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Topic: Well Ordering
Replies: 4   Last Post: Feb 14, 2013 10:48 AM

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Well Ordering
Posted: Feb 13, 2013 6:22 AM
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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 well-ordered. If S is not well-
orderable, its cardinality is incomparable with all sufficiently large
alephs. A Dedekind-finite infinite set is incomparable with all alephs.



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