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Topic: Trichotomy of Cardinals
Replies: 10   Last Post: Jul 4, 2014 5:33 PM

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Aatu Koskensilta

Posts: 2,633
Registered: 6/28/08
Re: Trichotomy of Cardinals
Posted: Jul 4, 2014 3:06 PM
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William Elliot <marsh@panix.com> writes:

> How is the trichotomy of cardinals equivalent to AxC?

In the form that says that given any two sets A and B, either A and B
are equipollent; A and B are not equipollent but A is equipollent to a
subset of B; or A and B are not equipollent but B is equipollent to a
subset of A.

> Defining the cardinals as done within ZF, as the the initial ordinals,
> aren't they well ordered?


Sure.

> Thusly a proof for AxC?

Observing that a class of ordinals is well-ordered falls somewhat
short of a proof of the axiom of choice.

--
Aatu Koskensilta (aatu.koskensilta@uta.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus



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