Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)

From: Eray Ozkural exa (examachine_at_gmail.com)
Date: 11/19/04 

Date: 19 Nov 2004 10:16:52 -0800

cbrown@cbrownsystems.com (Chas Brown) wrote in message news:<ba7dcfd7.0411190253.763df0f2@posting.google.com>...
> > You've indeed put it in a succinct
> > formal form, but formality should not avoid any problems, if there
> > were any to begin with. So, if your formalization is correct, which
> > seems to be the case, whatever Zenkin criticized must be there.
>
> "Must"? You seem to be putting the cart before the horse. You are
> assuming that Zenkin's _logic_ is correct (e.g., refers to something
> that exists). It is not.

I'm not assuming that. I'm simply saying that it will be a good test
of his arguments for Zenkin, in case your formalization is correct,
which seems to be the case. (Because neither I nor anybody else could
find an error in your post, although I merely skimmed over your
version of Cantor's diagonal proof, for it seemed similar enough to
other half-formal depictions of the same proof.)

You asked me whether I read any set theory or logic. Yes, I did, a lot
of it as a matter of fact. I will happily ignore your otherwise
condescending remarks for it is understandable that misunderstandings
occur on usenet, and ask you a *very* simple question. It's one of the
basic things in philosophy of mathematics, and just like the
actual/potential infinity debate, it has *no* definite answer. (That's
why it may also be called a paradox by some authors)

We use the concept of bijection to reason about the equivalence of the
"sizes" of supposedly infinite sets, like natural numbers. Under the
axioms of ZFC, we can comfortably talk about a bijection between even
and odd numbers, and even numbers and all natural numbers. However,
this would fail if we were to give the "subset" account of comparing
the magnitudes or sizes of supposedly infinite sets. Which one is
correct?

If you would like to see authoritative resources where this antinomy
is stated more clearly, I can send you some links, or you can google.

Looking forward to your reply.

Regards, 

--
Eray Ozkural
PS: It would be better if you could also explain your answer, rather
than referring to a common textbook you may have read.

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