Re: Proper classes

"Chip Eastham" <hardmath@xxxxxxxxx> writes:

> Jesse F. Hughes wrote:
>> "Jonathan Hoyle" <jonhoyle@xxxxxxx> writes:
>>
>> > In non-Well Founded theories, where K can be a set, you have a
>> > different problem. To be a set with cardinality, K must be in
>> > one-to-one correspondence with some cardinal number. This cardinal
>> > number must, in turn, be a member of the K. Since the K contains
>> > *all* cardinals, including itself, it must itself be "the largest"
>> > cardinal, since K+ = K U {K} = K. We are left with all sets are of
>> > cardinality <= K.
>>
>> By definition, cardinal numbers are ordinals and hence well-founded.
>> Even in set theories with an anti-foundation axiom.
>
> Jesse, I disagree that "by definition" cardinal numbers are ordinals.
>
> By definition, cardinal numbers are equivalence classes of sets,
> the equivalence relation being that of set isomorphism. (If you
> assume axiom of choice, the distinction becomes moot, but
> more on that in a moment.)

Alright. You are obviously considering more alternative set theories
than I was. I was just thinking in terms of ZFC - Foundation, and
there it is traditional to take cardinals to be particular ordinals
(as you mention). Certainly, I'll defer to your objection.

I've snipped the rest of your post for brevity's sake, but it was very
interesting. Thanks!

--
Jesse F. Hughes

One is not superior merely because one sees the world as odious.
-- Chateaubriand (1768-1848)
.

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