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

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

  • Next message: Kent Paul Dolan: "[OT]: Zenkin's paper on Cantor (reply of Dr. Zenkin)" 
    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
  • Next message: Kent Paul Dolan: "[OT]: Zenkin's paper on Cantor (reply of Dr. Zenkin)"

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