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?
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.