Orlow cardinality question
Randy Poe
are getting pretty overloaded.
This is a question about the self-consistency of your
intuitions of cardinality (and the theory you're trying
to develop). Here's a question. I'll admit in advance
that it's a trick question.
Consider a set whose elements are {a1, a2, a3}. We've
attached labels 1, 2, 3 to them. It's pretty obvious
that this has a cardinality equal to that of {1,2,3},
right? It doesn't matter whether these elements are
given in order, or whether they even have an order.
I could define a1 = elephant, a2 = tiger, a3 = monkey
and just talk about the set properties in terms of
the labels {a1,a2,a3}.
Obviously the same is true for a collection like
{a1,a2,a3,...,a10}. This is obviously the same
"size" as {1,2,3,...,10}, i.e., cardinality 10.
And it doesn't matter what order I label the
elements, or if they have an order.
Now suppose I have a set that I've labeled
{a1,a2,...} with exactly one label for every element
in the set. Doesn't it seem reasonable and consistent
to say that this is the same size as {1,2,3,...},
i.e. the set of natural numbers? Isn't the idea of
judging the set size by its labeling regardless
of what those labels stand for, an obvious
extension of judging the size of {a1,a2,a3}?
I said this is a trick question. The trick is
this: I have the set of rational numbers in mind
for {a1,a2,...}. Now I know you have a preconception
for what the size of that set is, but try if you
can to approach this without preconceptions, but
from the point of view of extending counting of
things without regard to order. Can you see how
"{a1,a2,a3} is the same size as {1,2,3}" and
"{a1,a2,a3,a4,a5} is the same size as {1,2,3,4,5}"
extends naturally to "{a1,a2,a3,...} is the same
size as {1,2,3,...}"?
In our natural view of counting, we don't think
that order matters. We just assign labels. We
arbitrarily point to the first object and
call it "1". We arbitrarily choose a second
object and call it "2". We don't expect that
shuffling a deck of cards will change its size.
Doesn't it seem reasonable not to impose order
all of a sudden when considering things like N, or Q,
or R?
- Randy
imagin...@despammed.com
Randy Poe wrote:
> New thread just for Tony, because some of those old ones
> are getting pretty overloaded.
Just a couple of comments - I think it really would be a much better
idea to use a different term: it's very clear that Tony's ideas are
incompatible with cardinality as normally defined. "Bigulosity" was a
nice idea.
<snip>
> In our natural view of counting, we don't think
> that order matters. We just assign labels. We
> arbitrarily point to the first object and
> call it "1". We arbitrarily choose a second
> object and call it "2". We don't expect that
> shuffling a deck of cards will change its size.
No, but in our natural view of counting (finite sets) we also expect
other things - for example, that if one set includes another, with some
"left over", then the one with the extra bits is larger. Choosing to
throw away this intuition and keep the ordering one is somewhat
arbitrary. (Of course I know why: because Tony's idea ain't actually
going anywhere, but that would be an abysmal reason for deterring him
from trying.)
Brian Chandler
http://imaginatorium.org
Tony Orlow
numeric elements that are defined by their natural order ala Peano's successor.
I explained long ago that one of the main fallacies in Cantor's approach is
that he uses these mapping functions to create his bijections, and then
entirely ignores the functions as they go to infinity. He makes the implicit
assumption that oo=oo=oo, period, when he inductively extends the bijection to
infinity, declaring both sets the same, because they are both infinite. For any
given number, finite or infinite, in the bijection between naturals and evens,
the corresponding even is twice as large as the natural. If there is some
particular aleph_0, if that term is any more descriptive than oo, then the set
of evens approaches that value at twice the rate and in half the time, as the
naturals. This is precisely the error that the idea of set density adresses,
and that the method I put forth using the inverses of the mapping functions
addresses even better, as a generalization of the concept behind set density,
which only works for linear mapping functions.
--
Smiles,
Tony
Tony Orlow
>
> Randy Poe wrote:
> > New thread just for Tony, because some of those old ones
> > are getting pretty overloaded.
>
> Just a couple of comments - I think it really would be a much better
> idea to use a different term: it's very clear that Tony's ideas are
> incompatible with cardinality as normally defined. "Bigulosity" was a
> nice idea.
word, so don't worry. I will stick to set size. I don't think I want to use
"bigulosity" permanently. Or, maybe I should.....
>
> <snip>
>
> > In our natural view of counting, we don't think
> > that order matters. We just assign labels. We
> > arbitrarily point to the first object and
> > call it "1". We arbitrarily choose a second
> > object and call it "2". We don't expect that
> > shuffling a deck of cards will change its size.
>
> No, but in our natural view of counting (finite sets) we also expect
> other things - for example, that if one set includes another, with some
> "left over", then the one with the extra bits is larger. Choosing to
> throw away this intuition and keep the ordering one is somewhat
> arbitrary. (Of course I know why: because Tony's idea ain't actually
> going anywhere, but that would be an abysmal reason for deterring him
> from trying.)
ideas have already covered much of the area that Cantor defaced. There are only
a couple of questions left before the truly interesting results emerge. This
has all been just a correction.
>
> Brian Chandler
> http://imaginatorium.org
>
>
--
Smiles,
Tony
Randy Poe
imagin...@despammed.com wrote:
> No, but in our natural view of counting (finite sets) we also expect
> other things - for example, that if one set includes another, with
some
> "left over", then the one with the extra bits is larger. Choosing to
> throw away this intuition and keep the ordering one is somewhat
> arbitrary. (Of course I know why: because Tony's idea ain't actually
> going anywhere, but that would be an abysmal reason for deterring him
> from trying.)
OK, I'll stick with "bigulosity".
Now, I think it might actually be interesting to develop
a self-consistent bigulosity, but I don't think what
he's got is self-consistent, and I think it violates
other intuitions.
He says that the bigulosity of the rationals is |N|^2. OK,
fine. But is that bigger than |N|? Well, it certainly looks
bigger because it's the square of |N|, right? But he
hasn't worked out his infinite arithmetic yet. As
a more trivial example, he wants to say that oo
is bigger than oo-1 and smaller than oo+1. But
for many informal proofs, we rely on an intuition
that oo+1 is equal to oo. For instance:
x = 0.3333....
10x = 3.3333... = 3 + x
Doesn't that rely on the notion that the decimal
part of 10x, which has an Orlow bigulosity of
oo-1, is equal to the decimal part of x?
- Randy
Tony Orlow
>
> imagin...@despammed.com wrote:
> > No, but in our natural view of counting (finite sets) we also expect
> > other things - for example, that if one set includes another, with
> some
> > "left over", then the one with the extra bits is larger. Choosing to
> > throw away this intuition and keep the ordering one is somewhat
> > arbitrary. (Of course I know why: because Tony's idea ain't actually
> > going anywhere, but that would be an abysmal reason for deterring him
> > from trying.)
>
> OK, I'll stick with "bigulosity".
>
> Now, I think it might actually be interesting to develop
> a self-consistent bigulosity, but I don't think what
> he's got is self-consistent, and I think it violates
> other intuitions.
intuitions, when you don't even claim your systems satisfies yours? What ARE
your intuitions about infinite sets?
>
> He says that the bigulosity of the rationals is |N|^2. OK,
> fine. But is that bigger than |N|? Well, it certainly looks
> bigger because it's the square of |N|, right? But he
> hasn't worked out his infinite arithmetic yet.
why it's displayed in a 2D grid of N by N terms, right? What does that
contradict, in your view? I mean, you would conclude that one line of N numbers
contains as many numbers as N such lines of N numbers? That is nonsensical bu
ANY intuition.
> As
> a more trivial example, he wants to say that oo
> is bigger than oo-1 and smaller than oo+1. But
> for many informal proofs, we rely on an intuition
> that oo+1 is equal to oo. For instance:
>
> x = 0.3333....
> 10x = 3.3333... = 3 + x
>
> Doesn't that rely on the notion that the decimal
> part of 10x, which has an Orlow bigulosity of
> oo-1, is equal to the decimal part of x?
The decimal part of x? That's a string representing a single value, not a set.
Do you mean the set of all reals in [0,1]? That would be 1/Nth of the entire
range of real numbers, so it would be R/N (aleph_1/aleph_0, to you). It also
appears by other thinking to be equal to N. If we tke both statements together,
we can derive that R=N^2, since R/N=N. This is some indication, perhaps, that
the rationals consitute some sort of enumeration on the reals, though it has
been pointed out that a mapping function of sqrt(n) will produce a set of size
N^2 over the range of N, and that this function does not densely populate the
reals, so that cannot be the size of the reals relative to the naturals. There
is no functional mapping between the discrete infinity and the continuous
infinity.
Now, I don't see how this has anything to do with adding a finite number to
infinity. Maybe you can explain your question better. is there a difference
between including zero in the naturals, and not? This is the difference between
oo and oo+1.
>
> - Randy
>
>
--
Smiles,
Tony
imagin...@despammed.com
Mathematics is open - anyone can play. If you can define something
interesting, people (mathematicians) will want to know. Assuming, of
course, that you understand what mathematics is, and what distinguishes
it from Belgian lace-making or English country dancing. But once you
start off on the "Cantor defaced" line, you mark yourself a crank,
vastly lessening your chances of ever persuading anyone capable of
being genuinely interested in new mathematics that you really have any
ideas at all.
Indidentally, grandiose predictions about the great results coming once
minor questions are resolved are another crank warning sign.
Meanwhile, of the many real mathematicians who have developed
non-standard models (Conway's "surreal numbers", Robinson's nonstandard
analysis) none found it necessary to be unable to understand Cantor's
definitions of cardinality and so on. They could all do real set
theory: the sort in which considering the set of all x such that P(x)
only includes elements x for which P(x) is true.
Brian Chandler
http://imaginatorium.org
Eckard Blumschein
> Randy Poe said:
> It is clearly infinitely bigger, as it has one more infinite dimension. That's
> why it's displayed in a 2D grid of N by N terms, right? What does that
> contradict, in your view? I mean, you would conclude that one line of N numbers
> contains as many numbers as N such lines of N numbers? That is nonsensical bu
> ANY intuition.
What you are calling intuition is noting but the stupid inability to
grasp the essence of the notion infinity. Cantor was also unable to do
so. He wonderd that the number of dimensions does not matter and the
whole universe does not contain more points than any tiny linear
interval: "Je le vois, mais je ne le crois pas."
>
>> As
>> a more trivial example, he wants to say that oo
>> is bigger than oo-1 and smaller than oo+1. But
>> for many informal proofs, we rely on an intuition
>> that oo+1 is equal to oo.
This is not simply an intuition but it exactly describes the essence of
the notion infinity.
> This is some indication, perhaps, that
> the rationals consitute some sort of enumeration on the reals,
Reals are uncountable, not just the number of reals but already every
"single" real "number". In other words, reals are not exactly
enumerable. Otherwise they are not reals but rationals.
Eckard Blumschein
Ed van der Meulen
But Tony plunges the infinity notion as a number in it. The normal numbers are in fact ordinal numbers. And the size is a cardinal number. But Tony doesn't recognize the difference.
For me you are clear. But Tony sees it from a physicist side with physical notions.
Like you are writing a computer program right away. Before thinking how the computer languages will work.
Tony could better post on a philosophy site. He likes to bring a general philosophy.
So he will not understand this.
But your efforts were a good.
Do you know that in the cardinals Aleph-0^n = Aleph_0 for each natural n.
That Aleph_0 is the smallest infinity so the infinity of the even numbers is the same and also of the 1000-folds.
The next infinity is Aleph-1. with nothing in between. That is the cardinality from R. So there's no 1-1 mapping from R to N.
But Tony thinks the natural ordinals are the same as the infinite cardinals.
I thought a cardinal number is normal thing.
You have cardinal numbers 1, 2 and 3 or three ones.
And ordinal numbers first, second, third and in math we write 1, 2, 3 but in fact we are just counting. Never coning to infinity.
You know of mathematics. But Tony speaks a physical language. That is not bad but the wrong language here.
Well oh well
David Kastrup
> You are right Randy.
>
> But Tony plunges the infinity notion as a number in it. The normal
> numbers are in fact ordinal numbers. And the size is a cardinal
> number. But Tony doesn't recognize the difference.
>
> For me you are clear. But Tony sees it from a physicist side with
> physical notions.
You are using "physicist" and "moron" equivalently. I don't think
this is warranted with a sample size that is, after all, rather small.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
Ed van der Meulen
This is an expected answer from you. You have to go to a philosophy site with your idea.
You want to change things that are also from others. Math is not only for you. You have to add things and not to change things.
Don't you understand me?
You can never come to an agreement.
Randy needs some who know more about mathematics. He knows too much of mathematics compared to you Tony.
He already knew you would see it as a trick.
Randy needs more educated mathematicians. You are a physicist thinking physically. And you have to learn much more of physics as well. That is not deductive.
Why don't you believe me?
Ed van der Meulen
In physics we call them labels and we call them ordinals in mathematics. The first object. We count the ordinals as first second third and we write 1, 2, 3. We mean the same.
Ordinals are like labels. Cardinals are the numbers as things. You van have five threes.
This is only a detail.
Robert Kolker
>
>
> What you are calling intuition is noting but the stupid inability to
> grasp the essence of the notion infinity. Cantor was also unable to do
> so. He wonderd that the number of dimensions does not matter and the
> whole universe does not contain more points than any tiny linear
> interval: "Je le vois, mais je ne le crois pas."
Cantor and Dediking get the essence of infinite sets and infite well
orderings just about right. What do YOU have to offer that is better?
If you are not part of the solution you are part of the problem, if
there is a problem
Bob Kolker
Ed van der Meulen
I don't compare scientists with morons I have many friends who are scientist. And we talk a lot about each other interests.
For instance quantum scientists with their complex quantum theories, they like often another theory as well.
Peaple who work on quantum computing. Still in an early stage. Very interesting. I could use that for my artificial inteligence project. But I am too early for them. All open source as well. Checkable.
Theories are models for them. No more that that.
The word Moron doesn't suit in a normal discussion.
I know also cosmoligists. In need for many other theories. Everything looks completrelty new there.
So I don't undersatand you david.
But surely I have a pure mathematical education as well.
So I understand Randy perfectly. He is right in all he has posted.
To Tony:
Tony
I mean it good for you.
You say: "You (that is me) don't want the information".
I don't understand you in the form you present it. I don't talk about the content.
You want to bring a philosophic idea that has its own language and to put in an exact way in mathematics.
In CS you could use it neither in this way. Try to code it in a program. You would at least need a developing phase and often more analyzing as well.
You have nothing of ideas that aren't realizable.
And the way to it is often rephrasing what your idea is not losing the content but adapting the form in which you present it. Using diagrams and other tools. And then arrives the developing phase. And in a real new project you can always come to surprises.
And you can trust people like Randy Poe and Dik de Winter. They know a lot of mathematics. More than you will learn.
So I don't know what your problem is with my postings.
Have a nice drink,
ed
Eckard Blumschein
> Eckard Blumschein wrote:
>
>> What you are calling intuition is noting but the stupid inability to
>> grasp the essence of the notion infinity. Cantor was also unable to do
>> so. He wonderd that the number of dimensions does not matter and the
>> whole universe does not contain more points than any tiny linear
>> interval: "Je le vois, mais je ne le crois pas."
>
> Cantor and Dediking get the essence of infinite sets and infite well
> orderings just about right. What do YOU have to offer that is better?
Fist of all a correct name: Richard Dedekind (1831-1916) and the first
name Georg to Cantor (in order to avoid confusion with M. C.).
I understood their fallacious thinking as follows:
The rational numbers do obviously not contain the irrationals. As also
assumed by Peirce, they are mere potentialities. If they cannot be
addressed numerically, then it should be possible to create them. There
are infinitely more irrational numbers than rational ones. But it is
always possible to find a rational number q between two real ones, etc.
Once again: This thinking is fallacious and ignores the peculiarities of
genuine continuum.
Meanwhile I would like to add a lot to what I already wrote in
http://iesk.et.uni-magdeburg.de/~blumsche/M280.html
Robert Kolker
>
> The rational numbers do obviously not contain the irrationals. As also
> assumed by Peirce, they are mere potentialities.
The notion of potentiality in a mathematical formal context is pure
metaphysical bullshit. Better save that for philosophy, which is a
failure than mathematics which is a success.
> Once again: This thinking is fallacious and ignores the peculiarities of
> genuine continuum.
Here and now. Define a genuine contiuum, carefully, rigorously and don't
leave out any steps.
Bob Kolker
Ed van der Meulen
Just a side product. Or better first rephrasing it. That is better for a philosophy. A philosophy is more abstract. And Tony wants to show a new philosphy based on his believe as Taoist.
And for me he can do that
Randy Poe
Tony Orlow (aeo6) wrote:
> Randy Poe said:
> >
> > imagin...@despammed.com wrote:
> > > No, but in our natural view of counting (finite sets) we also
expect
> > > other things - for example, that if one set includes another,
with
> > some
> > > "left over", then the one with the extra bits is larger. Choosing
to
> > > throw away this intuition and keep the ordering one is somewhat
> > > arbitrary. (Of course I know why: because Tony's idea ain't
actually
> > > going anywhere, but that would be an abysmal reason for deterring
him
> > > from trying.)
> >
> > OK, I'll stick with "bigulosity".
> >
> > Now, I think it might actually be interesting to develop
> > a self-consistent bigulosity, but I don't think what
> > he's got is self-consistent, and I think it violates
> > other intuitions.
> What other intuitions?
Some others you might hold.
> Do you want to claim my system must satisfy your
> intuitions,
No, but you are bothered by a system that doesn't satisfy
YOUR intuitions. There's no reason why it has to satisfy
anybody's. There's no reason why a naive intuition should
govern mathematics.
However the idea of lining things up next to other things,
the counting model of set size, is the motivating intuition
behind set counting by bijection.
What I suspect is that YOU hold intuitions which are
contradicted by your bigulosity notions, that you can't
construct a system which simultaneously satisfies
all of your own intuitions.
> when you don't even claim your systems satisfies yours? What ARE
> your intuitions about infinite sets?
That they're bigger than finite sets.
> > He says that the bigulosity of the rationals is |N|^2. OK,
> > fine. But is that bigger than |N|? Well, it certainly looks
> > bigger because it's the square of |N|, right? But he
> > hasn't worked out his infinite arithmetic yet.
> It is clearly infinitely bigger, as it has one more infinite
dimension.
That is not obvious. You say it is "clear" that
oo x oo > oo, but you are assuming an arithmetic of
oo which is not yet constructed.
> That's
> why it's displayed in a 2D grid of N by N terms, right? What does
that
> contradict, in your view? I mean, you would conclude that one line of
N numbers
> contains as many numbers as N such lines of N numbers?
Sure, because I can count them 1,2,3,...
> > As
> > a more trivial example, he wants to say that oo
> > is bigger than oo-1 and smaller than oo+1. But
> > for many informal proofs, we rely on an intuition
> > that oo+1 is equal to oo. For instance:
> >
> > x = 0.3333....
> > 10x = 3.3333... = 3 + x
> >
> > Doesn't that rely on the notion that the decimal
> > part of 10x, which has an Orlow bigulosity of
> > oo-1, is equal to the decimal part of x?
> The decimal part of x? That's a string representing a single value,
not a set.
Yes, very good. I spoke sloppily in an attempt to construct
an intuitive argument rather than a mathematical one.
Underlying 0.3333... is an infinite sequence, which
is a set of numbers.
0.3333... = lim{0.3, 0.33, 0.333, 0.3333, ...}
> Do you mean the set of all reals in [0,1]?
No. I mean this one infinitely length string. Can we assign
a number to its length, by numbering all of its digits?
Let us number them as a_1, a_2, a_3, ...
How many digits does 0.333... have past the decimal point?
I say it's |N|, because there's exactly one digit for
every n in N. Do you assign it a number?
Now multiply it by 10. We've shifted it to the left by
one digit. So beyond the decimal place, instead of
the digits {a_1, a_2, a_3, ....}, we now have
the digits that were number {a_2, a_3, ...}, with
a_1 being on the other side of the decimal place.
How many digits are left on the right if we take one
of them and shift it to the right?
How many digits are there in a string of length oo
if we delete the first one?
How big is a set {a_1, a_2, a_3, ...} after we delete
the first element?
How long is the sequence {0.3, 0.03, 0.003,...} if we
delete the first element to get {0.03, 0.003, ....}
(yes I know, that's not the same sequence I talked
about before, it's another one associated with the
same number).
- Randy
Ed van der Meulen
You say: "Define a genuine contiuum, carefully, rigorously and don't leave out any steps".
That is not strange, ne?
I am a pure mathematician. And I understand you. That is the way.
But now I look from a scientist position.
My goals is a working tool then. So what is a continuum for me then. Try to see that from a scientist position.
He wants results.
And we as mathematicians we stress make it exact.
I understand you, But I expect that many scientists don't fully agree with you.
have a nice day
Ed van der Meulen
You are right Randy.
Do you know what I would do Tony?
1. I would write a paper about it.
2. let that paper be commented by people who you trust
3. rephrase your paper
4. repeat this till you have a great result,
And you can everyone ask for help.
It's like the info analyzing phase from a project.
You have a kind of problem. You want to make it clear to everyone
I think that is a good way.
have a splendid day. The sun shines again,
ed
Ed van der Meulen
bri...@encompasserve.org
Note well that the phrase that you are responding to is a request for
clarification. It does not endorse or espouse any particular point of
view.
How you think people could disagree with a request for clarification
or how you can say that you understand it when you so blatantly do not
is a mystery.
Note also that you have once again failed to attribute properly.
Once again I request that you either learn to use your newsreader
or stop using it altogether.
John Briggs
Ed van der Meulen
Please don't become angry with me.
I defended here a physicist standpoint. Do you see that as bad from me? You are not the only one in this world.
Do you know what scimath means.
Please stay always correct.
Robert Kolker
>
> I am a pure mathematician. And I understand you. That is the way.
You don't know what formalizing a system is, specifying its primitive
terms and giving its axioms? What kind of a mathematician are you? I am
looking for an axiom system somewhat analogous to that which Hilbert
formulated for 3-d euclidean geometry. I wan't to see Bluemschein put up
or shut up.
A continuum is an abstract entity defined in the context of a
mathematical system have the components I have mentioned. I want to see
Bluemschiens definition of a "genuine continuum" not to be confused with
the continua currently defined.
I am sick unto death of people with philosophical pretensions taking
potshots at mathematicals theories that have been in use over a hundred
years, have been certified contradiction free by top mathematicians and
that have produced useful theorems and applications. These philosphical
weenies are dead weight and ballest on the human race.
Bob Kolker
Ed van der Meulen
Hou are you now?
You know that I never have heard of formalizing.
How could you know that? I have studied also proof theory. Also about our great friend Gödel.
But you know me better than I know me?
You have a special brain for that
Please have a feast. I have a birthday feast tonight.
Are you so sour about my posting? Scientists are also people. Don't you know them?
Ed van der Meulen
Mathematics is great subject. About what do you want to talk.
-Analyses
-number theory
-basics
-mathematical games.
Please tell me what you prefer.
Do you know there people who talk about nothing. And that all the day?
Do you like a drink?
Ed van der Meulen
And all those goods that mathematicians have produced will stay. No one can take that away. They can only add things.
So Bob I am also with you.
Ed van der Meulen
Give people some freedom. Everybody has his or her own interest. And you know a lot of me, that I don't know.
Please let us stay friends.
Nothing in mathematics will go away. It will only grow. Do you know of discrete math?
Look in Physics are the laws of Newton. But also the quantum theories. Certainly not so easy. Axiomatized in the meta theory above it. the so called MM. meta-model. So formalized. A very big job.
But what do you think they use in the life sciences? Just Newton.
Where are you afraid of.
For analysing jobs is our good math very needed.
For instance the tools that come from decrete math are for only simulations and teaching and that kind of things. But our traditinal math will be needed for sure.
Do you think we can throw away the analizing job?
I think everybody defends his here his or her interest.
Alan Morgan
Randy Poe <poespa...@yahoo.com> wrote:
>
>imagin...@despammed.com wrote:
>> No, but in our natural view of counting (finite sets) we also expect
>> other things - for example, that if one set includes another, with
>some
>> "left over", then the one with the extra bits is larger. Choosing to
>> throw away this intuition and keep the ordering one is somewhat
>> arbitrary. (Of course I know why: because Tony's idea ain't actually
>> going anywhere, but that would be an abysmal reason for deterring him
>> from trying.)
>
>OK, I'll stick with "bigulosity".
>
>Now, I think it might actually be interesting to develop
>a self-consistent bigulosity, but I don't think what
>he's got is self-consistent, and I think it violates
>other intuitions.
Can you compare the bigulousity of two sets that have no
elements in common? I'm thinking of the odd numbers and
the even numbers here. Or, even better, the positive odd
numbers and the negative odd numbers.
>He says that the bigulosity of the rationals is |N|^2. OK,
>fine. But is that bigger than |N|?
Since the rationals contain the integers doesn't that mean
that the bigulousity of the rationals is greater than the
bigulousity of the integers by definition?
>Well, it certainly looks
>bigger because it's the square of |N|, right? But he
>hasn't worked out his infinite arithmetic yet. As
>a more trivial example, he wants to say that oo
>is bigger than oo-1 and smaller than oo+1. But
>for many informal proofs, we rely on an intuition
>that oo+1 is equal to oo. For instance:
>
> x = 0.3333....
> 10x = 3.3333... = 3 + x
>
>Doesn't that rely on the notion that the decimal
>part of 10x, which has an Orlow bigulosity of
>oo-1, is equal to the decimal part of x?
So we have cardinal arithmetic, ordinal arithmetic, and bigulous
arithmetic. Just make sure you know which one you are doing.
Alan
--
Defendit numerus
Tony Orlow
some problems in Cantor, and some solutions. It's no coincidence that WM and I
were arguing the same point regarding the size of the set of naturals vs. their
values. Unfortunately I think he sees them both as finite, so we both disagree
with Cantor, and with each other at the same time. What fun!
>
> Indidentally, grandiose predictions about the great results coming once
> minor questions are resolved are another crank warning sign.
Perhaps, but you don't see what I see. It's hard to put my diagrams here for
you. I am working on it and have high hopes. I don't think I'm a crank. I
usually end up being right in such cases and people shake their heads, and I
move on. Such is life.
>
> Meanwhile, of the many real mathematicians who have developed
> non-standard models (Conway's "surreal numbers", Robinson's nonstandard
> analysis) none found it necessary to be unable to understand Cantor's
> definitions of cardinality and so on. They could all do real set
> theory: the sort in which considering the set of all x such that P(x)
> only includes elements x for which P(x) is true.
They were professional mathematicians, and obligated to study this wonder of
mathematics and be an expert. I am not a professional, and therefore am free to
express my distaste for whatever I want. I don't have to please the Oscar
committee. I learned the basics in school, enough to ace the test, but the
conclusions all seemed iffy at best, and downright incorrect at worst. Besides,
I really see all the logical loopholes and shell games in it as contributing to
the deliquency of mathematics. Conway and Robinson probably never tread on the
same ground as Cantor, so as not to contradict his established mathematics. My
territory apparently overlaps with Cantor's significantly, and in an
incompatible way. Oh well. Sorry.
>
> Brian Chandler
> http://imaginatorium.org
>
>
>
> > >
> > > Brian Chandler
> > > http://imaginatorium.org
> > >
> > >
> >
> > --
> > Smiles,
> >
> > Tony
>
>
--
Smiles,
Tony
Tony Orlow
> You are right Randy.
>
> But Tony plunges the infinity notion as a number in it. The normal numbers are in fact ordinal numbers. And the size is a cardinal number. But Tony doesn't recognize the difference.
>
> For me you are clear. But Tony sees it from a physicist side with physical notions.
wrong with checking math against the world.
>
> Like you are writing a computer program right away. Before thinking how the computer languages will work.
>
> Tony could better post on a philosophy site. He likes to bring a general philosophy.
mathematical subject matter.
>
> So he will not understand this.
>
> But your efforts were a good.
>
> Do you know that in the cardinals Aleph-0^n = Aleph_0 for each natural n.
a natural, like rational numbers are pairs of naturals, and therefore have
aleph_0^2 elements in its grid. This is my perspective.
>
> That Aleph_0 is the smallest infinity so the infinity of the even numbers is the same and also of the 1000-folds.
number equal to the upper bound of counting numbers, which is essentially an
infinite sum of 1's. The harmonic series is also infinite, but every term is
less than half the corresponding term in the sum of 1's, so that is a smaller
infinity. The smallest infinity I can imagine is the sum of the inverses of
primes. Gee, why didn't Cantor think of that?
>
> The next infinity is Aleph-1. with nothing in between. That is the cardinality from R. So there's no 1-1 mapping from R to N.
logically from anything, but simply declared, and wrongly so. I have shown that
there are many infinities between the size of the naturals and the size of the
reals, and smaller than the size of the naturals, and bigger than the set of
reals (but Cantor already knew that last part because he treated the continuum
doifferently from the discrete infinity).
>
> But Tony thinks the natural ordinals are the same as the infinite cardinals.
>
> I thought a cardinal number is normal thing.
>
> You have cardinal numbers 1, 2 and 3 or three ones.
>
> And ordinal numbers first, second, third and in math we write 1, 2, 3 but in fact we are just counting. Never coning to infinity.
like an unsigned integer in a computer used as an array index. The math is the
same. Ordinals define order, but cardinal derive their value from their place
int he order of counting. I have never seen any important operational
difference, but perhaps there is one.
>
> You know of mathematics. But Tony speaks a physical language. That is not bad but the wrong language here.
>
> Well oh well
>
--
Smiles,
Tony
Ed van der Meulen
a set S
Formally this is in a context. We could place braces around the texts. Talking in a meta language.
This is usual in proof theory.
N is a sleeping U, bottom left, from S. That means all natural numbers are also in S
There is one element with name q such that all n element from N: n < q.
In the James Bond Films was also a q.
We have a set that includes a member infinity. Too easy. I know. But effective.
Then we are looking at the consequences. Maybe we have to add more axioms. Beforehand we don't know it. So testing follows.
And as soon as have q we see aleph_0 is the same. Nobody can have problems with this. Please don't discuss anymore the possibility that axioms can give infinity. Choice we have now.
Would there be some scientist who isn't satisfied. Well, they can take the set without infinity.
And continuity. In a strict way look at articles about that. If you don't want to use the name of Cantor. Also Okay.
And if you want to use it. Great.
When I would be a scientists. I would take...
And I would reason computers are never precise. Floats they have with modulo. Losing information. A necessary consequence.
And our pi has there only a limited precision, it has been reduced to a rational number. Boy oh boy those bad computers.
But when you have to work with it. What then? I like also physics but I have a pure mathematical education. But I would do that in that way.
Do you see that Bob with margins. Not at all precise.
And they think, the throw away of information, isn't so bad. To-morrow we have a better model.
How would you do that in a somewhat shabby way Bob. I have told you how I would do it. At least the first step. Shabby is also easier. I know that very well. But when something doesn't need to be precise, why not?
For real analizing things you have to be precise. But in the adventurous landscape of many sciences. We walk through it, can you sing a song? And suddenly we have medicines, the plane, Missiles. Computers. And we can use them.
They measure everything with a plus/mines, a margin.
Who bakes our bread. What are all those people doing. Nobody can be missed. You neither, with your brain, Bob, be proud of it.
Bob do you know how the models of macro-economists work. That is a real horror. Full of assumptions. Please go help them. For they advice the politicians.
Maybe you are angry with the wrong people.
Tony Orlow
> On 5/18/2005 7:58 PM, Tony Orlow (aeo6) wrote:
> > Randy Poe said:
>
> > It is clearly infinitely bigger, as it has one more infinite dimension. That's
> > why it's displayed in a 2D grid of N by N terms, right? What does that
> > contradict, in your view? I mean, you would conclude that one line of N numbers
> > contains as many numbers as N such lines of N numbers? That is nonsensical bu
> > ANY intuition.
>
> What you are calling intuition is noting but the stupid inability to
> grasp the essence of the notion infinity. Cantor was also unable to do
> so. He wonderd that the number of dimensions does not matter and the
> whole universe does not contain more points than any tiny linear
> interval: "Je le vois, mais je ne le crois pas."
each line of N numbers and treat them like a unit. If I have N such units, that
is clearly infinitely larger than 1 such unit. Now, you can tell me I "can't"
do this, but you had better have some justification, besides "it's just not
done". What is wrong with this idea, besides the fact that you didn't think of
it yourself?
>
> >
> >> As
> >> a more trivial example, he wants to say that oo
> >> is bigger than oo-1 and smaller than oo+1. But
> >> for many informal proofs, we rely on an intuition
> >> that oo+1 is equal to oo.
>
> This is not simply an intuition but it exactly describes the essence of
> the notion infinity.
words, your position is not that arithmetic is different on infinite scales,
but that it is simply irrelevant. That's simply not true. Infinite units may be
defined and nested as much as desired, and finite differences can be
considered, even ifinitely far from the origin.
>
>
>
> > This is some indication, perhaps, that
> > the rationals consitute some sort of enumeration on the reals,
>
> Reals are uncountable, not just the number of reals but already every
> "single" real "number". In other words, reals are not exactly
> enumerable. Otherwise they are not reals but rationals.
rrepresented as an ifninite string of N bits, exactly. Define what you mean by
enumeration. On the subject of countability, it's irrelevant to my system,
because my system doesn't rely on anything so ill-suited to studying infinities
as counting.
>
> Eckard Blumschein
>
>
>
--
Smiles,
Tony
Tony Orlow
>
> I don't compare scientists with morons I have many friends who are scientist. And we talk a lot about each other interests.
I'm not. I am an information scientist with interest in many areas of science.
I have never been called a moron by anyone that knows me, not even remotely.
>
> For instance quantum scientists with their complex quantum theories, they like often another theory as well.
>
> Peaple who work on quantum computing. Still in an early stage. Very interesting. I could use that for my artificial inteligence project. But I am too early for them. All open source as well. Checkable.
>
> Theories are models for them. No more that that.
>
> The word Moron doesn't suit in a normal discussion.
>
> I know also cosmoligists. In need for many other theories. Everything looks completrelty new there.
>
> So I don't undersatand you david.
>
> But surely I have a pure mathematical education as well.
>
> So I understand Randy perfectly. He is right in all he has posted.
>
> To Tony:
> Tony
>
> I mean it good for you.
>
> You say: "You (that is me) don't want the information".
>
> I don't understand you in the form you present it. I don't talk about the content.
talking about. This seems to be an almost universal problem here.
>
> You want to bring a philosophic idea that has its own language and to put in an exact way in mathematics.
they're wrong.
>
> In CS you could use it neither in this way. Try to code it in a program. You would at least need a developing phase and often more analyzing as well.
>
> You have nothing of ideas that aren't realizable.
>
> And the way to it is often rephrasing what your idea is not losing the content but adapting the form in which you present it. Using diagrams and other tools. And then arrives the developing phase. And in a real new project you can always come to surprises.
>
> And you can trust people like Randy Poe and Dik de Winter. They know a lot of mathematics. More than you will learn.
>
> So I don't know what your problem is with my postings.
considered what I've said, since you never refer to it specifically.
>
> Have a nice drink,
You too.
>
> ed
>
--
Smiles,
Tony
Tony Orlow
and insisted that i must accept Cantor. He cut me off right here in sci.math
for not agreeing that a set of finite distinct naturals could be infinte. I
submitted the paper to a mathematician friend of mine, but I think his computer
couldn't handle the Word file. It is best for now to spar like this and prepare
myself for all the complaints, and perhaps make a buzz. I will be getting back
to the new axiom set soon, since people need to have those in order to use
their bitwise minds, as well as work on my spatially-oriented visual proofs and
demonstrations. The paper will come out eventually in better form. People will
still hate it. I don't care. I still have my day job. And Istill feel
accomplished, even if it's not recognized. That happens a lot. I'm not proud.
--
Smiles,
Tony
Tony Orlow
>
> Hou are you now?
>
> You know that I never have heard of formalizing.
>
> How could you know that? I have studied also proof theory. Also about our great friend Gödel.
>
> But you know me better than I know me?
isn't it?
>
> You have a special brain for that
>
> Please have a feast. I have a birthday feast tonight.
>
> Are you so sour about my posting? Scientists are also people. Don't you know them?
>
--
Smiles,
Tony
Ed van der Meulen
> Before thinking how the computer languages will work.
> I have been thinking about this for 25 years.
I also haver given you a well meant advice.
Okay. There was in the p20th century a grerat mathematisions Alan Church who developed the so called lambda calculus. Computer languages: Lisp and Haskel. (from Haskel B. Curry also a lambda calculas man)
And the lamdas calculus is equivalent to the normal set theory.
But now thge special thing. Chirch proved also that the lammda calculus equivalent was to the Turing machine. And I think you know what that means.
This meansd when you could make a computer program af waht you want to do. That is equivalent to make it in mathematics. And that is a test as well.
So could you make a software program that simulates what you want to reach. Then you have a consistent story as well. If not possible then you know that you have the way to rephrasing what you want. I could help you with that. But others can help you as well.
So you have now two possibilities.
I mean it good towards you.
ed
imagin...@despammed.com
> > > ideas have already covered much of the area that Cantor defaced.
> > There are only
> > > a couple of questions left before the truly interesting results
> > emerge. This
> > > has all been just a correction.
> >
> > Mathematics is open - anyone can play. If you can define something
> > interesting, people (mathematicians) will want to know. Assuming,
of
> > course, that you understand what mathematics is, and what
distinguishes
> > it from Belgian lace-making or English country dancing. But once
you
> > start off on the "Cantor defaced" line, you mark yourself a crank,
> > vastly lessening your chances of ever persuading anyone capable of
> > being genuinely interested in new mathematics that you really have
any
> > ideas at all.
> I don't think I ever used the term "defaced",
... OK: Look up to the line separated by asterisks. That's you.
> ... but I certainly have pointed out
> some problems in Cantor, and some solutions.
Well, yes. Perceived problems. But many people here have seen exactly
these same "problems" pointed out over and over again, with exactly the
same flaws (basically, there's *always* a quantifier swap) in the
logic. There was a chap called Phil a year or three back, who
approached things almost exactly as you have. Couldn't accept that
(our, finite) natural numbers are not a finite set, and drew uncannily
similar "numbers" consisting of digits and dots in ever-increasingly
baroque (and ill-defined) combinations. So if you think you have
already done something original, you are plainly mistaken.
It's no coincidence that WM and I
> were arguing the same point regarding the size of the set of naturals
vs. their
> values. Unfortunately I think he sees them both as finite, so we both
disagree
> with Cantor, and with each other at the same time. What fun!
>
> >
> > Indidentally, grandiose predictions about the great results coming
once
> > minor questions are resolved are another crank warning sign.
> Perhaps, but you don't see what I see. It's hard to put my diagrams
here for
> you. I am working on it and have high hopes. I don't think I'm a
crank. I
> usually end up being right in such cases and people shake their
heads, and I
> move on. Such is life.
Pity if you get so keyed up about it that you don't learn something out
of the debris. You've got as far as having "boundless sets". Why don't
you try to formalise the difference between a "boundless set" and an
"infinite set", instead of responding like you did to my making this
suggestion the other day.
> > Meanwhile, of the many real mathematicians who have developed
> > non-standard models (Conway's "surreal numbers", Robinson's
nonstandard
> > analysis) none found it necessary to be unable to understand
Cantor's
> > definitions of cardinality and so on. They could all do real set
> > theory: the sort in which considering the set of all x such that
P(x)
> > only includes elements x for which P(x) is true.
> They were professional mathematicians, and obligated to study this
wonder of
> mathematics and be an expert. I am not a professional, and therefore
am free to
> express my distaste for whatever I want. I don't have to please the
Oscar
> committee. I learned the basics in school, enough to ace the test,
but the
> conclusions all seemed iffy at best, and downright incorrect at
worst.
Hmm, you might well get crank points for that - "acing the test" while
knowing it's all 'wrong'. Look, Conway's numbers are a different system
from the conventional real numbers of set theory (they are a proper
class, for a start), so of course lots of things are true in surreals
and not in Cantor set theory, and vice versa. That's different from
claiming that propositions with a clear and simple proof in some system
are "wrong".
> Besides,
> I really see all the logical loopholes and shell games in it as
contributing to
> the deliquency of mathematics. Conway and Robinson probably never
tread on the
> same ground as Cantor, so as not to contradict his established
mathematics. My
> territory apparently overlaps with Cantor's significantly, and in an
> incompatible way. Oh well. Sorry.
So I think your claimed distinction here just isn't.
Brian Chandler
http://imaginatorium.org
Tony Orlow
>
> Tony Orlow (aeo6) wrote:
> > Randy Poe said:
> > >
> > > imagin...@despammed.com wrote:
> > > > No, but in our natural view of counting (finite sets) we also
> expect
> > > > other things - for example, that if one set includes another,
> with
> > > some
> > > > "left over", then the one with the extra bits is larger. Choosing
> to
> > > > throw away this intuition and keep the ordering one is somewhat
> > > > arbitrary. (Of course I know why: because Tony's idea ain't
> actually
> > > > going anywhere, but that would be an abysmal reason for deterring
> him
> > > > from trying.)
> > >
> > > OK, I'll stick with "bigulosity".
> > >
> > > Now, I think it might actually be interesting to develop
> > > a self-consistent bigulosity, but I don't think what
> > > he's got is self-consistent, and I think it violates
> > > other intuitions.
> > What other intuitions?
>
> Some others you might hold.
or conclusions violate your intuitions, in absence of Cantor.
>
> > Do you want to claim my system must satisfy your
> > intuitions,
>
> No, but you are bothered by a system that doesn't satisfy
> YOUR intuitions. There's no reason why it has to satisfy
> anybody's. There's no reason why a naive intuition should
> govern mathematics.
should not contradict each other logically, and derive different results for
the same problems, don't you think?
>
> However the idea of lining things up next to other things,
> the counting model of set size, is the motivating intuition
> behind set counting by bijection.
between the finite and infinite. I choose different approaches.
>
> What I suspect is that YOU hold intuitions which are
> contradicted by your bigulosity notions, that you can't
> construct a system which simultaneously satisfies
> all of your own intuitions.
conversations here I have been motivated to expand on them and state them
clearly, which has been surprisingly easy. What intutitions do you think I
might violate, in you or me?
>
> > when you don't even claim your systems satisfies yours? What ARE
> > your intuitions about infinite sets?
>
> That they're bigger than finite sets.
> > > He says that the bigulosity of the rationals is |N|^2. OK,
> > > fine. But is that bigger than |N|? Well, it certainly looks
> > > bigger because it's the square of |N|, right? But he
> > > hasn't worked out his infinite arithmetic yet.
> > It is clearly infinitely bigger, as it has one more infinite
> dimension.
>
> That is not obvious. You say it is "clear" that
> oo x oo > oo, but you are assuming an arithmetic of
> oo which is not yet constructed.
rectangle of size m by n units, the product being the area in square units. If
you multiply two infinities, it is not different. For instance, the grid or
real numbers which Cantor tries (unsuccessfully) to traverse diagonally is N
bits across, and R numbers long, so the number of digits in that grid is N*R. I
don't see any limit to this construction. Do you?
>
> > That's
> > why it's displayed in a 2D grid of N by N terms, right? What does
> that
> > contradict, in your view? I mean, you would conclude that one line of
> N numbers
> > contains as many numbers as N such lines of N numbers?
>
> Sure, because I can count them 1,2,3,...
diagonally. No matter how far you go, you can only have covered half of any
grid whose corners on top and left have been reached. You never get to the
bottom right corner. Does that ever bother you?
Better to take each linear direction in the grid as a separate dimension,
especially as each one corresponds to one natural number of the pair that
constitutes each rational number.
>
> > > As
> > > a more trivial example, he wants to say that oo
> > > is bigger than oo-1 and smaller than oo+1. But
> > > for many informal proofs, we rely on an intuition
> > > that oo+1 is equal to oo. For instance:
> > >
> > > x = 0.3333....
> > > 10x = 3.3333... = 3 + x
> > >
> > > Doesn't that rely on the notion that the decimal
> > > part of 10x, which has an Orlow bigulosity of
> > > oo-1, is equal to the decimal part of x?
> > The decimal part of x? That's a string representing a single value,
> not a set.
>
> Yes, very good. I spoke sloppily in an attempt to construct
> an intuitive argument rather than a mathematical one.
>
> Underlying 0.3333... is an infinite sequence, which
> is a set of numbers.
>
> 0.3333... = lim{0.3, 0.33, 0.333, 0.3333, ...}
>
> > Do you mean the set of all reals in [0,1]?
>
> No. I mean this one infinitely length string. Can we assign
> a number to its length, by numbering all of its digits?
> Let us number them as a_1, a_2, a_3, ...
>
> How many digits does 0.333... have past the decimal point?
> I say it's |N|, because there's exactly one digit for
> every n in N. Do you assign it a number?
Yes, i also assign it N.
>
> Now multiply it by 10. We've shifted it to the left by
> one digit. So beyond the decimal place, instead of
> the digits {a_1, a_2, a_3, ....}, we now have
> the digits that were number {a_2, a_3, ...}, with
> a_1 being on the other side of the decimal place.
>
> How many digits are left on the right if we take one
> of them and shift it to the right?
N. You have just multiplied by the number base which increases the significant
digit count by one. There are actually infinite digits to the left as well, but
in the first they are all zeroes. By mutiplying by the number base, you turned
a zero digit to the left to a non-zero digit. It didn't increase the overall
infinity of digits.
>
> How many digits are there in a string of length oo
> if we delete the first one?
oo-1, by that description, but that has nothing to do with multiplication.
>
> How big is a set {a_1, a_2, a_3, ...} after we delete
> the first element?
one less.
>
> How long is the sequence {0.3, 0.03, 0.003,...} if we
> delete the first element to get {0.03, 0.003, ....}
> (yes I know, that's not the same sequence I talked
> about before, it's another one associated with the
> same number).
They all have infinite digits to either side. You are simply changing the
values of the digits arithmetically.
>
> - Randy
>
>
--
Smiles,
Tony
Tony Orlow
aTuring machine. It seems to have far more qualitative properties and relations
ships than normal set theory. Can you say how they correspond, to you
understanding? I don't wuite get it. Thanks.
--
Smiles,
Tony
Tony Orlow
> In article <1116437898.7...@g14g2000cwa.googlegroups.com>,
> Randy Poe <poespa...@yahoo.com> wrote:
> >
> >imagin...@despammed.com wrote:
> >> No, but in our natural view of counting (finite sets) we also expect
> >> other things - for example, that if one set includes another, with
> >some
> >> "left over", then the one with the extra bits is larger. Choosing to
> >> throw away this intuition and keep the ordering one is somewhat
> >> arbitrary. (Of course I know why: because Tony's idea ain't actually
> >> going anywhere, but that would be an abysmal reason for deterring him
> >> from trying.)
> >
> >OK, I'll stick with "bigulosity".
> >
> >Now, I think it might actually be interesting to develop
> >a self-consistent bigulosity, but I don't think what
> >he's got is self-consistent, and I think it violates
> >other intuitions.
>
> Can you compare the bigulousity of two sets that have no
> elements in common? I'm thinking of the odd numbers and
> the even numbers here. Or, even better, the positive odd
> numbers and the negative odd numbers.
functions that define them are compared. Any two sets that are both defined as
functions on the naturals, or both as functions on the reals, can be compared
by comparing the inverses of their mapping functions. All the sets you
mentioned have size N/2, and are therefore equal in size.
>
> >He says that the bigulosity of the rationals is |N|^2. OK,
> >fine. But is that bigger than |N|?
>
> Since the rationals contain the integers doesn't that mean
> that the bigulousity of the rationals is greater than the
> bigulousity of the integers by definition?
infinitesimal fraction of the naturals, a tiny proper subset, and obviously not
equivalent in size, in my eyes.
>
> >Well, it certainly looks
> >bigger because it's the square of |N|, right? But he
> >hasn't worked out his infinite arithmetic yet. As
> >a more trivial example, he wants to say that oo
> >is bigger than oo-1 and smaller than oo+1. But
> >for many informal proofs, we rely on an intuition
> >that oo+1 is equal to oo. For instance:
> >
> > x = 0.3333....
> > 10x = 3.3333... = 3 + x
> >
> >Doesn't that rely on the notion that the decimal
> >part of 10x, which has an Orlow bigulosity of
> >oo-1, is equal to the decimal part of x?
>
> So we have cardinal arithmetic, ordinal arithmetic, and bigulous
> arithmetic. Just make sure you know which one you are doing.
LOL thanks Alan. It's really just a matter of choosing a unit infinity and
comparing things to that as one would do with finites. After all, don't we do
that on the finite level already, choose a unit and work with it consistently?
>
> Alan
>
--
Smiles,
Tony
Ed van der Meulen
-- DO NOT REPLY TO THIS MESSAGE --
A message has been posted to the discussion "sci.math".
Author: TonyBones
Subject: Re: Orlow cardinality question
Ed van der Meulen said:
> Don't make it to difficult for Tony. He is a good guy.
>
> You are right Randy.
>
> Do you know what I would do Tony?
>
> 1. I would write a paper about it.
>
> 2. let that paper be commented by people who you trust
>
> 3. rephrase your paper
>
> 4. repeat this till you have a great result,
>
> And you can everyone ask for help.
>
> It's like the info analyzing phase from a project.
>
> You have a kind of problem. You want to make it clear to everyone
>
> I think that is a good way.
>
> have a splendid day. The sun shines again,
>
> ed
>
I started out with a paper. Dave Rusin couldn't get past the first two pages, and insisted that i must accept Cantor. He cut me off right here in sci.math for not agreeing that a set of finite distinct naturals could be infinite.
First lets establish this. You have the natural numbers and then say that is N and N is infinite. Do you mean that?
We are going now very slow. Then we forget nothing. Step by step. Okay?
The fist step.
1. But what does it mean you HAVE such an infinite line. In which form do we HAVE that.
Do you understand me?
What is "having a line" that we can take it and make a bundle of it.
The line with numbers and stripes from your professor was an activity to do. You didn't HAVE that line. My peano staircase was also for go up the stairs. You didn't HAVE that Peano staircase.
Could we now concentrate on this one notion of HAVING A LINE.
How can we have that line in the way that we can work with it.
So only the question how would you describe that notion of having a line.
We are trying now to answer only this question. The rest has to wait.
Let the sun shine Tony,
ed
Tony Orlow
and thought I never claimed to deface him. Yes, I think he created a
problematic system that many feel is a blemish, but is very difficult to undo.
Sorry.
>
> > ... but I certainly have pointed out
> > some problems in Cantor, and some solutions.
>
> Well, yes. Perceived problems. But many people here have seen exactly
> these same "problems" pointed out over and over again, with exactly the
> same flaws (basically, there's *always* a quantifier swap)
If it was, I wouldn't mind seeing specifics.
> in the
> logic. There was a chap called Phil a year or three back, who
> approached things almost exactly as you have. Couldn't accept that
> (our, finite) natural numbers are not a finite set, and drew uncannily
> similar "numbers" consisting of digits and dots in ever-increasingly
> baroque (and ill-defined) combinations. So if you think you have
> already done something original, you are plainly mistaken.
does. I know now what a tenacious bunch you all are, and I can see how the
training in this system has adversely affected the way people think about these
things. it's time for the axiomatics to get a little picture training, and stop
thinking exclusively in words. That's not how the structure of benzene was
discovered.
>
> It's no coincidence that WM and I
> > were arguing the same point regarding the size of the set of naturals
> vs. their
> > values. Unfortunately I think he sees them both as finite, so we both
> disagree
> > with Cantor, and with each other at the same time. What fun!
> >
> > >
> > > Indidentally, grandiose predictions about the great results coming
> once
> > > minor questions are resolved are another crank warning sign.
> > Perhaps, but you don't see what I see. It's hard to put my diagrams
> here for
> > you. I am working on it and have high hopes. I don't think I'm a
> crank. I
> > usually end up being right in such cases and people shake their
> heads, and I
> > move on. Such is life.
>
> Pity if you get so keyed up about it that you don't learn something out
> of the debris. You've got as far as having "boundless sets". Why don't
> you try to formalise the difference between a "boundless set" and an
> "infinite set", instead of responding like you did to my making this
> suggestion the other day.
cheek response to the need to have infinite sets of distinct finite natural
numbers, which is mathematically impossible outside of set theory. My point was
this: if you want to violate the rules of numbers this way, please make up a
name that doesn't have a meaning outside of your theory, and resolve your
anomaly so we don't fight about this nonsense.
>
> > > Meanwhile, of the many real mathematicians who have developed
> > > non-standard models (Conway's "surreal numbers", Robinson's
> nonstandard
> > > analysis) none found it necessary to be unable to understand
> Cantor's
> > > definitions of cardinality and so on. They could all do real set
> > > theory: the sort in which considering the set of all x such that
> P(x)
> > > only includes elements x for which P(x) is true.
> > They were professional mathematicians, and obligated to study this
> wonder of
> > mathematics and be an expert. I am not a professional, and therefore
> am free to
> > express my distaste for whatever I want. I don't have to please the
> Oscar
> > committee. I learned the basics in school, enough to ace the test,
> but the
> > conclusions all seemed iffy at best, and downright incorrect at
> worst.
>
> Hmm, you might well get crank points for that - "acing the test" while
> knowing it's all 'wrong'.
rules. I can play chess. But, chess isn't supposed to be an exact description
of anything. If something is supposed to be "true", then I evaluate it against
other truths, bith for similarity and difference, and keep my eyes out for
contradictions in my understanding.
> Look, Conway's numbers are a different system
> from the conventional real numbers of set theory (they are a proper
> class, for a start), so of course lots of things are true in surreals
> and not in Cantor set theory, and vice versa. That's different from
> claiming that propositions with a clear and simple proof in some system
> are "wrong".
Mostly I take this stance because whenever I present my ideas, Cantorians tell
me I am "wrong". It has happened many times here. You don't want me to attack
your system? Don't call it "correct", expecially when it is at odds with so
many other areas of math, not to mention mathematical intuition, as well as the
rules cocnerning finite sets, like proper subsets are smaller. If you want to
claim your axioms are "correct" without justifying them, then I have the right
to show that ina larger context this little system is totally confused. I mean,
I am surprised you folks don't take it all the way into town, and prove that
there are MORE naturals than rationals. What a triumph that would be!
>
> > Besides,
> > I really see all the logical loopholes and shell games in it as
> contributing to
> > the deliquency of mathematics. Conway and Robinson probably never
> tread on the
> > same ground as Cantor, so as not to contradict his established
> mathematics. My
> > territory apparently overlaps with Cantor's significantly, and in an
> > incompatible way. Oh well. Sorry.
>
> So I think your claimed distinction here just isn't.
That sentence no objective clause. What means?
Ed van der Meulen
With set theory you can do also much. The same as with computers. They are also equavalent.
For many mathematicians all consists out of sets.
The is a very basic notion. And you can alo have very different types of sets. But that is another story.
Ed van der Meulen
For the real proof you have to read it. It's not a small proof from a few lines.
But informal I can tell you this. Look at the Turing machine with shelves or tapes or whatever. Do you have that picture?
Then think than all pieces of cake are trees that grow very fast. Each tick is a status of the whole machine and all the trees have grown and are to see.
You can make a picture for each tick. Then the Turing machine goes from one state to the following.
Can you follow me as well?
But now this. Compare the states with grown trees. And apes jump from old grown trees to new grown trees.
Are you following me?
Well. Informal we are ready. Do things please always first informal. You will get a better touch with it. The Bourbaki school wanted to do it formal in all details and they didn't succeed.
The apes are the lambda functions with an input "state" and an out put "state" tree.
Drop the apes and the Turing machine works.
Or drop the trees and you see the apes jump (together to a new state).
And the same pattern we see then.
It's like in photography. A negative and a positive picture. Or in paintings the figure and the rest figure. You can see it as a flip-flop. Turing-Lambda calculus.
This is a link with people who work on the lambda calculus. A huge job as well and it cost many years.
http://en.wikipedia.org/wiki/Lambda_calculus
and
http://ling.ucsd.edu/~barker/Lambda/
There's more on the net to find.
When you want to make the Turing machine compared with the lambda calculus formal. Then you describe first exactly what happens with the Turing machine and the fast growing trees. and you are using just the same language.
When you have done that you take a distance and you make a theory above this model and there you use a more mathematical and exact language. This is the real step we have to take when we make in formal. You will met prime events in the model. And you will notice compound actions (events) are built up from your prime actions.
Then you are on your way
This is proof theory.
The prime action soften will give rise to new axioms. No problem with that as soon as you know what you can do then. And you use the normal formal logic. And soon some light will shine. I can make it formal.
I tell this in short. But this is the center of formalization. The theory hangs above the model and has it's own more exact language. R=This language is no more that physical language. We call it the meta language. It's a more formal and parallel language to what happens in the model.
Important is you make it first very true in the model. An rephrasing what you have, Just putting in other word. Splitting it or combining you rephrase. And always you arrive at a result.
The rephrasing steps can cost a day. You have also to get used to new phrasing. A day is often enough.
In an information analyzing step you often start with some formulation but at the end you know it much better. I think you know that. Of course you can loose phrasings but that is not important. Phrasings are just the language we use. It concerns the content. What you really want.
When I am not clear, please ask,
ed
bri...@encompasserve.org
> Hi John,
>
> Please don't become angry with me.
Please learn to quote the posts you respond to.
>
> I defended here a physicist standpoint. Do you see that as bad from me?
> You are not the only one in this world.
You defended something that needed no defending against an attack
that did not exist. You wasted words on nothing.
If I thought I were the only one in this world, I probably would not
be wasting time talking to you. Come to think of it... why am I?
> Do you know what scimath means.
Do you know what non sequitur means?
John Briggs
Tony Orlow
in the set is spatially contiguous to two other points in the line. Okay?
--
Smiles,
Tony
Alan Morgan
Could you explain how this is done? The rational numbers can be defined as
a function on the naturals, but you don't think that the rational numbers
have the same bigulousity (whew! I'm really enjoying this word) as the
integers.
>All the sets you
>mentioned have size N/2, and are therefore equal in size.
What about the positive numbers? Negative numbers? I'm assuming that
both of them are size |N|/2. The postive numbers, negative numbers,
and 0 make up N, but that implies that |N|/2 + |N|/2 + 1 = |N|, which
doesn't make a lot of sense to me. The logical (?) conclusion is that
the bigulousity of the positive numbers is actually |N|/2 - 1/2.
I'd be interested in knowing what the bigulousity of the prime numbers
is. |N|/k won't work for any finite k.
All in all, the traditional definition of cardinality seems to work much
better.
Alan
--
Defendit numerus
Ed van der Meulen
What do you really have so you can take it as an input for a process.
Give just a provisional answer. I know this is the most dificult question I can ask. But why not right away.
Don't ever be afraid, we will find a solution.
I will tell you already the next question.
2. Do you know of lazy programming? Do you understand what that means for you at this moment.
Please don't answer this question. First a rough answer to the first question.
Back to the birthday party. Jippie.
ed
Tony Orlow
think of how to define the rationals as a proper function on a single natural
variable. They are represented by two naturals with a notation, which is why I
consider them to be a 2D natural number system. The grid of rationals is NxN,
so i calculate the area of that square as N^2 (N=aleph_0). Can you think of an
invertible function on a single natural that generates the rationals completely
in a linear order. I am not saying it can't be done. I susp[ect perhaps it can.
I'll think about it. One would hope that if one generated such a function, and
applied the inverse function method, that one would get N^2, but that would
make the function equivalent to sqrt(n), which doesn't seem to fit at all, at
first glance. So maybe it can't be done. If one can define the rationals as a
set defined by a function of a single natural variable, and the method doesn't
work on that function, then I would consider that the first genuine objection
and hole in my theory. Good thinking!! Thanks.
>
> >All the sets you
> >mentioned have size N/2, and are therefore equal in size.
>
> What about the positive numbers? Negative numbers? I'm assuming that
> both of them are size |N|/2. The postive numbers, negative numbers,
> and 0 make up N, but that implies that |N|/2 + |N|/2 + 1 = |N|, which
> doesn't make a lot of sense to me. The logical (?) conclusion is that
> the bigulousity of the positive numbers is actually |N|/2 - 1/2.
consistent. But, since you brought it up, I really think the unit infinity is
represented by a line, not a ray, and the whole number set (W) should be
everything from negative infinity ro positive infinity, like the reals. In this
case the positive whole numbers would be W/2. Really I am talking about the
integers, except that I don't exclude infinite whole numbers. Now, again, this
will look to you like we have an odd number of numbers, with 0 in the middle,
but remember that +0=-0, when I tell you that +oo=-oo, and the whole line can
be inverted so that -0 and +0 are the endpoints, and +-oo is the midpoint. The
number line is a circle. 0 and oo are both positive and negative, but guess
what? oo's odd, and almost surely prime. :D
D: Now I really won the crank contest. :D
>
> I'd be interested in knowing what the bigulousity of the prime numbers
> is. |N|/k won't work for any finite k.
Since the primes can't be formulated exactly, the functional approach won't
work. But, since we know the number of primes less than x is asymptotic to
x/log(x), our work is done for us. As x goes to |N|, the function goes to N/log
(N). I would say this function describes the size of the set relative to N. Log
(N) is of course infinite, so this is an infinite number, which is infinitely
smaller than N.
An interesting question I have. Is there an inverse function g(x) for f(x)
=x/log(x), such that g(f(x))=x? What is this g(x), and can it be used to
approximate primes?
>
> All in all, the traditional definition of cardinality seems to work much
> better.
For what? Take your time.
>
> Alan
>
--
Smiles,
Tony
Ed van der Meulen
You say: "You are asking what an infinite line is? A set of points such that each point in the set is spatially contiguous to two other points in the line. Okay?
That is your first formulation. We store of course the history. Always nice to have it.
This is a line.
And now in what form can you have it. How is it given to you that you can use it as an input for a process so an output can be produced.
In what way do you have that line.
I know this isn't an easy question. Please think a moment about it.
for instance... Can you put the line in your pocket. Can you throw it away. In what form do you have that line.
And again this is already the most difficult question.
Later comes that question about lazy programming. But first this.
In what form do you have that line.
You can always ask more questions.
I have to attend the birthday party. Till tomorrow.
ed
imagin...@despammed.com
Tony Orlow (aeo6) wrote:
> But there is good reason why different mathematical approaches to a
problem
> should not contradict each other logically, and derive different
results for
> the same problems, don't you think?
If I understand you correctly, then I think not. (It depends,
delicately, on quite what you mean by "contradict".)
For example, in the reals, the polynomial x^3-x^2+4x-4 has one root. In
the complex numbers it has two.
In the rationals, 3 divided by 7 has a solution; in the integers it
doesn't. In primary school, "7 into 3 won't go".
These claims are all true in their own terms, yet mutually
incompatible. One of the important features of real maths is that
people choose terminology* carefully, so that lots of claims in
different systems can be made without everyone losing track of what's
being said. That's also why it is (I submit) a vastly better idea to
call your ideas on set size something distinct, like 'bigulosity'.
* (Often fanciful, which upsets a few zealots obsessed with "True
Meaning" of words)
Brian Chandler
http://imaginatorium.org
imagin...@despammed.com
Tony Orlow (aeo6) wrote:
> Alan Morgan said:
> >
> > Can you compare the bigulousity of two sets that have no
> > elements in common? I'm thinking of the odd numbers and
> > the even numbers here. Or, even better, the positive odd
> > numbers and the negative odd numbers.
> Sure. They are all measured using functional mappings of the
naturals, and the
> functions that define them are compared. Any two sets that are both
defined as
> functions on the naturals, or both as functions on the reals, can be
compared
> by comparing the inverses of their mapping functions. All the sets
you
> mentioned have size N/2, and are therefore equal in size.
Are you sure? If the naturals (from 1) are N, then the integers must be
2N+1. I guess this means there are N+1 even integers and N odd ones? (I
think 'number' usually includes negative.)
> >
> > >He says that the bigulosity of the rationals is |N|^2. OK,
> > >fine. But is that bigger than |N|?
Is it OK? The rationals include negative values and zero. So I
calculate b(Q) as 2N^2+1, writing b() for bigulosity, and using Tony's
convention that N is the number of naturals.
> > So we have cardinal arithmetic, ordinal arithmetic, and bigulous
> > arithmetic. Just make sure you know which one you are doing.
> LOL thanks Alan. It's really just a matter of choosing a unit
infinity and
> comparing things to that as one would do with finites. After all,
don't we do
> that on the finite level already, choose a unit and work with it
consistently?
Yes, assuming [!!] that one can do these comparisons "as one would with
finites". Many of us think that's precisely what you _can't_ do, but
don't give up yet.
Brian Chandler
http://imaginatorium.org
imagin...@despammed.com
Tony Orlow (aeo6) wrote:
> imagin...@despammed.com said:
> > Tony Orlow (aeo6) wrote:
> > > ... but I certainly have pointed out
> > > some problems in Cantor, and some solutions.
> >
> > Well, yes. Perceived problems. But many people here have seen
exactly
> > these same "problems" pointed out over and over again, with exactly
the
> > same flaws (basically, there's *always* a quantifier swap)
> I remember WM being accused of that. I don;t think that was in
reference to me.
> If it was, I wouldn't mind seeing specifics.
OK. I hope you really do understand that "for all x" (written Ax)
really means "for absolutely any individual x you care to give me", and
not "for the set of xes". Urm, ok, I'll write "for any", because I
don't like using non-inverted 'A'.
For any (finite) set P of natnums, exists natnum m s.t. m > n for all n
in P. TRUE
Reverse the order of quantification:
Exists natnum m s.t. for any (finite) set P of natnums, m > n for all n
in P. FALSE
> > logic. There was a chap called Phil a year or three back, who
> > approached things almost exactly as you have. Couldn't accept that
> > (our, finite) natural numbers are not a finite set, and drew
uncannily
> > similar "numbers" consisting of digits and dots in
ever-increasingly
> > baroque (and ill-defined) combinations. So if you think you have
> > already done something original, you are plainly mistaken.
> I know people have been trying to make this work for a long time. I
think it
> does. I know now what a tenacious bunch you all are, and I can see
how the
> training in this system has adversely affected the way people think
about these
> things.
Then how did John Conway invent surreals? Your claims of brainwashing
are silly, and at least mildly offensive.
> ... it's time for the axiomatics to get a little picture training,
and stop
> thinking exclusively in words. That's not how the structure of
benzene was
> discovered.
Ironically, the inspiration here was realising that the benzene
molecule has no end!
> > It's no coincidence that WM and I
> > > were arguing the same point regarding the size of the set of
naturals
> > vs. their
> > > values. Unfortunately I think he sees them both as finite, so we
both
> > disagree
> > > with Cantor, and with each other at the same time. What fun!
You're not "disagreeing with Cantor", you're claiming that a huge body
of mathematics that has been thoroughly studied by tens(?) of thousands
of clever people just happens to be utterly wrong, because none of
these people noticed something that you have. Anyway, let's leave that
aside.
> > Pity if you get so keyed up about it that you don't learn something
out
> > of the debris. You've got as far as having "boundless sets". Why
don't
> > you try to formalise the difference between a "boundless set" and
an
> > "infinite set", instead of responding like you did to my making
this
> > suggestion the other day.
> Because "boundless" sets are not part of my theory. That term was a
tongue-in-
> cheek response to the need to have infinite sets of distinct finite
natural
> numbers, which is mathematically impossible outside of set theory.
So does your version of whatever corresponds to set theory
("bagulosity", perhaps?) allow one to consider the collection of all
objects x having property P for any well-formed property P? Such as
being either zero or a natnum plus one? In what way does this break
down, to force this collection to include some things that don't have
property P?
> > Look, Conway's numbers are a different system
> > from the conventional real numbers of set theory (they are a proper
> > class, for a start), so of course lots of things are true in
surreals
> > and not in Cantor set theory, and vice versa. That's different from
> > claiming that propositions with a clear and simple proof in some
system
> > are "wrong".
> Mostly I take this stance because whenever I present my ideas,
Cantorians tell
> me I am "wrong".
What is "wrong" is not whatever you have created (bigulosity, or the
beginnings thereof), but your posting of plainly unsupported claims
that this or that bit of standard theory is "wrong". I promise not to
call anything you *do* "wrong", if you agree to stop ranting about
Cantor being "wrong". That way we might get somewhere.
> > > Besides,
> > > I really see all the logical loopholes and shell games in it as
> > contributing to
> > > the deliquency of mathematics. Conway and Robinson probably never
> > tread on the
> > > same ground as Cantor, so as not to contradict his established
> > mathematics. My
> > > territory apparently overlaps with Cantor's significantly, and in
an
> > > incompatible way. Oh well. Sorry.
> >
> > So I think your claimed distinction here just isn't.
> That sentence no objective clause. What means?
"What you say is true isn't." I think that's grammatical (not
commenting on truth value for the moment). I meant: you appear to claim
that whereas Conway (for example) creates something "disjoint" from
Cantorian set theory, you are going to "create" something that somehow
"contradicts" it. This is not a valid distinction, IMO, because lots of
statements about surreal numbers are not true about reals; you may
establish facts about bigulosity that are not true of cardinality -
great, but this does not in itself make cardinality "wrong". The very
best thing that could ever happen for you is that universities abandon
the teaching of cardinality and start teaching bigulosity; they will
never give courses called "Why Cantor was wrong". (And not just because
he wasn't.)
Brian Chandler
http://imaginatorium.org
imagin...@despammed.com
> Ed van der Meulen said:
> > At 19-5-2005 14:42 -0400, you wrote:
> > -- DO NOT REPLY TO THIS MESSAGE --
Yeah, right, whoever you are.
Tony:
> You are asking what an infinite line is? A set of points such that
each point
> in the set is spatially contiguous to two other points in the line.
Okay?
Really? If the point and one of its adjacent points are distinct, what
is the distance between them?
(Tony, remember you're going to overturn set theory: you have a lot of
catching up to do first.)
Brian Chandler
http://imaginatorium.org
Tony Orlow
>
> Tony Orlow (aeo6) wrote:
> > But there is good reason why different mathematical approaches to a
> problem
> > should not contradict each other logically, and derive different
> results for
> > the same problems, don't you think?
>
> If I understand you correctly, then I think not. (It depends,
> delicately, on quite what you mean by "contradict".)
>
> For example, in the reals, the polynomial x^3-x^2+4x-4 has one root. In
> the complex numbers it has two.
component. The reals are a subset of complex numbers.
>
> In the rationals, 3 divided by 7 has a solution; in the integers it
> doesn't. In primary school, "7 into 3 won't go".
The complex numbers are the most generalized of the number systems you
mentioned. They include the integers, and every rational is equivalent to a
real, and so is a subset as well. What might be true for certain subsets of the
complex numbers may not hold for the most general case. If we say you can't
take the square root of a negative numbers, we have restricted the complex
plane to the real line. If you say you need to be able to count the value, then
you have restricted it to whole numbers. If you say you can't divide by zero,
then you have restricted your number set to finite values. 8D
I was looking at Hilbert's axioms of geometry a little, and in the axioms of
incidence, I notied that three or four could be boiled down to one axiom, if
dimension were allowed to be a dependent variable. Also 1.7 is only true in 3
dimensions. Beyond 3D, two planes can intersect at a single point. I kind of
think it would be nice to boil our axioms down and generalize them as much as
possible, so we end up with as few and as powerful laws as we can manage, kind
of like they do in physics. I guess I AM applying scientific thought to
mathematics, and if that offends anyone, I'm sorry, but maybe it's about time.
>
> These claims are all true in their own terms, yet mutually
> incompatible. One of the important features of real maths is that
> people choose terminology* carefully, so that lots of claims in
> different systems can be made without everyone losing track of what's
> being said. That's also why it is (I submit) a vastly better idea to
> call your ideas on set size something distinct, like 'bigulosity'.
Very well! I think I'll stick with Bigulosity, after all. Don't forget to
capitalize it, or I'll claim not to know what you're talking about. heh heh ;)
I do like terms such as "equibigulous" and "relatively semibigulous", or even
"cobigulously complementary". Mathematics needs terms like that. Terms that go
well with beer and nachos.
>
> * (Often fanciful, which upsets a few zealots obsessed with "True
> Meaning" of words)
>
> Brian Chandler
> http://imaginatorium.org
>
>
<long snippage>
--
Smiles,
Tony
Tony Orlow
>
> Tony Orlow (aeo6) wrote:
> > Alan Morgan said:
> > >
> > > Can you compare the bigulousity of two sets that have no
> > > elements in common? I'm thinking of the odd numbers and
> > > the even numbers here. Or, even better, the positive odd
> > > numbers and the negative odd numbers.
> > Sure. They are all measured using functional mappings of the
> naturals, and the
> > functions that define them are compared. Any two sets that are both
> defined as
> > functions on the naturals, or both as functions on the reals, can be
> compared
> > by comparing the inverses of their mapping functions. All the sets
> you
> > mentioned have size N/2, and are therefore equal in size.
>
> Are you sure? If the naturals (from 1) are N, then the integers must be
> 2N+1. I guess this means there are N+1 even integers and N odd ones? (I
> think 'number' usually includes negative.)
>
> > >
> > > >He says that the bigulosity of the rationals is |N|^2. OK,
> > > >fine. But is that bigger than |N|?
>
> Is it OK? The rationals include negative values and zero. So I
> calculate b(Q) as 2N^2+1, writing b() for bigulosity, and using Tony's
> convention that N is the number of naturals.
>
> > > So we have cardinal arithmetic, ordinal arithmetic, and bigulous
> > > arithmetic. Just make sure you know which one you are doing.
>
> > LOL thanks Alan. It's really just a matter of choosing a unit
> infinity and
> > comparing things to that as one would do with finites. After all,
> don't we do
> > that on the finite level already, choose a unit and work with it
> consistently?
>
> Yes, assuming [!!] that one can do these comparisons "as one would with
> finites". Many of us think that's precisely what you _can't_ do, but
> don't give up yet.
David Kastrup
> Tony Orlow (aeo6) wrote:
>> But there is good reason why different mathematical approaches to a problem
>> should not contradict each other logically, and derive different results for
>> the same problems, don't you think?
>
> If I understand you correctly, then I think not. (It depends,
> delicately, on quite what you mean by "contradict".)
>
> For example, in the reals, the polynomial x^3-x^2+4x-4 has one
> root. In the complex numbers it has two.
Sputter.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
Tony Orlow
>
> Tony Orlow (aeo6) wrote:
> > imagin...@despammed.com said:
> > > Tony Orlow (aeo6) wrote:
> > > > ... but I certainly have pointed out
> > > > some problems in Cantor, and some solutions.
> > >
> > > Well, yes. Perceived problems. But many people here have seen
> exactly
> > > these same "problems" pointed out over and over again, with exactly
> the
> > > same flaws (basically, there's *always* a quantifier swap)
>
> > I remember WM being accused of that. I don;t think that was in
> reference to me.
> > If it was, I wouldn't mind seeing specifics.
>
> OK. I hope you really do understand that "for all x" (written Ax)
> really means "for absolutely any individual x you care to give me", and
> not "for the set of xes". Urm, ok, I'll write "for any", because I
> don't like using non-inverted 'A'.
proof that the naturals as an infinite set must contain infinite values, but
that was because people got confused about what I was doing. I was using a set
defined by each successive natural, and using induction to show that if the set
generated by one natural had a property, thet the set generated by the
successive natural had that property. The property of defining a set that has a
certain property, IS a property of the natural number that defines the set. So,
that property would be true of any set defined in that same way on any natural
number. It wasn't that I was generalizing properties of element to sets, though
I am not sure that's always inappropriate.
>
> For any (finite) set P of natnums, exists natnum m s.t. m > n for all n
> in P. TRUE
conventional sense, and find one larger than that one.
>
> Reverse the order of quantification:
> Exists natnum m s.t. for any (finite) set P of natnums, m > n for all n
> in P. FALSE
>
>
> > > logic. There was a chap called Phil a year or three back, who
> > > approached things almost exactly as you have. Couldn't accept that
> > > (our, finite) natural numbers are not a finite set, and drew
> uncannily
> > > similar "numbers" consisting of digits and dots in
> ever-increasingly
> > > baroque (and ill-defined) combinations. So if you think you have
> > > already done something original, you are plainly mistaken.
> > I know people have been trying to make this work for a long time. I
> think it
> > does. I know now what a tenacious bunch you all are, and I can see
> how the
> > training in this system has adversely affected the way people think
> about these
> > things.
>
> Then how did John Conway invent surreals? Your claims of brainwashing
> are silly, and at least mildly offensive.
Okay, I am being too generally harsh here, Set theory has actually led to
excellent ways of thinking about things, notably including Conway's work on
surreals. It's a great alternative way to look at things, and worthwhile. I
guess my particular beef is with cardinality measures. That's the only part
that really bothers me. I don't mean to put down set theory in general, and I'm
sorry if I implied that. Cardinality of infinite sets is totally mishandled,
however, in my opinion, and has a few inconsistencies to blame for it's poor
conclusions.
>
> > ... it's time for the axiomatics to get a little picture training,
> and stop
> > thinking exclusively in words. That's not how the structure of
> benzene was
> > discovered.
>
> Ironically, the inspiration here was realising that the benzene
> molecule has no end!
And yet endless as it may be, it can be conceived of and understood. The snake
bit its tail in a dream, and benzene was understood. The snake biting its tail
is a sign of infinity. Why is this? The number line is only a straight line
when locally viewed. When you step back infinitely far away, it is a circle.
Positive and negative infinity are one. Irony indeed. Or symbolism for the
moment.
>
>
> > > It's no coincidence that WM and I
> > > > were arguing the same point regarding the size of the set of
> naturals
> > > vs. their
> > > > values. Unfortunately I think he sees them both as finite, so we
> both
> > > disagree
> > > > with Cantor, and with each other at the same time. What fun!
>
> You're not "disagreeing with Cantor", you're claiming that a huge body
> of mathematics that has been thoroughly studied by tens(?) of thousands
> of clever people just happens to be utterly wrong, because none of
> these people noticed something that you have. Anyway, let's leave that
> aside.
As I said above, i am not against set theory by any means and apologize if it
sounded like that. I disagree with cardinality measures as they applied to
infinite sets.
>
>
> > > Pity if you get so keyed up about it that you don't learn something
> out
> > > of the debris. You've got as far as having "boundless sets". Why
> don't
> > > you try to formalise the difference between a "boundless set" and
> an
> > > "infinite set", instead of responding like you did to my making
> this
> > > suggestion the other day.
> > Because "boundless" sets are not part of my theory. That term was a
> tongue-in-
> > cheek response to the need to have infinite sets of distinct finite
> natural
> > numbers, which is mathematically impossible outside of set theory.
>
> So does your version of whatever corresponds to set theory
to cardinality, really....
> ("bagulosity", perhaps?)
Ha ha Good one!
> allow one to consider the collection of all
> objects x having property P for any well-formed property P? Such as
> being either zero or a natnum plus one? In what way does this break
> down, to force this collection to include some things that don't have
> property P?
I am not sure what you are asking here.
>
>
> > > Look, Conway's numbers are a different system
> > > from the conventional real numbers of set theory (they are a proper
> > > class, for a start), so of course lots of things are true in
> surreals
> > > and not in Cantor set theory, and vice versa. That's different from
> > > claiming that propositions with a clear and simple proof in some
> system
> > > are "wrong".
> > Mostly I take this stance because whenever I present my ideas,
> Cantorians tell
> > me I am "wrong".
>
> What is "wrong" is not whatever you have created (bigulosity, or the
> beginnings thereof), but your posting of plainly unsupported claims
> that this or that bit of standard theory is "wrong". I promise not to
> call anything you *do* "wrong", if you agree to stop ranting about
> Cantor being "wrong". That way we might get somewhere.
Perhaps. It seems extremely hard for those who have accepted cardinality to
consider any other approach to set size measures for infinite sets, because
they feel they have "proved" their resluts, and I am simply wrong, a moron, a
crackpot or crank, retarded, uneducated, or a troll. I disagree with those
methods and those results, and think they really need to be uprooted, and that
won't happen until the flaws are understood and better method developed that
avoids them. I really feel that Cantor is dividing by zero and hiding it. But,
I'll try to ignore Cantor, as long as no one tries to prove to me that I am
wrong using those confounded bijections. Dagnabbit!
>
>
> > > > Besides,
> > > > I really see all the logical loopholes and shell games in it as
> > > contributing to
> > > > the deliquency of mathematics. Conway and Robinson probably never
> > > tread on the
> > > > same ground as Cantor, so as not to contradict his established
> > > mathematics. My
> > > > territory apparently overlaps with Cantor's significantly, and in
> an
> > > > incompatible way. Oh well. Sorry.
> > >
> > > So I think your claimed distinction here just isn't.
> > That sentence no objective clause. What means?
>
> "What you say is true isn't." I think that's grammatical (not
> commenting on truth value for the moment). I meant: you appear to claim
> that whereas Conway (for example) creates something "disjoint" from
> Cantorian set theory, you are going to "create" something that somehow
> "contradicts" it. This is not a valid distinction, IMO, because lots of
> statements about surreal numbers are not true about reals; you may
> establish facts about bigulosity that are not true of cardinality -
> great, but this does not in itself make cardinality "wrong". The very
> best thing that could ever happen for you is that universities abandon
> the teaching of cardinality and start teaching bigulosity; they will
> never give courses called "Why Cantor was wrong". (And not just because
> he wasn't.)
Point well taken. It's like hate speech in political campaigns. People get sick
of hearing it. I feel sorry for Cantor and the way things went down for him,
and he shouldn't be forgotten and more than Freud should. Set theory is robust,
but in this area I see a problem. I'll try to calm down, as long as I don't
have to go through the same arguments over and over, as if cardinality is the
last word on the subject. I mean, when I learned it 25 years ago, I kind of
went, "That doesn't sound right. I don't think I need to keep that in my
infinity bag. I think it goes with the trisected angles and squared circles,
and other curiosities" and left it at that. The infinity bag has a lot of other
stuff in it, but people don't seem to like my snakes and straight circles, or
multiplying zero by infinity. They all like the bag that's like the one's
everybody else has. That's human nature. I'm the kid in 5th grade with the
green mohawk. Oh no, actually, that's my son. The apple doesn't fall far from
the tree.
Tony Orlow
like on a circle, which is an infinity-sided regular polygon with finite
diameter (measured to the center of each side of the polygon, of course, not to
the vertices, for heaven's sake).
--
Smiles,
Tony
Tony Orlow
--
Smiles,
Tony
imagin...@despammed.com
David Kastrup wrote:
> imagin...@despammed.com writes:
> > For example, in the reals, the polynomial x^3-x^2+4x-4 has one
> > root. In the complex numbers it has two.
>
> Sputter.
Ah, but you see it has two roots. I'm right. But it also has a third,
nyah nyah!
(Sorry, brain-finger disconnect)
Brian Chandler
http://imaginatorium.org
Ed van der Meulen
We go furhter with the question you didn't answer. Now I will formulate it in this way. You've the natural numbers. All on one line. Never ending, for you can step further.
Here we understand it. But then...
You tell me we reach infinity in some way. I don't know how, But okay you have reached in finity.
And now my question:
Please put that line in the computer. How do you do that.
And please answer me that clearly. We have only the natural numbers no more but with your infinity now. You told before about them. So here we stick to.
And now the question how will you store all those numbers indeed reaching infity in a computer
And when you can't give me a proper anser you are wrong.
I know already you are wrong. For you can't do this. But try to answer this question. I haven't post it for nothing.
Infinite many things putting in a limited computer in a limited time. That are you wanted to do.
Please tell me how?
You may sleep a further night about this question.
ed
Ed van der Meulen
Maybe you need a more logically thinking brain.
Do you know of quantum scientists. Infinite axis and infinite many dimensions. A horror.
Cosmologists. Tony please stay where you are.
For computers you need the whole universe to work with that and even that it's not enough.
You make your philosopy for your own group of Taoists.
It's a horror for scientists and not logically in mathematics.
Look Randy hardly reacts. He knows your are wrong. Brian Chandler knows you are wrong. Dik de Winter knows you are wrong. People who know more than you know. Who is with you?
And your phylosophy is a horror for scientists. Infinity many dimensions.
You are some nice guy in a street who knows beforehand in the next street I will meet that person.
You are a scientist with a microscoop. But you don't use it. You know it already. Scientists are chldren of Aristotle. They have not such ideas as you have.
On the peano staircase you knew before at the top are blue steps. Hoever we didn't have blue paint.
And stubborn you go on without any support from reliable people.
You have to rephrase your idea Tony. Look at lazy programming.
That means do something only when you need it. Prepare nothing. So say that is the way to the program but the program only does what is needed to get a result. I hope you know this.
You may also try to make a computer program shwowing your ideas.
That it can produce infinities. Please try it with an existing computer.
Words are like written in water. You can tell everything an think still more but the reality there you live in.
Proof that your ideas are realizable please.
No scientist you get after your idea of infinite many dinmensions. Computers now already are often to small.
You live in dreams about the reality.
In the old time that believe of Taoists have started they didn't know of mathematics. And you want to introduce it. You have no chance on that.
Tony listen to others!
Robert Kolker
>
>
> Tony listen to others!
Van der Meulen.
Learn to write related sentence.
Brought togeither in paragraphs.
It makes for much easier reading.
Van der Meulen, listen to others.
Bob Kolker
Ed van der Meulen
Are you also a language lover. Finding your own threads in a story. Do you read literature books. Do you know of target groups? Writing for different people. Do you really like different languages. I do. To you like Latin and old Greek as well. Do you like stage plays. Who are you?
Or do you only like mathematics. But some have a broader interest than only mathematics. May I.
Robert Kolker
> Hi Bob again.
Will you please wrap your lines to 80 characters or less. There's a good
fellow.
Mathematics is my passion but it is not my only interest. I am
interested in history, particularly ancient history. Modern times are
tedious and boring.
I am fluent in Hebrew and Aramaic. I can muck around in Greek a little bit.
Bob Kolker
Ed van der Meulen
History for sure. A great subject.
Do you also have the wish to meet
some people from those times and
have a talk with them. Are you also
interested in archeology?
Modern history is still full of
one-sided opinions, more difficult
to convey, I think. It's like
flying over it with a long distance
view. While those happenings were
short ago. Strange, as well. Close
and more difficult.
Too much fuzz maybe. I know only
the first two letters of the Hebrew
alphabet, grin.
Greek is easier for me. And Aramaic is
also great. You are a special guy.
But very interesting for me.
Thanks Bob.
Oh Tony.
You have, what we can call, a
qualitative view of an infinite
line. You know it as symbol. But
you say you are a scientist. And
once they want a size, a weight, a
distance, a speed or whatsoever.
When we make a theory with algebra.
Scientists can use numbers for our
variables and it has to work then.
And to reach that you have to do
the step to a quantitative
expression.
Please, do look now at lazy
programming. A rephrasing step.
Then you can bridge the gap.
This a an applying step and when
you can't take that step you have
made your theory or your philosophy
for the monkeys.
What can a scientist do when you
say this is far.
What do you think Tony?
Bob
Mathematics is just great but what
parts do you like most, Bob?
I am now fond of discrete
mathematics as well. Full of
surprises.
Please have a good time
Ed van der Meulen
How are you god guy.
I know from the hiostory of TAO theuy never have done
someting for the population. Do you know that as well.
Maybe your parents werte more right. I don't know it. But
it could be?
Reprasing is your only option. And other wise you can't
bridge the gap. I showed you already a next step. Lazy
programming.
Pleaas have a splendid day
imagin...@despammed.com
> imagin...@despammed.com said:
<snip>
> > > > Why don't you try to formalise the difference between a
"boundless
> > > > set" and an "infinite set", ...
> > > Because "boundless" sets are not part of my theory. That term was
a
> > tongue-in-
> > > cheek response to the need to have infinite sets of distinct
finite
> > natural
> > > numbers, which is mathematically impossible outside of set
theory.
> > So does your version of whatever corresponds to set theory
> to cardinality, really....
> > ("bagulosity", perhaps?)
> Ha ha Good one!
> > allow one to consider the collection of all
> > objects x having property P for any well-formed property P? Such as
> > being either zero or a natnum plus one? In what way does this break
> > down, to force this collection to include some things that don't
have
> > property P?
> I am not sure what you are asking here.
Define pofnat to mean "plain old finite natural number, one which is
either zero or can be got by adding one to an existing pofnat". By
definition a pofnat is finite.
You seem to have agreed that if the property P is "being a pofnat", the
set of all of these is unbounded; that is, you accept that there is no
finite size to the set. Yet you say that your theory somehow discounts
the existence of boundless sets?
Here are a couple more bigulosity questions.
Q1: When you say N, do we understand that you are claiming that this is
the bigulosity of your set (also called N, though I think it should
have a subscript T) of "natural numbers", including some infinite ones.
Can you say anything about the bigulosity of the pofnats?
Q2: Consider the set B = {1, 10, 11, 100, 101, ... }. I believe that if
this *means* all naturals written in binary then b(B) = N. But if it
*means* the set of decimal numbers only containing the digits 0 and 1,
then its b() is a lot smaller. Can you say what it is? Is there
anything wrong with the following argument.
The density of the subset of set B within an initial segment of the
naturals from 1 to K is K / 5^(log10(K)).
(I hope that's right; someone cleverer can correct it, but up to 10
there are 2 = 10/5, up to 100 there are 4 = 100/25, and so on.)
So the limit as K -> oo of this density is plainly zero.
Therefore b(B) = 0.
This seems highly counterintuitive - bigulosity is supposed to
correspond intuitively to "size", yet we have a set with an unbounded
number of elements whose bigulosity is zero.
Brian Chandler
http://imaginatorium.org
Tony Orlow
I cetainly don't claim one can do or have an infinite number of finite things
or events in a finite space or time. That would be crazy. It would be like
claiming you can have an infinite number of distinct whole numbers and 1-unit
intervals, and have them all fit within a finite interval on the number line. I
would never claim any such thing! It would be absurd!
However, if you want to understand infinity in computer terms, the answer is
very simple. Do you know 2's complement, the common signed binary numbers used
every day in computers? If you subtract 1 from 0, which is a string of all 0's
(as many bits as your register allows), one gets a string of all 1's? In a way,
that string of all 1's is like the infinite natural numbers I was talking about
before. It can theoretically go one forever. But in this case the value is -1.
All the negatives start with 1 and the positives start with 0, but 0 isn't
considered positive, so we have one more negative than positive, a 1 followed
by all 0's, 10000.... This number is supposed to be the largest negative number
representable in those bits, but is it really a negative? I hope to have a web
page soon where I can demonstrate that this value really represents oo, the
opposite side of the number circle from 0, where adding 1 to the largest
positive number yields the largest negative number. Indeed, as is true with 0,
oo is both positive and negative. And, the uncountable zone in the strings of
digits is not at the top, 9999....99999, or at the bottom, 000....0000, but in
the middle between them, around 5000...00000 and 4999...99999. The fuzzy zone
between the largest positive and largest negative numbers, oo (or 1000..), is
that uncountable barrier on the opposite end from the origin, which is the
boundary between the smallest potitive and smallest negative numbers.
Did that answer your question?
--
Smiles,
Tony
Tony Orlow
being lazy, or am perhaps well prepared for this debate and not just blowing
smoke out my ass. If you are actually listening to what I have been saying,
perhaps you would like to commment on 100000.... as the binary representation
of oo, the opposite of 0, and the concept of the number circle which is
implemented in binary arithmetic, as addition becomes travel in one direction
around the circle, and subtraction is travel in the other. If you really look
at the various areas of math I am trying to unify and the kinds of conclusions
I am drawing without contradiction, you may start to pay attention, instead of
repeating that I am wrong and must agree with a system that in my mind is
deficient.
Bob, it does sound like you have some good other interests. I took Spanish,
Latin, and Greek in high school, and though I never pursued language as a
career, am glad to this day for what I learned by studying different languages.
By your choice of languages, I would guess there is biblical motivation for
them, perhaps? I am also very interested in religions. Perhaps one day I will
learn Sanskrit. We'll see what time permits in this life. PS - Thanks for
getting Ed to wrap his lines a little. ;)
>
>
> Bob
>
> Mathematics is just great but what
> parts do you like most, Bob?
>
> I am now fond of discrete
> mathematics as well. Full of
> surprises.
>
> Please have a good time
>
--
Smiles,
Tony
Tony Orlow
Taoist community, but believe in the concepts. I don't expect any organized
religion or party to remain uncorrupted as they gain power, which is why I
don't join them, but seek answers from whatever sources offer them. I brought
it up so you could put my position in some larger philosophical context, but
you don't seem to understand, or wnat to understand, what I mean when I say the
Tao, so, let's not discuss it, okay? Thanks, and have a nice day!
--
Smiles,
Tony
Tony Orlow
you cannot define the point where counting crosses the boundary between finite
and infinite. The definition is ala Peano, which doesn't explicitly state that
natural numbers are finite. Isn't that correct?
>
> You seem to have agreed that if the property P is "being a pofnat", the
> set of all of these is unbounded; that is, you accept that there is no
> finite size to the set. Yet you say that your theory somehow discounts
> the existence of boundless sets?
each member adds a constant finite unit to the range of values, and where the
number of elements is infinite, but the total range is finite. An infinite
number of elements each incrementing the overall range results in an infinite
overall range of values, which means at least one value must be infinite.
That's not MY theory. That's proper mathematics. Look up infinite series.
>
> Here are a couple more bigulosity questions.
>
> Q1: When you say N, do we understand that you are claiming that this is
> the bigulosity of your set (also called N, though I think it should
> have a subscript T) of "natural numbers", including some infinite ones.
> Can you say anything about the bigulosity of the pofnats?
no particular upper bound. In that sense it's almost a self-contradictory
concept. What saves it from being a self-contradictory concept is the
unspecified upper bound and resulting vagueness. If a value is declared to be
finite, but no value or upper bound is defined for it, then it is more like
"indefinite", being some unspecified finite quantitity.
>
> Q2: Consider the set B = {1, 10, 11, 100, 101, ... }. I believe that if
> this *means* all naturals written in binary then b(B) = N. But if it
> *means* the set of decimal numbers only containing the digits 0 and 1,
> then its b() is a lot smaller. Can you say what it is? Is there
> anything wrong with the following argument.
>
> The density of the subset of set B within an initial segment of the
> naturals from 1 to K is K / 5^(log10(K)).
>
> (I hope that's right; someone cleverer can correct it, but up to 10
> there are 2 = 10/5, up to 100 there are 4 = 100/25, and so on.)
>
> So the limit as K -> oo of this density is plainly zero.
>
> Therefore b(B) = 0.
>
> This seems highly counterintuitive - bigulosity is supposed to
> correspond intuitively to "size", yet we have a set with an unbounded
> number of elements whose bigulosity is zero.
Cantorians do like their strings tricks, you need to keep in mind the rules
concerning generation of strings. Much confusion is introduced when changing
the meanings of strings mid-proof. We can use strings to represent numbers, and
if we want to talk about the sets of numbers, we need to follow rules for the
numbers that are independent of their symbolic representations. If we want to
talk about the sets of strings, we need to follows rules about symbolic strings
that are independent of numbers. The representation and the repesented concept
are two different things and it should not be considered a problem if we arrive
at two different size measures for the set of symbolic strings and the set of
concepts that they represent, as long as the set of strings is at least
equivalent to the set of concepts.
So, if you are dealing with binary strings, that is, strings of symbols
constructed from a 2-symbol set, with lengths up to N, you have a set of
strings with 2^N members. If you are saying those strings represent binary
integers, then you have N numbers in that set, represented by these strings.
These two concepts are connected by the fact that a number n requires a string
of length log2(n) symbols, and a string of length n can represent 2^n different
numbers. In other words, we may say we have strings up to length N, since that
is smallest possible infinity, but if we discard that notion, and say that, for
N different numbers we need strings of length log2(N), this is an infinite
length for a string, which is less than the infinite numeric value represented
by that string, as it should be.
If we are interpreting the string as a decimal integer, then we have 10^n
strings of length n, and require strings of length log10(n) to represent n
numbers. Now, if we are talking about the first K naturals and saying there are
K/5^(log10(K)), which is K/(K*(1/2)^log10(K)), which is 2^log10(K), elements in
the set less than or equal to K, and our symbol set is restricted to 1's an
0's, so we have 2^log(n) possible strings.....Ummm, I'm going to have to mull
this over.
hanks for this question. it points me in the direction of an area that needs
development. The calculations given sets of strings and sets of numbers need to
be properly realted,a nd I haven't quite got that worked out yet. I am sure I
can come up with a proper relationship between the two that preserved their
respective measures while resolving any apparent conflict between the measure
of the sets or representation and represented concept. I'll think about this
and get back to you.
Tony Orlow
>
> With set theory you can do also much. The same as with computers. They are also equavalent.
>
> For many mathematicians all consists out of sets.
>
> The is a very basic notion. And you can alo have very different types of sets. But that is another story.
>
By the way, Ed, thanks for the suggesting of looking at Surreal Numbers. I have
begun readin an excellent paper on them at
http://www.tondering.dk/claus/sur15.pdf
They are a very important concept that is related, and may serve as a partial
axiomatic basis for what i am trying to acomplish.
Thanks!
--
Smiles,
Tony
Virgil
Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
> I cetainly don't claim one can do or have an infinite number of
> finite things or events in a finite space or time. That would be
> crazy. It would be like claiming you can have an infinite number of
> distinct whole numbers and 1-unit intervals, and have them all fit
> within a finite interval on the number line. I would never claim any
> such thing! It would be absurd!
Why would that stop you now?
> However, if you want to understand infinity in computer terms
Computers cannot even deal with sufficiently large finite
cardinalities, so why do you think they should be able to deal with
infinite cardinalities.
Virgil
Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
You continue to avoid the issue. What Peano does not state is not the
issue. Peano does say that all naturals, other than the first, are the
result of adding 1 to some previous natural.
> >
> > You seem to have agreed that if the property P is "being a pofnat", the
> > set of all of these is unbounded; that is, you accept that there is no
> > finite size to the set. Yet you say that your theory somehow discounts
> > the existence of boundless sets?
On the contrary, unbounded sets are quite acceptable, but that any
single member of such a set need be unbounded is not a legitimate
consequence of the set being unbounded.
> The mathematics of infinite series discounts the possibility of a set where
> each member adds a constant finite unit to the range of values, and where the
> number of elements is infinite, but the total range is finite.
The "total range" of an infinite series is an infinite sequence which,
for a series of allpositive terms, is always infinite even when it
converges to a finite value.
An infinite
> number of elements each incrementing the overall range results in an infinite
> overall range of values, which means at least one value must be infinite.
> That's not MY theory.
It is, in fact, no one's theory, and is bad mathematics, since it is
false.
A series of strictly increasing terms either has a finite limit or has
no limit at all, but does not, at least n standard mathematics, have an
infinite limit, and even if it did, that limit would not be a member of
the sequence.
> That's proper mathematics. Look up infinite series.
I did. You lose!
> >
> > Here are a couple more bigulosity questions.
> >
> > Q1: When you say N, do we understand that you are claiming that this is
> > the bigulosity of your set (also called N, though I think it should
> > have a subscript T) of "natural numbers", including some infinite ones.
> > Can you say anything about the bigulosity of the pofnats?
> It would be some indeterminate finite value. The set of all finite numbers
> has
> no particular upper bound. In that sense it's almost a self-contradictory
> concept. What saves it from being a self-contradictory concept is the
> unspecified upper bound and resulting vagueness. If a value is declared to be
> finite, but no value or upper bound is defined for it, then it is more like
> "indefinite", being some unspecified finite quantitity.
The above exhibits the self-contradictory nature that it attacks.
In an infinite universe, which ours might be, every physical object is a
finite distance from every other physical object, since anything
infinitely far away would be in a different universe.
Do you mean that, assuming a unique name for each number, the number of
names of these numbers is different from the number of numbers?
Any "counting system" so royally f***ed up is of no use to anybody.
Think about this: if you have any number of numbers you need the same
number of strings to name them, regardless what naming scheme is used.
The lengths of the names is irrelevant to how many names are needed.
Tony Orlow
> In article <MPG.1cfbbb031...@newsstand.cit.cornell.edu>,
> Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
>
> > I cetainly don't claim one can do or have an infinite number of
> > finite things or events in a finite space or time. That would be
> > crazy. It would be like claiming you can have an infinite number of
> > distinct whole numbers and 1-unit intervals, and have them all fit
> > within a finite interval on the number line. I would never claim any
> > such thing! It would be absurd!
>
> Why would that stop you now?
about the set of naturals? (sigh)
>
> > However, if you want to understand infinity in computer terms
>
> Computers cannot even deal with sufficiently large finite
> cardinalities, so why do you think they should be able to deal with
> infinite cardinalities.
don't you read the part you snipped, and see if you can understand it enough to
comment on it? You obviously don't want to learn anything.
>
--
Smiles,
Tony
Tony Orlow
still have finite values. When I asked for justification, I was given an
inductive proof, but when I used an inductive proof to prove that you cannot
have a set of naturals with a size that is larger than the value of any of its
members, I was told that only proved things true for finite n, so the proof I
was given amounts to a proof that all finite naturals are finite, not all
naturals. You guys run around in circles like this, pretending that if your
tail isn't actually down your throat, you're going somewhere.
Now, you said, "By definition a pofnat is finite." Care to point me to where
the definition states this, or you can derive it from the definition, without
assuming it true first?
> > >
> > > You seem to have agreed that if the property P is "being a pofnat", the
> > > set of all of these is unbounded; that is, you accept that there is no
> > > finite size to the set. Yet you say that your theory somehow discounts
> > > the existence of boundless sets?
>
> On the contrary, unbounded sets are quite acceptable, but that any
> single member of such a set need be unbounded is not a legitimate
> consequence of the set being unbounded.
That was Imaginatorium's comment. But, it is a rule of sets of any positive
values that are distinct and separated by a unit from their nearest neighbors,
that the set size cannot be bigger than all the members' values.
>
> > The mathematics of infinite series discounts the possibility of a set where
> > each member adds a constant finite unit to the range of values, and where the
> > number of elements is infinite, but the total range is finite.
>
> The "total range" of an infinite series is an infinite sequence which,
> for a series of allpositive terms, is always infinite even when it
> converges to a finite value.
No, you are confucing the number of elements with the range of their values.
{1,2,3} and {10,20,30} have the same number of elements, but a different range
of values. One can squeeze an infinite number of powers of 1/2 in a space of
1/2, but one cannot squeeze and infinite number of 1's in any finite number.
>
> An infinite
> > number of elements each incrementing the overall range results in an infinite
> > overall range of values, which means at least one value must be infinite.
> > That's not MY theory.
>
> It is, in fact, no one's theory, and is bad mathematics, since it is
> false.
Look up infinite series, you numbskull.
>
> A series of strictly increasing terms either has a finite limit or has
> no limit at all, but does not, at least n standard mathematics, have an
> infinite limit, and even if it did, that limit would not be a member of
> the sequence.
Wake up! An infinite series either converges or diverges. If it converges, the
sum is finite. If it diverges, the sum is either infinite, or oscillating and
therefore indeterminate. A sum of infinite 1's diverges in the sens that the
sum is continuously growing and becomes infinite. Look it up.
>
> > That's proper mathematics. Look up infinite series.
>
> I did. You lose!
Try looking again. Is the sum of 1's less than the sum of inverses of the
naturals (harmonic series), which are all less than 1? If all the terms in the
series are larger than all the terms of a series that diverges to infinity, can
it possibly converge? Any infinite series that converges must have terms that
approach zero as n increases. A series of 1's does not. By the well-established
mathematics of infinite series, your set of naturals is impossible.
> > >
> > > Here are a couple more bigulosity questions.
> > >
> > > Q1: When you say N, do we understand that you are claiming that this is
> > > the bigulosity of your set (also called N, though I think it should
> > > have a subscript T) of "natural numbers", including some infinite ones.
> > > Can you say anything about the bigulosity of the pofnats?
>
> > It would be some indeterminate finite value. The set of all finite numbers
> > has
> > no particular upper bound. In that sense it's almost a self-contradictory
> > concept. What saves it from being a self-contradictory concept is the
> > unspecified upper bound and resulting vagueness. If a value is declared to be
> > finite, but no value or upper bound is defined for it, then it is more like
> > "indefinite", being some unspecified finite quantitity.
>
> The above exhibits the self-contradictory nature that it attacks.
> In an infinite universe, which ours might be, every physical object is a
> finite distance from every other physical object, since anything
> infinitely far away would be in a different universe.
ROFL. That's very mathematical and logically exacting of you. Do you watch Star
Trek a lot? Sure, the universe is infinite, but anything infinitely far away
would be in a different universe. Whatever you say. You have a Cantorian tumor
in your forebrain.
Uh huh. Read on.
No comments? Interesting.
> Think about this: if you have any number of numbers you need the same
> number of strings to name them, regardless what naming scheme is used.
> The lengths of the names is irrelevant to how many names are needed.
Well, to express N numbers in binary, you need strings of length log2(N), so
why do you declare the strings of length N? You only need log2(N) bits. If you
use N bits, then you get 2^N numbers, when you only need N. There is a
difference between N numbers, and the number of numbers represented by N bits.
>
--
Smiles,
Tony
Virgil
Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
> Virgil said:
> > In article <MPG.1cfbbb031...@newsstand.cit.cornell.edu>,
> > Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
> >
> > > I cetainly don't claim one can do or have an infinite number of
> > > finite things or events in a finite space or time. That would be
> > > crazy. It would be like claiming you can have an infinite number of
> > > distinct whole numbers and 1-unit intervals, and have them all fit
> > > within a finite interval on the number line. I would never claim any
> > > such thing! It would be absurd!
> >
> > Why would that stop you now?
> You are so retarded, you don't even see the reference to your absurd claims
> about the set of naturals? (sigh)
I see your claims, but the only absurdity I see in in them.
> >
> > > However, if you want to understand infinity in computer terms
> >
> > Computers cannot even deal with sufficiently large finite
> > cardinalities, so why do you think they should be able to deal with
> > infinite cardinalities.
> Instead of making comments that are so obvious they make you look dim,
That I make obvious comments is because TO does not seem to see anything
less obvious.
> why don't you read the part you snipped, and see if you can
> understand it enough to comment on it?
That I did not care to comment should be obvious from my snipping. That
That I did not understand is an unwarranted assumption. An appropriate
assumption is that I did not agree.
> You obviously don't want to learn anything.
Not from you, at any rate, in part because you are so careful not to
learn from me.
imagin...@despammed.com
keep saying "look it up", but I've learned about this stuff from
textbooks, and recognise nothing you say. Either give up, or quote some
titles and page numbers of real books that say anything vaguely
resembling what you are claiming.
<snip>
aeo6 Tony Orlow wrote:
> ... But, it is a rule of sets of any positive
> values that are distinct and separated by a unit from their nearest
neighbors,
> that the set size cannot be bigger than all the members' values.
Book title and page number please.
> > > The mathematics of infinite series discounts the possibility of a
set where
> > > each member adds a constant finite unit to the range of values,
and where the
> > > number of elements is infinite, but the total range is finite.
Book title and page number please. (Actually maths books never talk
about "the number of elements being infinite".)
Do you mean that for any x, "sigma(n=1,oo) x diverges" ? Surely no-one
disagrees.
> > An infinite
> > > number of elements each incrementing the overall range results in
an infinite
> > > overall range of values, ***which means at least one value must
be infinite***.
What is "overall range of values"? Text book please. What on earth does
the last bit mean? One value of what?
> > > That's not MY theory.
> >
> > It is, in fact, no one's theory, and is bad mathematics, since it
is
> > false.
> Look up infinite series, you numbskull.
You really are being an idiot, you know. Where is anyone going to look
anything up to find your garbled misunderstanding? Virgil may have a
rather plodding style, but that is no reason to project your own
inadequacy on him.
I'll try to answer your long post more fully when I have time. I
recommend that in the mean time you might search the sci.math archives
for the discussion with "Phil" (try phil and other obvious names to
find the threads). You might be very depressed to see how stunningly
unoriginal your ideas are - Phil went through exactly the same phases:
first the digits-and-dots stuff, which we called 'worms', then the
"twilight zone" where finite numbers meld indistinguishably into
infinite ones, then the threat to write all this in a book/web page -
which never happened of course.
Brian Chandler
http://imaginatorium.org
Tony Orlow
> In article <MPG.1cfbe1f8a...@newsstand.cit.cornell.edu>,
> Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
>
> > Virgil said:
> > > In article <MPG.1cfbbb031...@newsstand.cit.cornell.edu>,
> > > Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
> > >
> > > > I cetainly don't claim one can do or have an infinite number of
> > > > finite things or events in a finite space or time. That would be
> > > > crazy. It would be like claiming you can have an infinite number of
> > > > distinct whole numbers and 1-unit intervals, and have them all fit
> > > > within a finite interval on the number line. I would never claim any
> > > > such thing! It would be absurd!
> > >
> > > Why would that stop you now?
>
> > You are so retarded, you don't even see the reference to your absurd claims
> > about the set of naturals? (sigh)
>
> I see your claims, but the only absurdity I see in in them.
> > >
> > > > However, if you want to understand infinity in computer terms
> > >
> > > Computers cannot even deal with sufficiently large finite
> > > cardinalities, so why do you think they should be able to deal with
> > > infinite cardinalities.
>
> > Instead of making comments that are so obvious they make you look dim,
>
> That I make obvious comments is because TO does not seem to see anything
> less obvious.
>
>
> > why don't you read the part you snipped, and see if you can
> > understand it enough to comment on it?
>
> That I did not care to comment should be obvious from my snipping. That
> That I did not understand is an unwarranted assumption. An appropriate
> assumption is that I did not agree.
quips and leave my argument there unchallenged for all to see. Bore.
>
> > You obviously don't want to learn anything.
>
> Not from you, at any rate, in part because you are so careful not to
> learn from me.
learned that junk 25 years ago. It's not in line with the rest of human
thought, or reality. No amount of repetition is going to chnage that. Do you
think it's some kind of catechism?
>
--
Smiles,
Tony
Tony Orlow
> Tony: I'm just picking out this bit about "infinite series". Look, you
> keep saying "look it up", but I've learned about this stuff from
> textbooks, and recognise nothing you say. Either give up, or quote some
> titles and page numbers of real books that say anything vaguely
> resembling what you are claiming.
>
> <snip>
>
> aeo6 Tony Orlow wrote:
> > ... But, it is a rule of sets of any positive
> > values that are distinct and separated by a unit from their nearest
> neighbors,
> > that the set size cannot be bigger than all the members' values.
>
> Book title and page number please.
>
> > > > The mathematics of infinite series discounts the possibility of a
> set where
> > > > each member adds a constant finite unit to the range of values,
> and where the
> > > > number of elements is infinite, but the total range is finite.
>
> Book title and page number please. (Actually maths books never talk
> about "the number of elements being infinite".)
>
> Do you mean that for any x, "sigma(n=1,oo) x diverges" ? Surely no-one
> disagrees.
an infinite set of distinct natural numbers which differ from their nearest
neighbors by 1 unit. Each of the infinity of elements adds on to the range and
to the set size. Infinite set size=infinite range of values=> some element is
infinite.
>
> > > An infinite
> > > > number of elements each incrementing the overall range results in
> an infinite
> > > > overall range of values, ***which means at least one value must
> be infinite***.
>
> What is "overall range of values"?
there is no identifiable last element, the overall range cannot be less than
the set size. if the set size is infinite, then there must be two elements in
the set the diference between which is infinite.
> the last bit mean?
> One value of what?
Of an element in the set of naturals. What do you think we're talking about?
>
> > > > That's not MY theory.
> > >
> > > It is, in fact, no one's theory, and is bad mathematics, since it
> is
> > > false.
> > Look up infinite series, you numbskull.
>
> You really are being an idiot, you know. Where is anyone going to look
> anything up to find your garbled misunderstanding? Virgil may have a
> rather plodding style, but that is no reason to project your own
> inadequacy on him.
said no one could possibly disagree, which is what both of you have been doing
for weeks. It's been amazing to me that you could disagree with me on this. I
am not the one being an idiot. At least YOU finally got an inkling. Virgil has
just been a doorknob on this topic.
>
> I'll try to answer your long post more fully when I have time. I
> recommend that in the mean time you might search the sci.math archives
> for the discussion with "Phil" (try phil and other obvious names to
> find the threads). You might be very depressed to see how stunningly
> unoriginal your ideas are - Phil went through exactly the same phases:
> first the digits-and-dots stuff, which we called 'worms', then the
> "twilight zone" where finite numbers meld indistinguishably into
> infinite ones, then the threat to write all this in a book/web page -
> which never happened of course.
institution somewhere. Thanks.
imagin...@despammed.com
Tony Orlow (aeo6) wrote:
> Virgil said:
> > You continue to avoid the issue. What Peano does not state is not
the
> > issue. Peano does say that all naturals, other than the first, are
the
> > result of adding 1 to some previous natural.
> And, you infer from that that you can do it an infinite number of
times, and
> still have finite values.
Nobody ever said you can "do anything an infinite number of times".
People have been telling you over and over again ad nauseam that none
of this stuff is defined by talking about "doing it an infinite number
of times". You do not get from 1/2 + 1/4 + 1/8... to 1 by "doing
anything infinitely many times". Please insert this fact into your
brain, by force if necessary. (Gentle force only, please.)
That is what the definitions of limits are set up for, PRECISELY to
avoid notions of "doing things infinitely many times".
Anyway, the simplified axioms for generating pofnats say: "Zero is a
pofnat, and if n is a pofnat, so is n. Do not ever do anything
infinitely many times." (The second sentence is, of course superfluous,
but might help you.)
Anyway, so you are allow to add 1+1+1... +1 for ANY FINITE number of
1s. There is, obviously, no limit to this finite number, but it is
always a finite number. That's all.
If there is no limit to something, then it is unlimited. Boundless.
There is no end. They are endless. Endless means not that the end is
"vague" and "indistinguishable", it means that the end does not exist.
Like this thread. Except eventually you will tire, and yep, sooner or
later the next one will come along. Amazing, isn't it! (Imagine 130
years of sci.math archives...)
Brian Chandler
http://imaginatorium.org
Tony Orlow
> Sorry - I missed this bit...
>
> Tony Orlow (aeo6) wrote:
> > Virgil said:
> > > You continue to avoid the issue. What Peano does not state is not
> the
> > > issue. Peano does say that all naturals, other than the first, are
> the
> > > result of adding 1 to some previous natural.
> > And, you infer from that that you can do it an infinite number of
> times, and
> > still have finite values.
>
> Nobody ever said you can "do anything an infinite number of times".
> People have been telling you over and over again ad nauseam that none
> of this stuff is defined by talking about "doing it an infinite number
> of times". You do not get from 1/2 + 1/4 + 1/8... to 1 by "doing
> anything infinitely many times". Please insert this fact into your
> brain, by force if necessary. (Gentle force only, please.)
you have an infinite number of them. How do you get this infinite number of
counting numbers? Force yourself to address this question critically,
expecially with respect to your assertion that the counting numbers themselves
never achieve infinite vlaues because you say you cannot count to infinity.
>
> That is what the definitions of limits are set up for, PRECISELY to
> avoid notions of "doing things infinitely many times".
what we were talking about?), but to avoid the need to repeat the iteration,
and provde the ability to jump to infinite and derive the final answer. That's
why it's a GOOD tool for dealing with infinities.
>
> Anyway, the simplified axioms for generating pofnats say: "Zero is a
> pofnat, and if n is a pofnat, so is n. Do not ever do anything
> infinitely many times." (The second sentence is, of course superfluous,
> but might help you.)
due to their inability to count through the haze between finite and infinite.
Such is the nature of counting.
>
> Anyway, so you are allow to add 1+1+1... +1 for ANY FINITE number of
> 1s. There is, obviously, no limit to this finite number, but it is
> always a finite number. That's all.
claim to have an infinite number of naturals in your set. You do still claim
the set is infinite, right? Then stop complaining about me talking about
infinite sets and counting to infinity. That's YOUR claim.
> If there is no limit to something, then it is unlimited. Boundless.
> There is no end. They are endless. Endless means not that the end is
> "vague" and "indistinguishable", it means that the end does not exist.
"No end". Isn't that what "infinite" means? Look up the etymology. That applies
to the set size, and also the element values.
>
> Like this thread. Except eventually you will tire, and yep, sooner or
> later the next one will come along. Amazing, isn't it! (Imagine 130
> years of sci.math archives...)
Imagine 130 years of foolishness finally coming to an end, and discovering too
late that you have been fighting to save something that is worse than useless.
imagin...@despammed.com
assume you agree you cannot find any "authoritative" support. That need
not stop you from arguing, but kindly desist from using the expression
"Look it up!" in this context.
Right, so far. Now, as always, your standard, ginormous non sequitur:
> Infinite set size=infinite range of values=> some element is
> infinite.
Why? Is there an axiom somewhere that says that the limit of an
infinite series must appear somewhere in the set? No, there isn't.
What's more, it's plainly not true. Yet this single nonfact must be
responsible for the vast majority of crank mathematics, I would guess.
Want to prove 0.999... != 1 ? Easy: no element of the sequence 0.9,
0.99, 0.999, ... actually equals 1, and therefore 1 is not present in
the sequence, But by Krankaxiom 1, the limit must be in there
somewhere, therefore 1 is not the limit.
(Can't remember: what's your position on this?)
> > > > An infinite
> > > > > number of elements each incrementing the overall range
results in
> > an infinite
> > > > > overall range of values, ***which means at least one value
must
> > be infinite***.
> >
> > What is "overall range of values"?
> The difference between the vlaues of the largest and smallest
elements.
There is no last element.
Even if
> there is no identifiable last element,
Ah, there's a last element, but it can't be "identified"? What sort of
element is that? What happens when you add 1 to it? Hmm, can't be a
natural number then, can it. So it can't actually be in the set, can
it. Look, you have intuitions about finite sets, which tell you that
you can always string them out so there's one at This End, and another
at That End. Your entire enterprise is rather obviously predicated on
the idea that even when there isn't a That End, you are just going to
manage, somehow, to treat things as thought That End was there, in some
ineffable, can't-quite-be-identified, way. This is going to fail,
because when things are not there, building logic on the supposition
that they are leads to contradiction.
the overall range cannot be less than
> the set size. if the set size is infinite, then there must be two
elements in
> the set the diference between which is infinite.
This is obviously false. OK, supposing the left one of these elements
is at '1'. The right element, hmm, can't be at the end, because the end
doesn't exist. But it can't actually be in the "middle" (in the sense
of the part that is none of the end or ends however many there are),
because all of the numbers in the middle are simply ordinary finite
numbers. Would it be near where the end would be if it did exist?
Roughly how far to the right would that be? Ah, yes, the Twilight
Zone. (This is a parody, Tony. There is no Twilight Zone in anything
even vaguely resembling mathematics.)
Brian Chandler
http://imaginatorium.org
Virgil
Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
> > You continue to avoid the issue. What Peano does not state is not the
> > issue. Peano does say that all naturals, other than the first, are the
> > result of adding 1 to some previous natural.
> And, you infer from that that you can do it an infinite number of times, and
> still have finite values.
We do not 'infer' any such thing. We take induction as an axiom to see
where it will lead. There is no justification either necessary or
possible for taking it as an axiom except that it gives us the easiest
known basis for constructing models for the number systems we are
accustomed to using.
> When I asked for justification, I was given an
> inductive proof, but when I used an inductive proof to prove that you cannot
> have a set of naturals with a size that is larger than the value of any of
> its
> members, I was told that only proved things true for finite n, so the proof I
> was given amounts to a proof that all finite naturals are finite, not all
> naturals. You guys run around in circles like this, pretending that if your
> tail isn't actually down your throat, you're going somewhere.
Perhaps it is your failure to read what the inductive axiom and the
other Peano axioms say that leads you to misunderstand what they say.
>
> Now, you said, "By definition a pofnat is finite." Care to point me to where
> the definition states this, or you can derive it from the definition, without
> assuming it true first?
AS I have never defined the finiteness of anything but sets in this
discussion, the finiteness of natural numbers, unless they represent
sets, is irrelevant. So let us consider a situation in which each
natural "is" a set.
Using the von Neumann definition, so that naturals will be sets, we
start with 0 = {}, the empty set, and define inductively
n+1 = (n union {n}).
It then transpires that 1 = {0}, 2 = {0,1}, 3 = {0,1,2}, and so on, so
that each n is a set containing n members, all proper subsets of itself.
Now if one the Cantor definition of finiteness for sets, a set S will be
finite unless one can find an injection from S to some proper subset of
S.
Once one accepts that definition, one can then prove that 0 is finite,
and prove that n+1 is finite whenever n is finite, so that each natural,
by induction, is finite.
Note that none of this says anything at all about the set of all
naturals, and, in fact, there has not yet been anything to claim that
the collection of all of them actually is a set.
>
> > > >
> > > > You seem to have agreed that if the property P is "being a pofnat", the
> > > > set of all of these is unbounded; that is, you accept that there is no
> > > > finite size to the set. Yet you say that your theory somehow discounts
> > > > the existence of boundless sets?
> >
> > On the contrary, unbounded sets are quite acceptable, but that any
> > single member of such a set need be unbounded is not a legitimate
> > consequence of the set being unbounded.
> That was Imaginatorium's comment. But, it is a rule of sets of any positive
> values that are distinct and separated by a unit from their nearest
> neighbors,
> that the set size cannot be bigger than all the members' values.
"Bigger than all the members values" is ambigiuous. It must be bigger
than set of all values, or bigger than a particular value.
And if it is bigger than a particular value it must be bigger than
infinitely many of them, and you should be able to mane at least one of
them out of so many.
> >
> > > The mathematics of infinite series discounts the possibility of a set
> > > where
> > > each member adds a constant finite unit to the range of values, and where
> > > the
> > > number of elements is infinite, but the total range is finite.
> >
> > The "total range" of an infinite series is an infinite sequence which,
> > for a series of allpositive terms, is always infinite even when it
> > converges to a finite value.
> No, you are confucing the number of elements with the range of their values.
> {1,2,3} and {10,20,30} have the same number of elements, but a different
> range
> of values. One can squeeze an infinite number of powers of 1/2 in a space of
> 1/2, but one cannot squeeze and infinite number of 1's in any finite number.
What you are saying is that a series cannot converge unless the sequence
of its terms has limit zero. Hardly news.
> >
> > An infinite
> > > number of elements each incrementing the overall range results in an
> > > infinite
> > > overall range of values, which means at least one value must be infinite.
> > > That's not MY theory.
The hell it ain't! The "limit" of a divergent series DOES NOT EXIST AT
ALL. There is no such thing. You are basing your argument on the
existence of the non-existent.
> >
> > It is, in fact, no one's theory, and is bad mathematics, since it is
> > false.
> Look up infinite series, you numbskull.
I have, and found TO wrong! Divergent series do not have limit values.
> >
> > A series of strictly increasing terms either has a finite limit or has
> > no limit at all, but does not, at least n standard mathematics, have an
> > infinite limit, and even if it did, that limit would not be a member of
> > the sequence.
> Wake up! An infinite series either converges or diverges. If it converges,
> the
> sum is finite. If it diverges, the sum is either infinite, or oscillating and
> therefore indeterminate.
According to all my tests, divergent series do not have limits at all,
only convergent series have limits.
A sum of infinite 1's diverges in the sens that the
> sum is continuously growing and becomes infinite. Look it up.
> >
> > > That's proper mathematics. Look up infinite series.
> >
> > I did. You lose!
> Try looking again.
I did again, you lose again. Divergent series do not have limits.
> No comments? Interesting.
> > Think about this: if you have any number of numbers you need the same
> > number of strings to name them, regardless what naming scheme is used.
> > The lengths of the names is irrelevant to how many names are needed.
> Well, to express N numbers in binary, you need strings of length
> log2(N)
You need N strings to express N numbers. The string lengths can be 1
digit each if you have enough different digits.
> so why do you declare the strings of length N?
I don't, try reading what I say , not what you think I say.
You keep talking about how long while I am talking about how many.
How long the strings are is not the same as how many strings there are.
guenther vonKnakspot
Tony Orlow (aeo6) wrote:
> Yes, that is precisely what I mean, and that is precisely what is
happening in
> an infinite set of distinct natural numbers which differ from their
nearest
> neighbors by 1 unit. Each of the infinity of elements adds on to the
range and
> to the set size. Infinite set size=infinite range of values=> some
element is
> infinite.
Say Orlow, what is an infinite element of the natural numbers? What is
an infinite number? And please mark my words, I am not asking what an
infinite amount is.
Sorry I could not get back to this same question in the other thread,
but I was away for a while, and here you are clearly talking of
infinite numbers.
> Smiles,
>
> Tony
Regards
Tony Orlow
> Tony: I see you name no book or website (wikipedia, for example). I
> assume you agree you cannot find any "authoritative" support. That need
> not stop you from arguing, but kindly desist from using the expression
> "Look it up!" in this context.
Do you know how to use Google? Do you know how to spell "infinite series"?
That infinite series that don't have a limit of zero for terms, as n goes to
infinity, diverge:
http://mathworld.wolfram.com/DivergenceTests.html
Series in general:
http://mathworld.wolfram.com/Series.html
Happy reading!
I have been through this umpteen times. Each added natural adds 1 to the range
of element values in the set. Agree?
There are an infinite number of elements in the set. Agree?
The range of values is the sum of that infinite number of increments of 1 in
the element value. Agree?
If we want to have an infinite number of points on a finite line segment, then
those points must be dense on that line at least at some point in the segment.
Agree?
> What's more, it's plainly not true. Yet this single nonfact must be
> responsible for the vast majority of crank mathematics, I would guess.
> Want to prove 0.999... != 1 ? Easy: no element of the sequence 0.9,
> 0.99, 0.999, ... actually equals 1, and therefore 1 is not present in
> the sequence, But by Krankaxiom 1, the limit must be in there
> somewhere, therefore 1 is not the limit.
> (Can't remember: what's your position on this?)
So, 0.99999........<>1? I thought that was considered crankdom.
>
>
> > > > > An infinite
> > > > > > number of elements each incrementing the overall range
> results in
> > > an infinite
> > > > > > overall range of values, ***which means at least one value
> must
> > > be infinite***.
> > >
> > > What is "overall range of values"?
> > The difference between the vlaues of the largest and smallest
> elements.
>
> There is no last element.
> Even if
> > there is no identifiable last element,
>
Try finishing the sentence before responding......
> Ah, there's a last element, but it can't be "identified"? What sort of
> element is that? What happens when you add 1 to it? Hmm, can't be a
> natural number then, can it. So it can't actually be in the set, can
> it. Look, you have intuitions about finite sets, which tell you that
> you can always string them out so there's one at This End, and another
> at That End. Your entire enterprise is rather obviously predicated on
> the idea that even when there isn't a That End, you are just going to
> manage, somehow, to treat things as thought That End was there, in some
> ineffable, can't-quite-be-identified, way. This is going to fail,
> because when things are not there, building logic on the supposition
> that they are leads to contradiction.
>
....because it looks dumb to complain when the response is in the sentence
already:
>
> the overall range cannot be less than
> > the set size. if the set size is infinite, then there must be two
> elements in
> > the set the diference between which is infinite.
>
> This is obviously false.
You mean obviously true. You said so already.
> OK, supposing the left one of these elements
> is at '1'. The right element, hmm, can't be at the end, because the end
> doesn't exist. But it can't actually be in the "middle" (in the sense
> of the part that is none of the end or ends however many there are),
> because all of the numbers in the middle are simply ordinary finite
> numbers. Would it be near where the end would be if it did exist?
> Roughly how far to the right would that be? Ah, yes, the Twilight
> Zone. (This is a parody, Tony. There is no Twilight Zone in anything
> even vaguely resembling mathematics.)
What gloppy thought patterns! You claim to have an infinite number of naturals,
each 1 unit greater than the last, so you are counting an infinite number of
1's, and you claim that final sum, that final distance from the starting point,
is less than infintiely far away. Either you get through the twilite zone and
achieve infinity, in which case you have an infinite number of naturals which
include infinite values, or you don't, in which case you have a set that is
still finite with values that are still finite. Please refer to infinite series
and to the definition of the natural numbers and see if you can't make sense of
this statement. That this should take so long to get through to you folks is a
clear indication that correction is needed. If you think I will tire out like
Phil (what was his name?), don't hold your breath. These symptoms of ill-logic
make me all the more committed to eradicating this malady.
imagin...@despammed.com
ISTB. Faced with the notion that there's 1, and 2, and 3, and 10^64
(which has a name in Japanese: muryou-daisuu, lit.
"unmeasurable-large-number'"), and 10^64 + 1, and as Wolf's grandchild
can see, there ain't no limit to this, it ain't going to end, however
long we keep counting genuine (finite) numbers, then there are two
things you can do:
1. Give up, essentially. Mutter the expression "potential infinity",
and assert that you are unable to consider the totality of such
numbers, since, after all, the sequence never ends, and you cannot,
even in principle ever look at every one. Totally consistent position.
2. Use a Leap: an axiom, actually. Say "Well, I _am_ prepared to
consider the totality of these numbers, even though I could never
complete the job of working through them." This is what set theory
does, and it is in fact consistent - at least in so far as your
misperceived contradictions are just that.
You appear to be trying course 2a: "Well, I know I couldn't really ever
get to the end of counting em, but I'm darn well going to try. I know,
I'll pretend there's a special place I can go, called Infinity, and
when I get there, I'll be able to finish counting them." This is not a
consistent position, and indeed makes no sense at all.
> > That is what the definitions of limits are set up for, PRECISELY to
> > avoid notions of "doing things infinitely many times".
> Not to avoid the notion, which persists in the term "infinite series"
(remember
> what we were talking about?), but to avoid the need to repeat the
iteration,
> and provde the ability to jump to infinite and derive the final
answer. That's
> why it's a GOOD tool for dealing with infinities.
"Jump to infinite" - does that mean the death-defying leap that
actually gets to the end of the infinite series? Have you actually
tried looking up "infinite series" yourself, and reading the
definitions rather carefully?
> >
> > Anyway, the simplified axioms for generating pofnats say: "Zero is
a
> > pofnat, and if n is a pofnat, so is n. Do not ever do anything
> > infinitely many times." (The second sentence is, of course
superfluous,
> > but might help you.)
> Not at all superfluos, nor necessary. Peano doesn't say that. People
infer it
> due to their inability to count through the haze between finite and
infinite.
> Such is the nature of counting.
Such is precisely _not_ the nature of counting. What is this haze? How
come you know about it? Have you been somewhere and seen it? Or did you
create a whole new set of axioms and derive it? Or is there just the
Axiom of the Twilight Zone?
> >
> > Anyway, so you are allow to add 1+1+1... +1 for ANY FINITE number
of
> > 1s. There is, obviously, no limit to this finite number, but it is
> > always a finite number. That's all.
> Not in infinite series, which is exactly what you are talking about
when you
> claim to have an infinite number of naturals in your set. You do
still claim
> the set is infinite, right? Then stop complaining about me talking
about
> infinite sets and counting to infinity. That's YOUR claim.
No, I'm carefully avoiding the I-word, because it causes your eyeballs
to rotate counterintuitively. I talked only of this bunch of ANY FINITE
number of 1s which you are allowed to add. I have said that there is no
limit to this finite number, but that's all.
>
> > If there is no limit to something, then it is unlimited. Boundless.
> > There is no end. They are endless. Endless means not that the end
is
> > "vague" and "indistinguishable", it means that the end does not
exist.
> "No end". Isn't that what "infinite" means? Look up the etymology.
That applies
> to the set size, and also the element values.
So you mean if I say I have this bunch of finite numbers. A lot of
them, but they're all finite, and I say, even that there is simply no
specific limit to how big these finite numbers can be, one of them
instantly morphs into something different? What if I keep quite about
the limit: I've got a set of finite numbers.
> Imagine 130 years of foolishness finally coming to an end, and
discovering too
> late that you have been fighting to save something that is worse than
useless.
Yeah, amazingly, that's *exactly* what Phil said. (Sadly, I am quite
sure that in another 130 years time there will _still_ be 0.999... != 1
threads.)
Brian Chandler
http://imaginatorium.org
Ross A. Finlayson
Far be it for me to step in for Virgil, but he's often quite right.
When he says I'm wrong, which he does less these days, he's wrong,
unless I agree, but otherwise he's quite expert. He doesn't reply to
me anymore anyways.
Please don't read this as discouragement. On the contrary, you have
some solid and correct intuitive notions about things infinite and
infinite sets.
One consideration you have is to compare infinite sets of integers. As
numbers, that is directly addressed by number theory. The asymptotic
density of the even integers within the integers, for example, is one
half, half of the integers are even. There are vagaries in the number
theoretic definition, for example the standard number theory does not
address comparable infinitesimal ratios.
When you're dealing with collections of integers, or other numbers,
instead of collections of pure sets, there are some differences.
In comparing sets, their size or cardinality or relative densities
within a superset, there are a variety of methods. For finite sets,
with distinct set member elements, they're very obvious methods.
One notion of the set size comparison is that a proper subset A of a
set B is smaller than that set: A < B => |A| < |B|, using the pipes
notation to indicate the contextual set size operand. Directly, A\B,
that's A setminus B, is empty, and B\A is nonempty. People here on
sci.math used to complain extendedly that that was no good, and now
instead it's said that is so and as necessary it is referred to Fred
Katz' MIT dissertation.
That doesn't help so much when the sets are partially or completely
disjoint, instead then each is compared to their union, and a
quantitatve, or qualitative, measure is often simply determined from
the sets' elements "distribution" as elements and often numbers.
Another way that a set is B larger than another A is when B has
infinitely many proper subsets that are proper supersets of A. For
example, the rational numbers have infinitely many subsets that contain
all the integers and some non-integer rationals, where an integer is a
rational, real, complex, and dully hypercomplex number and the origin
is 0 and (0, 0, 0, ...) and even (0, 0, 0, ...) x (0, 0, 0, ...).
That's the same thing as 0 at a high (or low, based on perspective)
level.
The powerset, for a set that is not its own powerset, is obviously
larger than the set. In the realm of pure sets, that is basically what
is discussed when we're talking Cantorian set theories transfinite
cardinals or lack thereof. If you want to argue against them you must
become familiar with the powerset and antidiagonal results.
As well, if you want to consider the bijections between the set of
integers and real numbers, then you must be familiar with the
antidiagonal and nested intervals results.
That might seem obvious, those results might seem obvious, but there
are some basically contradictions to Cantorian set theory as it is
often practiced. For example, the cardinality of the reals, or c, has
been shown equivalent, that a bijection exists, to Aleph_1, Aleph_2,
Aleph_3, .... (As well, where infinite sets are equivalent, for
example as in my null-axiom theory, Aleph_0.) Hausdorff says: a
countable union of coutable sets might be uncountable. ZF is
inconsistent because of the Domain Principle, and the reals are
equivalent to the natural integers because of the Well-Ordering
Principle.
Basically, and this might seem counterintuitive, in pure sets no
infinite set is regular.
I approached infinite sets as basically an extension of the
consideration of the rationalization of the limit, delta-epsilon,
Cauchy-Weierstrass. Newton and Leibniz's infinitesimal calculus, while
railed against as not the summation of infinitely many infinitesimally
small areas, is. In some modern nonstandard treatments of the real
numbers, for example the as-called Finlayson numbers or as I claim to
show them the real numbers of the continuous real number line, being
able to put aside the notion that infinite sets are not equivalent,
which is wrong, leads to some what I think are going to be useful
analytical properties and methods of the real numbers as points on a
line, or not.
Ross F.
--
"Also, consider this: the unit impulse function times
one less twice the unit step function times plus/minus
one is the mother of all wavelets."
Tony Orlow
> In article <MPG.1cfbe74ab...@newsstand.cit.cornell.edu>,
> Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
>
> > > You continue to avoid the issue. What Peano does not state is not the
> > > issue. Peano does say that all naturals, other than the first, are the
> > > result of adding 1 to some previous natural.
>
>
> > And, you infer from that that you can do it an infinite number of times, and
> > still have finite values.
>
> We do not 'infer' any such thing. We take induction as an axiom to see
> where it will lead. There is no justification either necessary or
> possible for taking it as an axiom except that it gives us the easiest
> known basis for constructing models for the number systems we are
> accustomed to using.
including infinite naturals, is flawed. You cannot maintain finiteness as a
property that is preserved over an infinite domain where the value is
increasing constantly. On the other hand, if you claim induction only holds for
finite iterations, based on your pre-assumption that naturals are finite, then
all you have proven is that finite naturals are finite.
>
> > When I asked for justification, I was given an
> > inductive proof, but when I used an inductive proof to prove that you cannot
> > have a set of naturals with a size that is larger than the value of any of
> > its
> > members, I was told that only proved things true for finite n, so the proof I
> > was given amounts to a proof that all finite naturals are finite, not all
> > naturals. You guys run around in circles like this, pretending that if your
> > tail isn't actually down your throat, you're going somewhere.
>
> Perhaps it is your failure to read what the inductive axiom and the
> other Peano axioms say that leads you to misunderstand what they say.
I have read and understood. You are being inconsistent in using them. It is my
understanding that induction proves things, when used properly, for all n in N,
to infinite iterations. There is no restriction of finiteness either stated
explicitly, or justifiably inferred. You argument that there is is entirely
circular nonsense.
> >
> > Now, you said, "By definition a pofnat is finite." Care to point me to where
> > the definition states this, or you can derive it from the definition, without
> > assuming it true first?
>
> AS I have never defined the finiteness of anything but sets in this
> discussion, the finiteness of natural numbers, unless they represent
> sets, is irrelevant. So let us consider a situation in which each
> natural "is" a set.
HUH??? The main issue here is an infinite set of finite natural numbers. You
insist all naturals are finite, do you not? If not, and you are willing to
allow infinite whole numbers in the infinite set, then we have no quarrel on
that issue. But you have repeatedly held that position, until you just began to
waffle on it.
By the way, my inductive proof used sets generated by a ntural, as you are
about to do, and you seemed entirely confused, and I heard complaints about
drawing conclusions for theset based on its elements, which is not what I was
doing. Shall I accuse you of the same? let's see what you got.....
>
> Using the von Neumann definition, so that naturals will be sets, we
> start with 0 = {}, the empty set, and define inductively
> n+1 = (n union {n}).
>
> It then transpires that 1 = {0}, 2 = {0,1}, 3 = {0,1,2}, and so on, so
> that each n is a set containing n members, all proper subsets of itself.
yeah, that's exactly what I did, starting with 1.
>
> Now if one the Cantor definition of finiteness for sets, a set S will be
> finite unless one can find an injection from S to some proper subset of
> S.
That definition sux. Prove your case without resorting to Cantor's axioms. Use
ANY other established area of math besides that nonsense. ANY other area.
>
> Once one accepts that definition, one can then prove that 0 is finite,
> and prove that n+1 is finite whenever n is finite, so that each natural,
> by induction, is finite.
No, that inductive proof relies on an infinite number of additions of a unit
having a finite sum, which is clearly false. The sum of an infiite number of
1's is infinite. This is the point where you folks claimed induction only holds
for finite values, which blows a hole in your own argument that all values are
finite, since now you are claiming that all finite naturals are finite, which
is less than tautological.
>
> Note that none of this says anything at all about the set of all
> naturals, and, in fact, there has not yet been anything to claim that
> the collection of all of them actually is a set.
Induction ala Peano states that the property will hold for all n in N, if p(0)
is true (or p(1)) and if p(n)->p(n+1). Induction proves a property true for the
set by definition.
>
> >
> > > > >
> > > > > You seem to have agreed that if the property P is "being a pofnat", the
> > > > > set of all of these is unbounded; that is, you accept that there is no
> > > > > finite size to the set. Yet you say that your theory somehow discounts
> > > > > the existence of boundless sets?
> > >
> > > On the contrary, unbounded sets are quite acceptable, but that any
> > > single member of such a set need be unbounded is not a legitimate
> > > consequence of the set being unbounded.
> > That was Imaginatorium's comment. But, it is a rule of sets of any positive
> > values that are distinct and separated by a unit from their nearest
> > neighbors,
> > that the set size cannot be bigger than all the members' values.
>
> "Bigger than all the members values" is ambigiuous. It must be bigger
> than set of all values, or bigger than a particular value.
(sigh) there exists n in N s.t n>=|N|. At least one member must have a value as
large as the size of the set. There is nothing remotely ambiguous here, but
nice try.
>
> And if it is bigger than a particular value it must be bigger than
> infinitely many of them, and you should be able to mane at least one of
> them out of so many.
I am saying, if you learn how to read, that the set size is LESS THAN OR EQUAL
TO the value of AT LEAST ONE MEMBER of the set.
>
>
> > >
> > > > The mathematics of infinite series discounts the possibility of a set
> > > > where
> > > > each member adds a constant finite unit to the range of values, and where
> > > > the
> > > > number of elements is infinite, but the total range is finite.
> > >
> > > The "total range" of an infinite series is an infinite sequence which,
> > > for a series of allpositive terms, is always infinite even when it
> > > converges to a finite value.
> > No, you are confucing the number of elements with the range of their values.
> > {1,2,3} and {10,20,30} have the same number of elements, but a different
> > range
> > of values. One can squeeze an infinite number of powers of 1/2 in a space of
> > 1/2, but one cannot squeeze and infinite number of 1's in any finite number.
>
> What you are saying is that a series cannot converge unless the sequence
> of its terms has limit zero. Hardly news.
No, it's not news at all. Why has it taken three weeks and twenty repetitions
to get you to even acknowledge what I am talking about?
Does the sequence (1,1,1,1.....) have a limit of zero? No? Good, then you get
my point. No?
> > >
> > > An infinite
> > > > number of elements each incrementing the overall range results in an
> > > > infinite
> > > > overall range of values, which means at least one value must be infinite.
> > > > That's not MY theory.
>
>
> The hell it ain't! The "limit" of a divergent series DOES NOT EXIST AT
> ALL. There is no such thing. You are basing your argument on the
> existence of the non-existent.
The limit of the values, or the sum of its terms? Infinite series come in three
varieties: convergent on a finite sum, divergent with oscillating sum, or
divergent with infinite sum. The infinite series sigma(x) x=0->oo, neither
oscillates, nor decreases, so the sum of that series is infinite. That's not MY
theory. I learned it in high school.
> > >
> > > It is, in fact, no one's theory, and is bad mathematics, since it is
> > > false.
> > Look up infinite series, you numbskull.
>
> I have, and found TO wrong! Divergent series do not have limit values.
I can't tell if you are referring to sums when you mean limit values. Certainly
(1,1,1,1....) has a limit value of...1! The sum is infinite. What are you
talking about?
> > >
> > > A series of strictly increasing terms either has a finite limit or has
> > > no limit at all, but does not, at least n standard mathematics, have an
> > > infinite limit, and even if it did, that limit would not be a member of
> > > the sequence.
> > Wake up! An infinite series either converges or diverges. If it converges,
> > the
> > sum is finite. If it diverges, the sum is either infinite, or oscillating and
> > therefore indeterminate.
>
> According to all my tests, divergent series do not have limits at all,
> only convergent series have limits.
Is the sum of an infinite series of 1's infinite or not?
Just answer that one simple question.
> A sum of infinite 1's diverges in the sens that the
> > sum is continuously growing and becomes infinite. Look it up.
> > >
> > > > That's proper mathematics. Look up infinite series.
> > >
> > > I did. You lose!
>
> > Try looking again.
>
> I did again, you lose again. Divergent series do not have limits.
KWocckk!! Poly wanna cracker?
>
>
> > No comments? Interesting.
> > > Think about this: if you have any number of numbers you need the same
> > > number of strings to name them, regardless what naming scheme is used.
> > > The lengths of the names is irrelevant to how many names are needed.
>
> > Well, to express N numbers in binary, you need strings of length
> > log2(N)
>
> You need N strings to express N numbers. The string lengths can be 1
> digit each if you have enough different digits.
Duh! In binary we have two symbols in our set. That's what binary means. Would
you like a link that explains this?
>
> > so why do you declare the strings of length N?
>
> I don't, try reading what I say , not what you think I say.
>
> You keep talking about how long while I am talking about how many.
>
> How long the strings are is not the same as how many strings there are.
>
They are quite realted, as I have repeatedly pointed out. If we say thestrings
go to maximum length log2(N), that gives us sufficient strings to represent N
different numbers. The strings don't need to be of length N.
There are very fine measures possible. Why do you want to keep things vague?
--
Smiles,
Tony
Tony Orlow
digits. I have shown how an infinite number of strings that are generated from
a finite set of symbols must logically require infinitely long strings, if each
is unique. If this is a set of digital numbers, digits extending infinitely to
the left of the digital point represent infinite values. It is my position that
any infinite set of natural numbers must contain such strings pursuant to
informaton theory and combinatorics.
--
Smiles,
Tony
Tony Orlow
little thing and argue about infinite series, you might understand a little
more than you seem to about what I am talking about. There are areas of this I
have barely touched on.
>
>
> > > That is what the definitions of limits are set up for, PRECISELY to
> > > avoid notions of "doing things infinitely many times".
> > Not to avoid the notion, which persists in the term "infinite series"
> (remember
> > what we were talking about?), but to avoid the need to repeat the
> iteration,
> > and provde the ability to jump to infinite and derive the final
> answer. That's
> > why it's a GOOD tool for dealing with infinities.
>
> "Jump to infinite" - does that mean the death-defying leap that
> actually gets to the end of the infinite series? Have you actually
> tried looking up "infinite series" yourself, and reading the
> definitions rather carefully?
if we could add them aup all the way to infinity.
>
>
> > >
> > > Anyway, the simplified axioms for generating pofnats say: "Zero is
> a
> > > pofnat, and if n is a pofnat, so is n. Do not ever do anything
> > > infinitely many times." (The second sentence is, of course
> superfluous,
> > > but might help you.)
> > Not at all superfluos, nor necessary. Peano doesn't say that. People
> infer it
> > due to their inability to count through the haze between finite and
> infinite.
> > Such is the nature of counting.
>
> Such is precisely _not_ the nature of counting. What is this haze? How
> come you know about it? Have you been somewhere and seen it? Or did you
> create a whole new set of axioms and derive it? Or is there just the
> Axiom of the Twilight Zone?
infinity, because every next number after a finite number is finite. I
acknowledge this zone. I don't concentrate on it as the only salient feature of
the set.
>
> > >
> > > Anyway, so you are allow to add 1+1+1... +1 for ANY FINITE number
> of
> > > 1s. There is, obviously, no limit to this finite number, but it is
> > > always a finite number. That's all.
> > Not in infinite series, which is exactly what you are talking about
> when you
> > claim to have an infinite number of naturals in your set. You do
> still claim
> > the set is infinite, right? Then stop complaining about me talking
> about
> > infinite sets and counting to infinity. That's YOUR claim.
>
> No, I'm carefully avoiding the I-word, because it causes your eyeballs
> to rotate counterintuitively. I talked only of this bunch of ANY FINITE
> number of 1s which you are allowed to add. I have said that there is no
> limit to this finite number, but that's all.
your position that it WAS.
>
> >
> > > If there is no limit to something, then it is unlimited. Boundless.
> > > There is no end. They are endless. Endless means not that the end
> is
> > > "vague" and "indistinguishable", it means that the end does not
> exist.
> > "No end". Isn't that what "infinite" means? Look up the etymology.
> That applies
> > to the set size, and also the element values.
>
> So you mean if I say I have this bunch of finite numbers. A lot of
> them, but they're all finite, and I say, even that there is simply no
> specific limit to how big these finite numbers can be, one of them
> instantly morphs into something different? What if I keep quite about
> the limit: I've got a set of finite numbers.
successor by a constant finite value, they must achieve infinite values. If you
tell me they are all finite, then I can conclude logically and correctly that
there are a finite number of them, no more than the greatest finite value they
can have, and certainly not infinite.
>
>
> > Imagine 130 years of foolishness finally coming to an end, and
> discovering too
> > late that you have been fighting to save something that is worse than
> useless.
>
> Yeah, amazingly, that's *exactly* what Phil said. (Sadly, I am quite
> sure that in another 130 years time there will _still_ be 0.999... != 1
> threads.)
working on it for 25 years, on and off. What was Phil's name? Do you remember?
Do you know where I can find his stuff on archive?
Tony Orlow
I have considered these and other areas. Looking at surreals now, and working
on constructons on the number circle. Thanks for the encouraging words. There
isn't discussion about this topic because it's obviously correct. There are
problems that many perceive, though cardinality is ahrd knot to untie.
--
Smiles,
Tony
imagin...@despammed.com
to delete a chunk of it, but forgot. For the record, my position is
exactly the same as that of every mathematician on sci.math, every
teacher of mathematics in a real university department, and every
textbook on the subject. There is an infinite set of (finite) natural
numbers. This position is not going to change, so if you think it has,
look out for irony, or mistyping brought on by weariness.
Tony Orlow (aeo6) wrote:
> imagin...@despammed.com said:
> > You appear to be trying course 2a: "Well, I know I couldn't really
ever
> > get to the end of counting em, but I'm darn well going to try. I
know,
> > I'll pretend there's a special place I can go, called Infinity, and
> > when I get there, I'll be able to finish counting them." This is
not a
> > consistent position, and indeed makes no sense at all.
> That is not my position or approach at all.
OK, well everywhere below that I see you apparently "reaching infinity"
I'll mark it with a special symbol ^oo^ (maybe a road sign meaning
'infinity ahead!' or a pair of spectacles).
> > > > That is what the definitions of limits are set up for,
PRECISELY to
> > > > avoid notions of "doing things infinitely many times".
> > > Not to avoid the notion, which persists in the term "infinite
series"
> > (remember
> > > what we were talking about?), ...
Just to point out that an "infinite series" is one that does not end.
Not one in which you get to the non-existent end by jumping.
but to avoid the need to repeat the
> > iteration,
> > > and provde the ability to jump to infinite and derive the final
> > answer. That's
> > > why it's a GOOD tool for dealing with infinities.
> >
> > "Jump to infinite" - does that mean the death-defying leap that
> > actually gets to the end of the infinite series? Have you actually
> > tried looking up "infinite series" yourself, and reading the
> > definitions rather carefully?
> Uh huh. Is 1 exactly the sum of 1/2^n for n=1 to oo? Yep, exactly
what we'd get
> if we could add them aup all the way to infinity.[^oo^]
So you agree that every next number after a finite number is finite.
That's great. What happens after that? Is there an "after that"? I
mean, these finite numbers - each one is followed by another finite
number, isn't it? So really speaking, there's, er, no end to them? No
end at all. So how, please does one get to the "next zone"? No good
just charging ahead, because you will simply get the next pofnat, and
the next. Do you have to leap slightly sideways? Can you give a
rigorous explanation of how you know you won't crash into a sequence of
chocolate sundaes?
> > > > If there is no limit to something, then it is unlimited.
Boundless.
> > > > There is no end. They are endless. Endless means not that the
end
> > is
> > > > "vague" and "indistinguishable", it means that the end does not
> > exist.
> > > "No end". Isn't that what "infinite" means? Look up the
etymology.
> > That applies
> > > to the set size, and also the element values.
No, it doesn't apply to the element values, because I'm talking only
about the unending supply of pofnats. Remember, there's always a next
one. I can charge ahead in an unending sequence of finite numbers, and
you can't stop me. (Well, I suppose if you *stopped* me, I'd stop at a
finite number in the "middle" somewhere. As soon as you let go, I'd
just carry on. This process would never end.)
> > > Imagine 130 years of foolishness finally coming to an end, and
> > discovering too
> > > late that you have been fighting to save something that is worse
than
> > useless.
> >
> > Yeah, amazingly, that's *exactly* what Phil said. (Sadly, I am
quite
> > sure that in another 130 years time there will _still_ be 0.999...
!= 1
> > threads.)
> Perhaps Phil will beat me to publication. Of course, he probably
hasn't been
> working on it for 25 years, on and off. What was Phil's name? Do you
remember?
> Do you know where I can find his stuff on archive?
phil + brian + chandler + worms : turns up masses. He must have written
a medium to heavy book, being incredibly verbose. Of course there's no
publication. The conspiracy sees to that. Oh, but hang on, how did ONAG
get published? (Conway's On Numbers and Games) Curiously, I've just
realised I learned epsilon-delta from Conway himself, in Analysis I. I
remember it took a while to sink in.
Brian Chandler
http://imaginatorium.org
Virgil
Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
> > > why don't you read the part you snipped, and see if you can
> > > understand it enough to comment on it?
> >
> > That I did not care to comment should be obvious from my snipping. That
> > That I did not understand is an unwarranted assumption. An appropriate
> > assumption is that I did not agree.
> That's a lie. You snipped because it made you look bad to respond with your
> quips and leave my argument there unchallenged for all to see.
And does TO claim to have the power to read my mind at a distance?
Virgil
I am too much the agnostic to have any catechism. And I certainly do not
support TO's version of thought police.
Virgil
Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
> > What is "overall range of values"?
> The difference between the vlaues of the largest and smallest elements. Even
> if
> there is no identifiable last element, the overall range cannot be less than
> the set size. if the set size is infinite, then there must be two elements in
> the set the diference between which is infinite.
Then please point out two such marvelous wonders among the naturals,
which, by definition, consist only of a first natural and a successor
for each natural.
Virgil
Tony Orlow (aeo6) <ae...@cornell.edu> wrote:
> Gee you seem to have understood what I was saying about infinite series, and
> said no one could possibly disagree, which is what both of you have been
> doing
> for weeks. It's been amazing to me that you could disagree with me on this. I
> am not the one being an idiot. At least YOU finally got an inkling. Virgil
> has
> just been a doorknob on this topic.
TO has kept asking me to look up infinite series, supposedly to find
that an unbounded such series must contain an infinite value, but I
cannot find any such reference. Perhaps if TO could point me towards any
reference that says what he claims...