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

From: Eray Ozkural exa (examachine_at_gmail.com)
Date: 11/21/04 

Date: 20 Nov 2004 18:03:41 -0800

stephen@nomail.com wrote in message news:<cnmolo$1gp8$1@msunews.cl.msu.edu>...
> In sci.math Eray Ozkural exa <examachine@gmail.com> wrote:
> : stephen@nomail.com wrote in message news:<cnldto$b2e$1@msunews.cl.msu.edu>...
> :> In sci.math Eray Ozkural exa <examachine@gmail.com> wrote:
> :> : 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?
> :>
> :> That is a meaningless question. Two sets have the same cardinality
> :> if there exists a bijection between them. That is the definition.
> :> How can you claim that the definition is not correct?
> :>
> :> If we defined "same cardinality" differently then of course
> :> sets that had the same cardinality under the old definition
> :> might no longer have the same cardinality under the new definition.
> :> No surprise there. The only interesting question is which
> :> definitions are more useful.
>
> : And that is exactly the question in philosophy of mathematics!
> : Bijection is apparently not seen as the only sensible way to define
> : "same cardinality"! I bet you never heard that!
>
> I have heard that. The point is that that is how "same cardinality"
> is defined. Yes, you can consider other definitions, but to avoid
> confusion it would make sense to call it something else, as
> "cardinality" has already been defined.

It is obvious what cardinality means, and it is not a definite thing,
because THIS PARADOX EXISTS. If you do not understand WHY this is a
paradox this is your problem.

Do you accept that there is a paradox or not?

Yes or no?

> : You may want to read these slides. It's called the "Paradoxes of the
> : Infinitely Big"
> : http://ls.poly.edu/~jbain/philmath/philmathlectures/M05.Cantor.pdf
>
> : Obviously subset criterion is one of two criteria for comparing size
> : of sets in a prominent philosophy of mathematics textbook. Perhaps you
> : never touched one?
>
> This from someone who claimed that a program could only
> correctly answer a finite number of instances of the halting
> problem, and resorted to insults when folks disagreed with him.
> Again, you resort to insults. Why is that?

What a blunder have I made. No, I should have said that a program can
only correctly detect instances of the halting up to a finite
complexity. That was the idea I had, but I misrepresented it.

I am not resorting to insults, I am simply returning the favor.

> :> Like so many of the people who seem to object to Cantor's
> :> proof, you are apparently arguing with the definitions used in the
> :> proof, not the proof itself.
>
> : A truly brilliant observation! I am most impressed!
>
> : Cheers,
>
>
> I am assuming you are being sarcastic. Given your past
> performance of mistatements and misunderstandings you
> might consider dropping the attitude.

Actually, you do not understand.

My comment above is not sarcastic. Definitions are exactly that which
are contested. (although Zenkin wants to extend it to *proof*,
however, I think you can understand this: the validity of a proof-step
is also a definitional issue)

Regards, 

--
Eray Ozkural

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