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

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Posts: 11,749
Registered: 12/4/04
Re: Well Ordering
Posted: Feb 13, 2013 1:03 PM
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On Wednesday, February 13, 2013 4:17:41 AM UTC-6, William Elliot 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...

How do you define "cardinality aleph_nu"?

Usually, the cardinals/alephs are the ordinals that are not bijectable with any strictly smaller ordinal, and we say that a set S has "cardinality aleph_nu" if and only if it is bijectable with the cardinal/ordinal aleph_nu.

The statement "every set can be bijected with an aleph" is equivalent to the Axiom of Choice.

In the absence of Choice, some sets may not be bijectable with any aleph; of course, any set that is well-orderable can be bijected with an ordinal, and hence with an aleph; conversely, any set that can be bijected with an aleph is well-orderable.

Hence, what you wrote is basically equivalent to "In the absence of AC, if a set S can be well-ordered, then S can be well-ordered." To which I can only say "Well done."

Arturo Magidin

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