Condensable models of set theory

Abstract

A model {mathcal {M}} of ZF is said to be condensable if {mathcal {M}}cong {mathcal {M}}(alpha )prec _{mathbb {L}_{{mathcal {M}}}} {mathcal {M}} for some “ordinal” alpha in mathrm {Ord}^{{mathcal {M}}}, where mathcal {M}(alpha ):=(mathrm {V}(alpha ),in )^{{mathcal {M}}} and mathbb {L}_{{mathcal {M}}} is the set of formulae of the infinitary logic mathbb {L}_{infty ,omega } that appear in the well-founded part of {mathcal {M}}. The work of Barwise and Schlipf in the 1970s revealed the fact that every countable recursively saturated model of ZF is cofinally condensable (i.e., {mathcal {M}}cong {mathcal {M}}(alpha ) prec _{mathbb {L}_{{mathcal {M}}}}{mathcal {M}} for an unbounded collection of alpha in mathrm {Ord}^{{mathcal {M}}}). Moreover, it can be readily shown that any omega -nonstandard condensable model of mathrm {ZF} is recursively saturated. These considerations provide the context for the following result that answers a question posed to the author by Paul Kindvall Gorbow.Theorem A.Assuming a modest set-theoretic hypothesis, there is a countable model {mathcal {M}}of ZFC that is bothdefinably well-founded (i.e., every first order definable element of {mathcal {M}} is in the well-founded part of mathcal {M)}andcofinally condensable. We also provide various equivalents of the notion of condensability, including the result below.Theorem B.The following are equivalent for a countable model{mathcal {M}}of mathrm {ZF}: (a) {mathcal {M}}is condensable. (b) {mathcal {M}}is cofinally condensable. (c) {mathcal {M}}is nonstandard and mathcal {M}(alpha )prec _{mathbb {L}_{{mathcal {M}}}}{mathcal {M}}for an unbounded collection of alpha in mathrm {Ord}^{{mathcal {M}}}.