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

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: Well Ordering
Posted: Feb 13, 2013 9:07 AM
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On Wed, 13 Feb 2013 02:17:41 -0800, William Elliot <marsh@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, if you begin by _assuming_ that S has cardinality aleph_nu
then you don't need AC to well-order S.

Now, given an arbitrary set S, how do you show that it _does_
have cardinality aleph_nu for some nu?




>
>





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