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

From: Eray Ozkural exa (examachine_at_gmail.com)
Date: 11/20/04 

Date: 19 Nov 2004 22:14:37 -0800

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!

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?

> 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, 

--
Eray Ozkural

Relevant Pages

  • Re: infinity
    ... >> a real bijection, it is a pseudo-bijection. ... I want only to speak alittle bit about what cardinality mean. ... At finite level Cardinality goes hand by hand with Ordinality ... Ord A' =w ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... :>: be preferred in mathematics or science simply because it is more ... :>:> implications of cardinality. ... definition of number of elements that gave rise to the bijection approach ... The "size" of the sets of octal and decimal strings remain the same ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... :>: be preferred in mathematics or science simply because it is more ... :>:> implications of cardinality. ... definition of number of elements that gave rise to the bijection approach ... The "size" of the sets of octal and decimal strings remain the same ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... :>: be preferred in mathematics or science simply because it is more ... :>:> implications of cardinality. ... definition of number of elements that gave rise to the bijection approach ... The "size" of the sets of octal and decimal strings remain the same ...
    (sci.physics)
  • Re: Cantors proof that #(Evens) = #(Naturals) is inconsistent
    ... of cardinality? ... the bijection which you have defined. ... this assignment of values and simultaneously define x_i = ffor all ... f iff is an element of f iff ...
    (sci.math)