Which models of ZFA are permutation models?
by
Eric J. Hall
University of Missouri---Kansas City

Let ZFAC^K denote the theory of ZF modified to allow a set of atoms (but not a proper class of atoms), with the Axiom of Choice holding in the kernel (pure part). Let N be a model of ZFAC^K. The following are equivalent: (a) N is a permutation model, and (b) AC holds in a generic extension of N that does not add new pure sets. I have recently realized that these are equivalent to (c): SVC (Small Violations of Choice) holds in N and N has some extension (not assumed to be generic), with the same atoms and pure sets, in which AC holds. I'll give some definitions, summarize the proof of either c⇒b or c⇒a, and mention open questions (as far as I know, these equivalent conditions may simply be true for models of ZFAC^K.)

Date received: March 17, 2006