[FOM] paradox and circularity
Thomas Forster
T.Forster at dpmms.cam.ac.uk
Sun Sep 15 10:32:12 EDT 2002
Stephen Yablo's observation about set-theoretic analogues of his
puzzle is well made. Consider Simple typed set theory, one
type for each natural or, better, typed set theory a la Wang (MIND
1952?) where there is a type for each integer. This theory is
consistent by compactness. for every n \in Z, V_{n+1} is the power
set of V_n (``internally''). All consistent, no probs. Notice tho',
that in no such model can W_n, the set of (externally) wellfounded
sets of type n ever be a set of the model. For if it were,
W_{n+1} would be a set too, being the power set of W_n, and W_{n-1}
would be a set, being \bigcup W_n. But then one has an infinite
descending \in-chain of supposedly wellfounded sets. So W_n was not
a set of the model after all.
I trust that the other NF-istes on this mailing list can confirm
my suspicion that this is folklore. I've know it for a while, but
i'm pretty sure i've never seen a published proof.
Thomas Forster
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