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

stephen_at_nomail.com
Date: 11/20/04

  • Next message: Chas Brown: "Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)" 
    Date: 20 Nov 2004 06:36:40 GMT

    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.

    : 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?

    :> 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.

    Stephen

  • Next message: Chas Brown: "Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)"

    Relevant Pages

    • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
      ... :> In sci.math Eray Ozkural exa wrote: ... :>: We use the concept of bijection to reason about the equivalence of the ... :>: "sizes" of supposedly infinite sets, ... Two sets have the same cardinality ...
      (sci.math)
    • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
      ... :> In sci.math Eray Ozkural exa wrote: ... :>:>: We use the concept of bijection to reason about the equivalence of the ... Two sets have the same cardinality ... paradox this is your problem. ...
      (comp.theory)
    • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
      ... :> In sci.math Eray Ozkural exa wrote: ... :>:>: We use the concept of bijection to reason about the equivalence of the ... Two sets have the same cardinality ... paradox this is your problem. ...
      (sci.math)
    • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
      ... We use the concept of bijection to reason about the equivalence of the ... "sizes" of supposedly infinite sets, ... If we defined "same cardinality" differently then of course ...
      (sci.math)
    • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
      ... We use the concept of bijection to reason about the equivalence of the ... "sizes" of supposedly infinite sets, ... If we defined "same cardinality" differently then of course ...
      (comp.theory)