MODELS OF SET THEORY: EXTENSIONS AND DEAD-ENDS
Ali Enayat · Journal of Symbolic Logic · 2026
Abstract This article is a contribution to the study of extensions of arbitrary models of $\mathsf {ZF}$ (Zermelo–Fraenkel set theory), with no regard to countability or well-foundedness of the models involved. Our main results include the theorems below; in Theorems A and B, ${\mathcal {N}}$ is said to be a conservative elementary extension of $\mathcal {M}$ if $\mathcal { N}$ elementarily extends $\mathcal {M}$ , and the intersection of every $ {\mathcal {N}}$ -definable set with the universe of $\mathcal {M}$ is $\mathcal {M} $ -definable (parameters allowed). In Theorem B, $\mathsf {ZFC}$ is the result of augmenting $\mathsf {ZF}$ with the axiom of choice. Theorem A. Every model $\mathcal {M}$ of $\mathsf {ZF}+\exists p\left ( \mathrm {V}=\mathrm {HOD}(p)\right ) $ has a conservative elementary extension ${\mathcal {N}}$ that contains an ordinal above all of the ordinals of $\mathcal {M}$ . Theorem B. If ${\mathcal {N}}$ is a conservative elementary extension of a model $\mathcal {M}$ of $ \mathsf {ZFC}$ , and ${\mathcal {N}}$ has the same natural numbers as $\mathcal {M}$ , then $\mathcal {M}$ is cofinal in