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

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Registered: 2/15/09
Re: Trichotomy of Cardinals
Posted: Jul 4, 2014 5:20 PM
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On 7/4/2014 12:06 PM, Aatu Koskensilta wrote:
> 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.

2 <-> 3

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