Condensable models of set theory

Abstract

AbstractA model $${\mathcal {M}}$$ M of ZF is said to be condensable if $$ {\mathcal {M}}\cong {\mathcal {M}}(\alpha )\prec _{\mathbb {L}_{{\mathcal {M}}}} {\mathcal {M}}$$ M ≅ M ( α ) ≺ L M M for some “ordinal” $$\alpha \in \mathrm {Ord}^{{\mathcal {M}}}$$ α ∈ Ord M , where $$\mathcal {M}(\alpha ):=(\mathrm {V}(\alpha ),\in )^{{\mathcal {M}}}$$ M ( α ) : = ( V ( α ) , ∈ ) M and $$\mathbb {L}_{{\mathcal {M}}}$$ L M is the set of formulae of the infinitary logic $$\mathbb {L}_{\infty ,\omega }$$ L ∞ , ω that appear in the well-founded part of $${\mathcal {M}}$$ 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., $${\mathcal {M}}\cong {\mathcal {M}}(\alpha ) \prec _{\mathbb {L}_{{\mathcal {M}}}}{\mathcal {M}}$$ M ≅ M ( α ) ≺ L M M for an unbounded collection of $$\alpha \in \mathrm {Ord}^{{\mathcal {M}}}$$ α ∈ Ord M ). Moreover, it can be readily shown that any $$\omega $$ ω -nonstandard condensable model of $$\mathrm {ZF}$$ 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 $${\mathcal {M}}$$ M of ZFC that is bothdefinably well-founded (i.e., every first order definable element of $${\mathcal {M}}$$ M is in the well-founded part of $$\mathcal {M)}$$ 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$${\mathcal {M}}$$ M of $$\mathrm {ZF}$$ ZF : (a) $${\mathcal {M}}$$ M is condensable. (b) $${\mathcal {M}}$$ M is cofinally condensable. (c) $${\mathcal {M}}$$ M is nonstandard and $$\mathcal {M}(\alpha )\prec _{\mathbb {L}_{{\mathcal {M}}}}{\mathcal {M}}$$ M ( α ) ≺ L M M for an unbounded collection of $$ \alpha \in \mathrm {Ord}^{{\mathcal {M}}}$$ α ∈ Ord M .