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Re: Some Questions about Infinite Sets
Posted:
May 10, 2011 3:31 PM
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On May 10, 11:05 am, FredJeffries <fredjeffr...@gmail.com> wrote:
> On May 10, 8:40 am, Transfer Principle <lwal...@lausd.net> wrote:
> > This is a question apparently not answered in Katz, since he
> > mainly focuses on subsets of N. It is a _real_ question about
> > Katz, not a "pretend" question.
> It's a very good question, and the best answer that I can come up
> with is that he presents a pure existence argument. He shows that
> his notion of Class Size must exist. In fact, if I understand, he
> shows that infinitely many class size mappings exist. And that, it
> seems to me, is the problem: how do you choose which one to use and
> how do you know if Harold is using the same one?
Thanks.
Of course, here's where a cardinality supporter is likely to argue
that cardinality is inferior to standard size as follows: we can
only prove that the sizes of the three sets must _exist_, but with
cardinality, we _prove_ all three sets to have the same cardinality,
aleph_0 to be precise.
But in that case, what is the cardinality of the set R of reals? Is
it aleph_1? aleph_2? aleph_3? In ZFC, we prove that the cardinality
of R _exists_ and equals some aleph, but ZFC itself doesn't tell us
which aleph it is. So standard cardinality is hardly free of mere
existence results as well.
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