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

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Posts: 1,948
Registered: 2/15/09
Re: Trichotomy of Cardinals
Posted: Jul 4, 2014 5:33 PM
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On 7/4/2014 2:25 PM, Aatu Koskensilta wrote:
> "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.

2 <-> 3, yes these are the numbers,
why 2 and 3 are so, here in trichotomy
where there is otherwise only dichotomy,
for example.

Why yes it is.

Mathematics is so advanced,
we're figuring out 2 and 3.

Really this is the 21'st
century mathematics.

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