February 27
Elliot Glazer,
Harvard University
Explicit models of arithmetic do not have full standard system
It is well-known under ZFC that there is a nonstandard model of PA which has a full standard system, i.e. every subset of this model's standard cut is the intersection of the standard cut with some subset of the model which is definable from parameters. We show that the use of Choice here cannot be avoided. More precisely, we prove that it is consistent relative to ZF that no model of PA has full standard system, and it is provable in ZF (or just a fragment of second-order arithmetic) that no Borel model has full standard system. The proof is measure-theoretic in nature, and as a simpler first argument, we will prove from Projective Determinacy the stronger claim that no projective model has full standard system.
Since the power set of the naturals is trivially a Scott set, an immediate corollary of this result is that Scott's problem is independent of ZF.