting
> clear to him that ordinals and surreal numbers arent the BIG
> answers.
The "BIG answers" to what?
mike3
wrote:
$50? Ouch. Wish I had access to a university library... :(((
And what about that last bit though about it being "what is often
meant by infinity"? This would seem to suggest that the infinite
ordinals
are _bigger_ than "infinity" (I guess the "infinity" used in
analysis?)?
Now it makes more sense.
mike3
You can do *anything* when you're allowed to just redefine
terms willy-nilly. I could define that "zero" actually means
"1", and so "prove" that 0 = 1.
mike3
Unlike you, I never thought it invalidated the theory.
> although " holes " are perhaps not the best arguments , its getting clear to him that ordinals and surreal numbers arent the BIG answers.
>
What are the "BIG" answers? What are the "BIG" questions they
are supposed to answer?
> i have been objecting to ordinals and surreals since the beginning of time ...
>
> calculus is just fine , even better.
>
How does the "existence" of ordinal and surreal numbers destroy
"calculus", anyway?
amy666
not quite.
the infinitesimals have the right to be there too.
although its been objected they are not reals , how then do you take real derivatives of real functions ?
f'(x) = f(x + h) - f(x) / h = REAL !!
so you see , the infinitesimals are on the continuum , their job is to connect the reals to eachother.
tommy1729
Bill Dubuque
For in-depth discussion of this and related matters see
Ehrlich, P. The Absolute Arithmetic Continuum and its Peircean Counterpart
http://oak.cats.ohiou.edu/~ehrlich/Peirce.pdf
http://oak.cats.ohiou.edu/~ehrlich
--Bill Dubuque
David C. Ullrich
>On Mar 23, 10:44 am, David C. Ullrich <dullr...@sprynet.com> wrote:
>> On Mon, 23 Mar 2009 01:55:23 -0700 (PDT), mike3 <mike4...@yahoo.com>
>> wrote:
>>
>> >Hi.
>> >[...]
>>
>> >But what *are* these "gaps" or "holes", anyway?
>>
>> Well, that's an informal way of putting it.
>>
>> The real numbers have the following property:
>>
>> (*) If A and B are nonempty sets of reals, x < y for
>> every x in A and y in B, and A union B = R, then
>> either A has a largest element or B has a smallest
>> element.
>>
>> When people talk about this or that ordered set
>> having "gaps" they mean that (*) does not hold.
>> You have two sets A and B as in (*) but A does
>> not have a largest element and B does not have
>> a smallest element. So conceptually there's this
>> gap between A and B, a place where there might
>> be a number but no number's there.
>>
>> >And just as
>> >perplexing is the statement that one particular hole "is the one
>> >often meant by "infinity"". What does this mean?
>>
>> If A is the set of finite surreals and B the set of infinite
>> surreals then (*) fails.
>
>I don't like this last sentence because the class of infinite surreals
>is not a set. When discussing foundations, the fact that surreal
>numbers form a class rather than a set is quite important to note.
That's certainly true. I wasn't aware that we _were_ discussing
foundations, I thought the question was just what various
people _meant_ by saying this or that structure had a "gap".
>Also, if by finite surreals, you mean surreals x such that -n < x <
>n for some positive integer n, then the class of finite surreals is
>not a set either.
>
>Paul Epstein
>
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
paulde...@att.net
Ok, but the surreal numbers do have the following non-gap property:
Suppose you partition the surreal numbers S into two disjoint and non-
empty sets A and B such that A U B = S and such that every member of A
< every member of B.
Then either A has a largest member or B has a smallest member.
This property holds vacuously because the class of surreals can't be
partitioned into two sets.
But these are quibbles, I agree. Your post did a good job of
explaining the difference between surreals and reals with respect to
gap-like behaviour but Denis and I would both have preferred it if you
didn't say "set".
Paul Epstein
Jhonas
::not quite.
::the infinitesimals have the right to be there too.
::although its been objected they are not reals , how
::then do you take real derivatives of real ::functions ?
::f'(x) = f(x + h) - f(x) / h = REAL !!
::so you see , the infinitesimals are on the
::continuum , ::their job is to connect the reals to eachother.
quite right tommy, they are the nexus between reals!
David R Tribble
>> So indeed, referring to the reals as "*THE* *continuum" (as opposed to
>> merely "a continuum") makes a lot of sense. As there aren't any
>> *complete* (in the sense of having no gaps of the type under
>> discussion) non-Archimedean ordered fields.
>
amy666 wrote:
> not quite. the infinitesimals have the right to be there too.
> although its been objected they are not reals , how then do you take
> real derivatives of real functions ?
> f'(x) = f(x + h) - f(x) / h = REAL !!
> so you see , the infinitesimals are on the continuum , their job is
> to connect the reals to eachother.
Please explain how f'(x) is both a real and an infinitesimal.
Please explain how the infinitesimals "connect" the reals when
the reals do not contain any infinitesimals.
Or were you trying to be more allegorical than mathematical?
Tim Little
> how then do you take real derivatives of real functions ?
>
> f'(x) = f(x + h) - f(x) / h = REAL !!
You are missing the most important part: the limit.
f'(x) = lim(h->0) (f(x+h) - f(x)) / h.
Limits are how you take real derivatives of real functions.
Your equation is not even true in nonstandard analysis with
infinitesimals. Instead of a limit, you have a "standard part"
operator.
- Tim
lwa...@lausd.net
> On Mar 23, 11:54 am, Denis Feldmann <denis.feldmann.sanss...@neuf.fr>
> wrote:
> > Please buy ONAG ; you will find all the answers there. Anyway, gaps are
> > Dedekind cuts between *classes* (and not sets) of surreal numbers, and
> > they shar"e some (but not all) properties of true surreal numbers
> $50? Ouch. Wish I had access to a university library... :(((
And here we go again with someone suggesting that another poster
purchase a book that they can't afford.
As mike3 mentions, not all of us are lucky enough to have access
to university libraries -- unlike the professors here who
obviously do have such access. It's unreasonable, therefore, to
expect someone to spend that much money just to be able to post
on Usenet.
Most it's the standard analysts who suggest that their opponents
start spending money on books. It's ironic, therefore, that the
book suggested here isn't a book on standard analysis but on the
surreals, which are definitely nonstandard. Actually, $50 is
reasonably priced -- many books on nonstandard mathematics can
be _more_ expensive and elusive than their standard counterparts!
> And what about that last bit though about it being "what is often
> meant by infinity"? This would seem to suggest that the infinite
> are _bigger_ than "infinity" (I guess the "infinity" used in
> analysis?)?
If I recall correctly (from websites about surreals, not from any
pricey books, of course), Conway considers "what is often meant by
infinity" -- given the symbol oo -- to be at the gap between the
finite and infinite surreals. He then derives a strange-looking
equation, namely that oo is the On-th root of omega. So he
connects the three types of infinity (the infinity ordinal omega,
the absolute infinity On, and the extended real infinity oo).
> > No, read more carefully : what is said is that 1+w = w, which is
> > perfectly true...
> Now it makes more sense.
By 1+omega = omega, are these ordinals or surreals? In the
ordinals, 1+omega is definitely omega. But the surreals are
supposed to be a (proper class) Field, and so we must have
commutativity: 1+omega = omega+1, and neither can equal omega.
lwa...@lausd.net
> > and infinite by declaring 242!+1 to be infinite?
> SIGH.
> tell me something , how do you rime
> AP - adic 354131874....34513854351,3548713...134835
> and finitism ??
> its amazing that AP wants to EXTEND THE NATURALS WHILE SUPPORTING FINITISM ???!!!???!!!???
I don't consider AP to be a finitist. But in trying to
find the smallest infinite AP-adic, AP unwittingly
ended up making an ultrafinitist-like argument about
there existing a largest finite number.
I've said it before, and I'll say it again -- I think
that sometimes AP gives an answer that even I don't
believe makes sense, but only because the standard
analysts pressured (tricked?) him into it. Someone
asked AP what the smallest infinite AP-adic is. But
instead of saying that there is no smallest infinite
AP-adic (which I think would've made more sense), he
started coming up with 242!+1, in reference to the
isotope Plutonium-242.
In the hyperreals, there is no smallest hyperreal. In
the surreals, oo isn't really the smallest infinite
surreal, but more like a gap (as Feldmann describes
as a sort of Dedekind cut) between the surreals.
Also, notice that AP has made another argument that
sounds similar to WM/other ultrafinitists. WM argues
that a set can't be infinite unless it has at least
one infinite element. AP writes that there can't
exist infinite lines in Euclidean geometry, but only
line segments, for otherwise they would require
infinitely long naturals (or reals that are infinitely
long to the left of the decimal point) as coordinates
for some of their points.
> i have been objecting to ordinals and surreals since the beginning of time ...
So has AP.
> give me back my epsilons !
> newton was RIGHT.
Ironically, Newton never used epsilons/deltas, but these
strange objects called "fluxions." Some nonstandard
analysts believe that their infinitesimals are actually
closer to what Newton and Leibniz orginally had in mind,
than the standard epsilons/deltas.
Denis Feldmann
> On Mar 23, 2:23 pm, mike3 <mike4...@yahoo.com> wrote:
>> On Mar 23, 11:54 am, Denis Feldmann <denis.feldmann.sanss...@neuf.fr>
>> wrote:
>>> Please buy ONAG ; you will find all the answers there. Anyway, gaps are
>>> Dedekind cuts between *classes* (and not sets) of surreal numbers, and
>>> they shar"e some (but not all) properties of true surreal numbers
>> $50? Ouch. Wish I had access to a university library... :(((
>
> And here we go again with someone suggesting that another poster
> purchase a book that they can't afford.
>
Hard to believe . It is expensive, agreed. But how long will it last?
> As mike3 mentions, not all of us are lucky enough to have access
> to university libraries -- unlike the professors here who
> obviously do have such access. It's unreasonable, therefore, to
> expect someone to spend that much money just to be able to post
> on Usenet.
>
No, the suggestion is that this book has all the answers to his
questions, and is one of the best books ever written...
> Most it's the standard analysts who suggest that their opponents
> start spending money on books. It's ironic, therefore, that the
> book suggested here isn't a book on standard analysis but on the
> surreals, which are definitely nonstandard. Actually, $50 is
> reasonably priced -- many books on nonstandard mathematics can
> be _more_ expensive and elusive than their standard counterparts!
>
>> And what about that last bit though about it being "what is often
>> meant by infinity"? This would seem to suggest that the infinite
>> are _bigger_ than "infinity" (I guess the "infinity" used in
>> analysis?)?
>
> If I recall correctly (from websites about surreals, not from any
> pricey books, of course), Conway considers "what is often meant by
> infinity" -- given the symbol oo -- to be at the gap between the
> finite and infinite surreals.
Yes
He then derives a strange-looking
> equation, namely that oo is the On-th root of omega.
Well, it is smpbolic, but OK
So he
> connects the three types of infinity (the infinity ordinal omega,
> the absolute infinity On, and the extended real infinity oo).
Fun way to put it , but correct
>
>>> No, read more carefully : what is said is that 1+w = w, which is
>>> perfectly true...
>> Now it makes more sense.
>
> By 1+omega = omega, are these ordinals or surreals?
Read the thread. It was ordinals
In the
> ordinals, 1+omega is definitely omega. But the surreals are
> supposed
No : are proved
to be a (proper class) Field, and so we must have
> commutativity: 1+omega = omega+1, and neither can equal omega.
Sure. All this is explained in ONAG too...
Jesse F. Hughes
> And here we go again with someone suggesting that another poster
> purchase a book that they can't afford.
>
> As mike3 mentions, not all of us are lucky enough to have access
> to university libraries -- unlike the professors here who
> obviously do have such access. It's unreasonable, therefore, to
> expect someone to spend that much money just to be able to post
> on Usenet.
Mike3 didn't claim to be oppressed this way. Mike3 didn't whine that
others were criticizing his posts unfairly because he's poor.
No one requires that you buy expensive textbooks. But if you do not
find some way to learn the mathematics you want to talk about, then
people will point out where you go wrong and suggest how you can learn
the subject (though not necessarily for free).
That's not oppression or censorship.
--
Sam Vimes could parallel process. Most husbands can. They learn to
follow their own line of thought while /at the same time/ listening to
what their wives say.... At any time they could be challenged and
must be ready to quote the last sentence in full. -- Terry Pratchett
Aatu Koskensilta
> That's not oppression or censorship.
Bah. It's just that. That not all mathematical texts are available for
download for free is nothing but a facet to the underhanded campaign
to prevent non-standard mathematicians from posting on Usenet.
mike3
> lwal...@lausd.net writes:
> > And here we go again with someone suggesting that another poster
> > purchase a book that they can't afford.
>
> > As mike3 mentions, not all of us are lucky enough to have access
> > to university libraries -- unlike the professors here who
> > obviously do have such access. It's unreasonable, therefore, to
> > expect someone to spend that much money just to be able to post
> > on Usenet.
>
> Mike3 didn't claim to be oppressed this way. Mike3 didn't whine that
> others were criticizing his posts unfairly because he's poor.
>
> No one requires that you buy expensive textbooks. But if you do not
> find some way to learn the mathematics you want to talk about, then
> people will point out where you go wrong and suggest how you can learn
> the subject (though not necessarily for free).
>
Which of course then raises the question of how exactly I can make the
large amounts of money I'd need to pursue this. But I've been told
that
money isn't necessarily a barrier, so how do I make it not one?
mike3
> On Mar 23, 2:23 pm, mike3 <mike4...@yahoo.com> wrote:
>
> > On Mar 23, 11:54 am, Denis Feldmann <denis.feldmann.sanss...@neuf.fr>
> > wrote:
> > > Please buy ONAG ; you will find all the answers there. Anyway, gaps are
> > > Dedekind cuts between *classes* (and not sets) of surreal numbers, and
> > > they shar"e some (but not all) properties of true surreal numbers
> > $50? Ouch. Wish I had access to a university library... :(((
>
> And here we go again with someone suggesting that another poster
> purchase a book that they can't afford.
>
> As mike3 mentions, not all of us are lucky enough to have access
> to university libraries -- unlike the professors here who
> obviously do have such access. It's unreasonable, therefore, to
> expect someone to spend that much money just to be able to post
> on Usenet.
>
I'd agree.
> Most it's the standard analysts who suggest that their opponents
> start spending money on books.
I'm not an "opponent" of standard analysis. So I do not get the point
here.
Plus I don't see why an interest in "nonstandard" analysis makes one
"oppose" standard analysis. The book may be useful, it may help answer
my questions, I just do not have the access to it.
> It's ironic, therefore, that the
> book suggested here isn't a book on standard analysis but on the
> surreals, which are definitely nonstandard. Actually, $50 is
> reasonably priced -- many books on nonstandard mathematics can
> be _more_ expensive and elusive than their standard counterparts!
>
> > And what about that last bit though about it being "what is often
> > meant by infinity"? This would seem to suggest that the infinite
> > are _bigger_ than "infinity" (I guess the "infinity" used in
> > analysis?)?
>
> If I recall correctly (from websites about surreals, not from any
> pricey books, of course), Conway considers "what is often meant by
> infinity" -- given the symbol oo -- to be at the gap between the
> finite and infinite surreals. He then derives a strange-looking
> equation, namely that oo is the On-th root of omega. So he
> connects the three types of infinity (the infinity ordinal omega,
> the absolute infinity On, and the extended real infinity oo).
>
I.e. this infinity is the infinity of analysis and calculus, which is
at
some sort of "nexus" point between the reals and the transfinite
numbers, and this "nexus" point is not a surreal number?
mike3
> If I recall correctly (from websites about surreals, not from any
> pricey books, of course), Conway considers "what is often meant by
> infinity" -- given the symbol oo -- to be at the gap between the
> finite and infinite surreals. He then derives a strange-looking
> equation, namely that oo is the On-th root of omega. So he
> connects the three types of infinity (the infinity ordinal omega,
> the absolute infinity On, and the extended real infinity oo).
>
Is this why the "hole" is considered "gaping" in one of the websites
I mentioned? As to "get inside" it you have to take a root of omega
that is as strong as the entire collection of ordinals is large?
lwa...@lausd.net
> "Jesse F. Hughes" <je...@phiwumbda.org> writes:
> > That's not oppression or censorship.
> Bah. It's just that. That not all mathematical texts are available for
> download for free is nothing but a facet to the underhanded campaign
> to prevent non-standard mathematicians from posting on Usenet.
Consider a message board or online forum devoted to your
favorite sports team. Now suppose that, after you post
there, someone tells you that you can't post there unless
you've spent money on season tickets for the team.
Or consider a forum devoted to your favorite musical artist,
but you can't post there unless you own all of the albums
produced by that artist.
And so on. No one insists that you spend money to post on
most other forums on the Internet. So why should sci.math
be different?
lwa...@lausd.net
> On Mar 24, 11:50 pm, lwal...@lausd.net wrote:
> > Most it's the standard analysts who suggest that their opponents
> > start spending money on books.
> Plus I don't see why an interest in "nonstandard" analysis makes one
> "oppose" standard analysis. The book may be useful, it may help answer
> my questions, I just do not have the access to it.
Many posters on sci.math, including WM (Wolfgang Muckenheim),
AP (Archimedes Plutonium), and MR (Mitch Raemsch), are
nonstandard analysts who do oppose standard analysis. They
devote numerous threads to their nonstandard beliefs and
trying to prove standard analysis or ZFC to be inconsistent.
In return, the standard analysts and set theorists consider
such posters to be "cranks" or "trolls."
If you (mike3) are a nonstandard analyst who doesn't oppose
standard analysis, then more power to you. We need more
posters on sci.math like you. You're less likely to be
called a "crank."
It was once stated that the so-called "cranks" don't want
to spend the energy to learn mathematics, and this is one
key distinguishing trait in "cranks." But based on the
attitude I've seen in this thread and others, it's the
unwillingness to spend the _money_ to learn mathematics
that makes one called a "crank."
> > If I recall correctly (from websites about surreals, not from any
> > pricey books, of course), Conway considers "what is often meant by
> > infinity" -- given the symbol oo -- to be at the gap between the
> > finite and infinite surreals. He then derives a strange-looking
> > equation, namely that oo is the On-th root of omega. So he
> > connects the three types of infinity (the infinity ordinal omega,
> > the absolute infinity On, and the extended real infinity oo).
> I.e. this infinity is the infinity of analysis and calculus, which is
> some sort of "nexus" point between the reals and the transfinite
> numbers, and this "nexus" point is not a surreal number?
Something like that.
> Is this why the "hole" is considered "gaping" in one of the websites
> I mentioned? As to "get inside" it you have to take a root of omega
> that is as strong as the entire collection of ordinals is large?
Something like that. I'm curious as to what site you (mike3)
found, since it's hard to find free sites on surreals. (Of
course there's always Wikipedia, but the Wikipedia page
doesn't discuss the gaps in the surreal line.)
mike3
> On Mar 25, 5:43 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>
> > "Jesse F. Hughes" <je...@phiwumbda.org> writes:
> > > That's not oppression or censorship.
> > Bah. It's just that. That not all mathematical texts are available for
> > download for free is nothing but a facet to the underhanded campaign
> > to prevent non-standard mathematicians from posting on Usenet.
>
> Consider a message board or online forum devoted to your
> favorite sports team. Now suppose that, after you post
> there, someone tells you that you can't post there unless
> you've spent money on season tickets for the team.
>
> Or consider a forum devoted to your favorite musical artist,
> but you can't post there unless you own all of the albums
> produced by that artist.
>
Who's saying I *can't post* unless I've got this book -- heck, unless
I've got "ALL" the books?
mike3
> On Mar 26, 6:23 pm, mike3 <mike4...@yahoo.com> wrote:
>
> > On Mar 24, 11:50 pm, lwal...@lausd.net wrote:
> > > Most it's the standard analysts who suggest that their opponents
> > > start spending money on books.
> > I'm not an "opponent" of standard analysis. So I do not get the point.
> > Plus I don't see why an interest in "nonstandard" analysis makes one
> > "oppose" standard analysis. The book may be useful, it may help answer
> > my questions, I just do not have the access to it.
>
> Many posters on sci.math, including WM (Wolfgang Muckenheim),
> AP (Archimedes Plutonium), and MR (Mitch Raemsch), are
> nonstandard analysts who do oppose standard analysis. They
> devote numerous threads to their nonstandard beliefs and
> trying to prove standard analysis or ZFC to be inconsistent.
>
> In return, the standard analysts and set theorists consider
> such posters to be "cranks" or "trolls."
>
> If you (mike3) are a nonstandard analyst who doesn't oppose
> standard analysis, then more power to you. We need more
> posters on sci.math like you. You're less likely to be
> called a "crank."
>
Actually I don't have a preference for nonstandard vs
standard analysis.
> It was once stated that the so-called "cranks" don't want
> to spend the energy to learn mathematics, and this is one
> key distinguishing trait in "cranks." But based on the
> attitude I've seen in this thread and others, it's the
> unwillingness to spend the _money_ to learn mathematics
> that makes one called a "crank."
>
I'm not sure where someone said that because one lacks the
money, one is automatically some sort of "crank". Usually it
seems that "crank" is acquired when one goes and arrogantly
declares to have "TRUTH" and that they cannot possibly be
wrong at all, refuses to listen to anyone's comments or critique,
calls the opponents "liars", has paranoid conspiracy theories,
etc.
> > > If I recall correctly (from websites about surreals, not from any
> > > pricey books, of course), Conway considers "what is often meant by
> > > infinity" -- given the symbol oo -- to be at the gap between the
> > > finite and infinite surreals. He then derives a strange-looking
> > > equation, namely that oo is the On-th root of omega. So he
> > > connects the three types of infinity (the infinity ordinal omega,
> > > the absolute infinity On, and the extended real infinity oo).
> > I.e. this infinity is the infinity of analysis and calculus, which is
> > some sort of "nexus" point between the reals and the transfinite
> > numbers, and this "nexus" point is not a surreal number?
>
> Something like that.
>
Hmm.
> > Is this why the "hole" is considered "gaping" in one of the websites
> > I mentioned? As to "get inside" it you have to take a root of omega
> > that is as strong as the entire collection of ordinals is large?
>
> Something like that. I'm curious as to what site you (mike3)
> found, since it's hard to find free sites on surreals. (Of
> course there's always Wikipedia, but the Wikipedia page
> doesn't discuss the gaps in the surreal line.)
I don't really have a list offhand, but I posted some sources for the
quotes I gave in the original post here.
amy666
thanks.
regards
tommy1729
Dik T. Winter
...
> Consider a message board or online forum devoted to your
> favorite sports team. Now suppose that, after you post
> there, someone tells you that you can't post there unless
> you've spent money on season tickets for the team.
>
> Or consider a forum devoted to your favorite musical artist,
> but you can't post there unless you own all of the albums
> produced by that artist.
>
> And so on. No one insists that you spend money to post on
> most other forums on the Internet. So why should sci.math
> be different?
Who has told you or mike3 not to post here?
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Dik T. Winter
...
> Many posters on sci.math, including WM (Wolfgang Muckenheim),
> AP (Archimedes Plutonium), and MR (Mitch Raemsch), are
> nonstandard analysts who do oppose standard analysis.
"nonstandard analysists" is really too much.
> They
> devote numerous threads to their nonstandard beliefs and
> trying to prove standard analysis or ZFC to be inconsistent.
>
> In return, the standard analysts and set theorists consider
> such posters to be "cranks" or "trolls."
Yes, because of their futile attempts to prove ZFC inconsistent.
O, btw, do you know that AP claims that Fermats last theorem is false
because there are solutions to a^n + b^n = c^n for every n? Or has
he dropped that claim by now?
Jesse F. Hughes
> On Mar 25, 5:43 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>> "Jesse F. Hughes" <je...@phiwumbda.org> writes:
>> > That's not oppression or censorship.
>> Bah. It's just that. That not all mathematical texts are available for
>> download for free is nothing but a facet to the underhanded campaign
>> to prevent non-standard mathematicians from posting on Usenet.
>
> Consider a message board or online forum devoted to your
> favorite sports team. Now suppose that, after you post
> there, someone tells you that you can't post there unless
> you've spent money on season tickets for the team.
You've never been told you can't post here unless you buy books.
It has been gently suggested that learning about a subject before
speaking on that subject is a good idea. This is far more obviously
true in mathematics than in your examples, since there is a clear
means of judging correct reasoning and true mathematical claims.
No one will prevent you from posting here, of course. And, if you say
something that betrays ignorance, you will surely see comments to that
effect.
Why do you think that this is bad?
> Or consider a forum devoted to your favorite musical artist,
> but you can't post there unless you own all of the albums
> produced by that artist.
>
> And so on. No one insists that you spend money to post on
> most other forums on the Internet. So why should sci.math
> be different?
--
Jesse F. Hughes
"Women aren't that unpredictable."
"Well, I can't guess what you're getting at, honey."
-- Hitchcock's _Rear Window_
Jesse F. Hughes
> Which of course then raises the question of how exactly I can make
> the large amounts of money I'd need to pursue this. But I've been
> told that money isn't necessarily a barrier, so how do I make it not
> one?
Sorry, I teach philosophy. I know nothing about making money.
--
I thought the wreck was over. I thought the fire was out.
I thought the storm had passed and I was safe at last.
I thought the wreck was over, but here she comes again.
--The Flatlanders
Jesse F. Hughes
> Many posters on sci.math, including WM (Wolfgang Muckenheim),
> AP (Archimedes Plutonium), and MR (Mitch Raemsch), are
> nonstandard analysts who do oppose standard analysis. They
> devote numerous threads to their nonstandard beliefs and
> trying to prove standard analysis or ZFC to be inconsistent.
>
> In return, the standard analysts and set theorists consider
> such posters to be "cranks" or "trolls."
You like straw man arguments, don't you?
You've been told again and again what distinguishes a crank from
respectable finitists, NSA mathematicians and so on, and yet you still
pretend it's some odd sort of religious war.
Your delusions are only somewhat more respectable than JSH's.
--
"At some point in the future history of humanity, AP will eclipse even
Jesus." -- Archimedes Plutonium, 10/21/07
"I wrote those lines because I am not a megalomania [sic] but rather
very humble and down to earth." -- Archimedes Plutonium, 10/22/07
plutonium....@gmail.com
Dik T. Winter wrote:
> In article <2dd0ef2f-a2ff-4e56...@e18g2000yqo.googlegroups.com> lwa...@lausd.net writes:
> ...
> > Many posters on sci.math, including WM (Wolfgang Muckenheim),
> > AP (Archimedes Plutonium), and MR (Mitch Raemsch), are
> > nonstandard analysts who do oppose standard analysis.
>
> "nonstandard analysists" is really too much.
>
> > They
> > devote numerous threads to their nonstandard beliefs and
> > trying to prove standard analysis or ZFC to be inconsistent.
> >
> > In return, the standard analysts and set theorists consider
> > such posters to be "cranks" or "trolls."
>
> Yes, because of their futile attempts to prove ZFC inconsistent.
>
And look at Dik Winter, who believes in Absolute Continuity, yet
Planck proved
in 1900s that Space has holes to the tune of Planck's Constant.
And look at Dik Winter when the Reals are defined as finite leftwards
yet he
never understood that such means the Cartesian Coordinate System and
ZF
are wrong.
No wonder you are old and dumb Dik, for you are too stupid to realize
any of
your mistakes. So hold on to your decayed ZF
> O, btw, do you know that AP claims that Fermats last theorem is false
> because there are solutions to a^n + b^n = c^n for every n? Or has
> he dropped that claim by now?
> --
> dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
> home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
By the way, do you know that Dik cannot even get a correct valid
Euclid Infinitude of Primes
proof for Dik thinks that Euclid's number has several values as his
last attempt several years
ago.
Dik, your just a calculator but when it comes to any theory, you fall
apart.
plutonium....@gmail.com
> --
> "At some point in the future history of humanity, AP will eclipse even
> Jesus." -- Archimedes Plutonium, 10/21/07
> "I wrote those lines because I am not a megalomania [sic] but rather
> very humble and down to earth." -- Archimedes Plutonium, 10/22/07
Quoting out of context, repetitiously is very bad manners, especially
for some "adjunct professor"
allegedly at Bentley College
posting this to news admin for there must be some rules about
signature blocks and quoting out of
context.
So far, Hughes has been too obtuse and stupid to know his bad manners
amy666
yes i was allegorical.
Dik T. Winter
> Dik T. Winter wrote:
...
> > Yes, because of their futile attempts to prove ZFC inconsistent.
>
> And look at Dik Winter, who believes in Absolute Continuity, yet
> Planck proved
> in 1900s that Space has holes to the tune of Planck's Constant.
>
> And look at Dik Winter when the Reals are defined as finite leftwards
> yet he never understood that such means the Cartesian Coordinate System and
> ZF are wrong.
Mathematics is *not* an empirical science.
> By the way, do you know that Dik cannot even get a correct valid
> Euclid Infinitude of Primes proof for Dik thinks that Euclid's number
> has several values as his last attempt several years ago.
What is Euclid's number? And indeed, you are the *only* person in the whole
history of mathematics who is able to get a correct valid proof of the
infinitude of primes.