Definability in models of set theory

Abstract

Call a set A of ordinals “definable” over a theory T if T is some brand of set theory and whenever A appears in the standard part of a (not necessarily standard) model of T, A is “definable”. Two kinds of “definability” are considered, for each of which is provided a complete (or almost complete) characterization of the hereditarily countable sets of ordinals “definable” over true finitely axiomatizable set theories: (1) there is a single formula ϕ such that in any model of T containing A, A is the unique solution to ϕ; (2) the defining formula is allowed to vary from model to model. (Note. The restrictions “finitely axiomatizable”, and “true” are largely for the sake of convenience: such theories provably have lots of models.)There are few allusions to what a model theorist would regard as his subject—the methods coming from recursion theory and set theory; but the treatment is intended to be intelligible to nonspecialists. The referee's criticisms have greatly improved the exposition.I would like to thank Leo Harrington for several discussions, both helpful and hapless, and especially for a clever and timely proof which rescued this project from a moribund state. (Further thanks are due to the Movshon family, as a result of whose New Year's Eve party it became clear that the only really magic formulas are Σ1 formulas.)