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

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William Elliot

Posts: 1,530
Registered: 1/8/12
Re: Trichotomy of Cardinals
Posted: Jun 29, 2014 1:56 AM
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On Sat, 28 Jun 2014, scattered wrote:
> On Saturday, June 28, 2014 5:12:02 AM UTC-4, William Elliot wrote:

> > How is the trichotomy of cardinals equivalent to AxC?
> > Defining the cardinals as done within ZF, as the
> > the initial ordinals, aren't they well ordered?
> > Thusly a proof for AxC?

>
> Cardinals are defined as equivalence classes under the equinumerable
> equivalence relation. Without AC these equivalence classes can fail
> trichotomy. In this case they are not automatically equivalent to
> certain ordinals.
>

What set theory? NF, NBG?
Certainly not ZF. How could such classes be defined as sets?



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