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Re: Well Ordering
Posted:
Feb 13, 2013 6:22 AM


On Feb 13, 4:17 am, William Elliot <ma...@panix.com> wrote: > Let S be a set with cardinality aleph_nu. > Since S is equinumerous with omega_nu, there's > . . a bijection h:S > omega_nu. > > Thus S is well ordered by x <= y when h(x) <= h(y); > . . well ordered without using AxC. Hm...
Yes, of course. The alephs are the cardinalities of the *well orderable* infinite sets. By saying that card S = aleph_{nu} you are implicitly saying that S can be wellordered. If S is not well orderable, its cardinality is incomparable with all sufficiently large alephs. A Dedekindfinite infinite set is incomparable with all alephs.



