# [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