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

From: Shmuel (Seymour J.) Metz (spamtrap_at_library.lspace.org.invalid)
Date: 11/23/04 

Date: Mon, 22 Nov 2004 20:28:55 -0500

In <320e992a.0411210919.77f4930b@posting.google.com>, on 11/21/2004
   at 09:19 AM, examachine@gmail.com (Eray Ozkural exa) said:

>That would be mathematical solipsism. You can't define things
>completely freely in mathematics.

Wrong; you can and do. What is prohibited is to use your definitions
inconsistently. Having freely chosen a definition, you have to stick
with the same definition every time that you use the defined term.

>Once you define "number" you have very little space to move...

That doesn't contradict what he wrote. However, it seems to be a
strange thing for *you* to write, since you are the one wanting to use
a nonstandard definition.

>I think "cardinality" must be synonymous with "size of a set"

That statement is meaningless without a definition of "size of a set".

>I'm saying that there are two approaches for measuring the size of
>sets, and in general the bijection account does not seem to be
>satisfactory to reason about the size of infinite sets.

It doesn't matter whether there are two approaches, or 42. What does
matter is that Cantor's result is about the existence of bijections.
No hypothetical alternative approach is relevant to Cantor's results.

>Do you accept that there is a paradox of the infinitely big?

Why would he accept the claim when you have produced no such paradox,
nor defined what you mean by "paradox"? Of course, if by paradox you
simply mean something that is valid but that you initially failed to
understand, then I would expect him to accept that there is a paradox. 

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