Re: Cantor's diagonal proof wrong?
From: Todd Trimble (trimble1_at_optonline.net)
Date: 11/16/04
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Date: Tue, 16 Nov 2004 13:40:44 +0000 (UTC)
On 15 Nov 2004, Curt Welch wrote:
>trimble1@optonline.net (Todd Trimble) wrote:
>> On 14 Nov 2004, Curt Welch wrote:
>> >Let me demonstrate.
>> >
>> >I claim that there is only one type of infinity. That there are just as
>> >many integers as there real numbers. (or more accurately, that the
>> >concept of the size of an infinite set is a contradiction in itself).
>> >
>>
>> So you claim that all infinite sets have the same cardinality?
>
>I had to do some studying just to try and understand what were were saying
>below (like what a power set was), so I didn't reply to your message until
>I could understand what you were asking.
>
>What I don't understand is what the notion of cardinality really is. It
>seems to me to be something, by definition, that can not exist.
Cardinality is an equivalence relation on sets; by definition,
two sets X and Y have the same cardinality if there exists a
bijection f: X --> Y.
There is also a more refined notion of cardinal number, which
in view of the fact that you have (by your own admission) very
little experience with basic notions of mathematical foundations,
would require a long explanation; the short definition is that
it's an ordinal which is least in its cardinality class, and thus
is a certain type of set. For details, you might consult the
book by Suppes which another poster recommended.
I am afraid there are some basic confusions at work when you
insist here and elsewhere that something "can not exist". Perhaps
you are confusing formal mathematical statements with philosophical
or metaphysical assertions. In any event, I've just defined
cardinality; what in thunder does it mean to say that a formal
concept like cardinality "by definition, can not exist"?
It's one thing to give a formal inconsistency proof of some set
of axioms; it's quite another to make philosophical claims about
existence. More on this below.
>
>> Given a bijection f: X --> P(X) between X and its power set P(X),
>
>Ok, I've read enough that I think I understand that at least. But maybe
>not.
>
>> what do you say about A = {x in X: x not in f(x)}?
>
>Ok, I'm a bit lost with your notation there. Are you defining a set called
>A, which contains a single member x, which is in X and not in f(x), or a
>set A which contains all the members of X, which are not in f(x)?
>
>Oh, wait, I finally see what you are saying. You are creating A, which is
>a subset of X, which contains all the elements of x which are not f(x).
Should read: Not ->in<- f(x), i.e., are not members of the set f(x).
>The fact that A, by the way you defined it had to be a subset of X is the
>key thing I was missing.
>
>So since it's a subset of X, if must be in P(X).
>
>> Since f is a
>> bijection, there exists y in X such that f(y) = A.
>
>Ah, I couldn't follow that logic until just now when I realized A, by how
>it was defined, was also a subset of X.
>
>So it's really more like, "beacuse A is a subset of X, and f is a bijection
>from X to P(X), there must exist a y such that f(y) = A. Yeah, sure, I see
>that now.
>
>> Does y belong
>> to A? If so, then y is not in f(y) [cf. defn. of A], i.e., y is
>> not in A. Does y not belong to A? If that's the case, then it
>> is false that y is not in f(y); therefore, it's true that y is
>> in f(y), and thus y is in A. Either way we reach a contradiction;
>> therefore there is no bijection between X and P(X).
>
>Sure, that's all fine and dandy. You start with a simple and obvious
>contradiction and then show that after valid manipulations, you still end
>up with a contradiction.
No, you've missed my point entirely. The point is that I start
with what *you* said,
>I claim that there is only one type of infinity.
formulate this more precisely as an assertion that any two
infinite sets X, Y are equivalent (meaning that there exists
a bijection f: X --> Y), and show that this flies in the face
of this well-known result of Cantor.
>
>> This is a version of the diagonal argument. Please point out
>> why it's "wrong".
>
>I see no problems when the domain of X is limited to all finite sized sets.
>
Please observe that Cantor's result makes no reference whatever to
finiteness. So this is a non sequitur.
>But if you try to substitute the set of a natural numbers as X into that
>argument, it seems to me to fall apart. This is because the idea of
>infinity to me means "the thing which can not exist". And if X doesn't
>exist, then you can not use the argument above to prove something about the
>properties of this item which has already been defined to not exist.
Again, this business of an *idea* being a "thing" which "can not
exist". More bizarrely, "defined to not exist." Whatever you're
on about, it's not mathematics, it's a confession of faith (and
not one which has been at all coherently expressed).
But logically, the statement "for any set X, there is no bijection
between X and P(X)" trivially implies the weaker statement "for any
infinite set X, there is no bijection between X and P(X)". This
trivial implication is logically independent of the issue of
whether infinite sets "exist" (whatever that might mean).
Since the time of Aristotle, beginners in logic have gotten hung up
on this aspect of implication, where the hypothesis might not be
satisfied but the implication is nevertheless true. I'll let you
think about it on your own, if this is your problem (as the next
quoted passage would strongly suggest).
>Things that do not exist, do not have properties. Things that do not
>exist, can not be manipulated by functions, or processes. Things that do
>not exist can not be compared to one another to see if some property they
>posses (cardinality) is the same, or different. This is where I get
>totally lost with all this notion of cardinality.
>
>Or, in our universe, the set which contains the set of all natural numbers
>is the null set, because the set of all natural numbers can not exist in
>our universe. What does, and can exist in our universe, is the set which
>contains the first N natural numbers, for any value of N.
>
This business of existence you're on about is a question of
philosophy, not mathematics. You seem to want to commit formal
axiom systems for set theory to a declaration of ontology, and
a crude one at that, since apparently you are considering
existence only in a physical sense.
See, mathematics per se has nothing to say on whether an infinite
set "really exists", or on whether "axioms" in one system or another
are "really true". All that belongs to an ancient mode of thinking.
Perhaps I can illustrate with an example: once upon a time, people
believed that Euclidean geometry was "really true", that the
parallel postulate held, that there really were points, lines,
planes, with self-evident properties. It took a long time to
disengage and see that non-Euclidean geometry was perfectly
viable, as was Euclidean geometry, and that neither was "true"
or "false", and that the whole question "is it true" or "does
it really exist" was based on a misunderstanding of what
Mathematics actually is.
Similarly, it's true that people had long wrestled with the issue
that you're wrestling with now: whether there were actual completed
infinities like N (as opposed to "potential infinities"). We no
longer worry about existential issues like this, at least not
as mathematicians trying to prove the next theorem. We accept
math as a study of formal patterns which makes no necessary claims
on ontology, and derive consequences from axioms which make no
claim to Truth. It's quite true that the vivid imagination
conjured up to contemplate mathematical forms often make them
"feel real", but then so do characters in novels, and it makes
no more sense to say that numbers like 2, 3, etc. "exist in our
universe" [to use your words] than to say such fictional characters
exist in the universe. Surely you cannot mean physical existence?!
>Has anyone developed a theory of math based on this type of foundation?
>That is, one where it is ok to define a successor function which, though
>recursion, will create a never ending stream of new objects, but where it's
>not ok, to assume you can apply it recursively an infinite number of times?
>
If you remove the axiom of infinity from Zermelo-Fraenkel set
theory, you obtain the theory of hereditarily finite sets, but
this is not strong enough to support much of what we do.
>This gets down to my problem of what "exists" means, and in how important
>the definition of that idea is in relation to understanding what the brain
>does, and how it does what it does, and in what that translates to how we
>understand what the universe itself is.
Maybe you should take it up with a philosophy newsgroup. But
regarding the title of your thread, I hope you are beginning to
see that Cantor's diagonal argument is correct (is logically
airtight in drawing consequences from premises), quite aside
from issues as to whether "infinity really exists" or whatever.
One last thing: while mathematicians are naturally and quite
rightly skeptical of claims from novices that mathematicians have
overlooked something for centuries, I don't think anyone worth
his or her salt would insist that our axioms are written in stone.
After all, we can't rule out the possibility that the axioms of
set theory are inconsistent. But mathematics is very very deep,
and to earn the right of even entering the discussion with
something so huge, one had better have a thorough mastery of
what came before (think Einstein, Godel), or accept the
consequences of being dismissed as a crackpot. That's the
way it is and should be.
Todd Trimble
- Next message: mindy: "a question on continuous functions, please help..."
- Previous message: Philip Lavers: "Re: Intuition about Lebesgue measure of the interval"
- Maybe in reply to: Curt Welch: "Cantor's diagonal proof wrong?"
- Next in thread: Shmuel (Seymour J.) Metz: "Re: Cantor's diagonal proof wrong?"
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