Re: Proper classes

"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.

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