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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 5:25 PM

"Ross A. Finlayson" <ross.finlayson@gmail.com> writes:

> 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

This is no doubt a pertinent observation, if somewhat obscure.

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

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