On certain elementary extensions of models of set theory

Abstract

In § 1 \S 1 we study two canonical methods of producing models of ZFC \operatorname {ZFC} with no elementary end extensions. § 2 \S 2 is devoted to certain "completeness" theorems dealing with elementary extensions, e.g., using ♢ ω 1 {\diamondsuit _{{\omega _1}}} we show that for a consistent T ⊇ ZFC T \supseteq \operatorname {ZFC} the property "Every model A \mathfrak {A} of T T has an elementary extension fixing ω A {\omega ^\mathfrak {A}} " is equivalent to T ⊢ T\vdash "There exists an uncountable measurable cardinal". We also give characterizations of T ⊢ T\vdash " κ \kappa is weakly compact" and T ⊢ T\vdash " κ \kappa is measurable" in terms of elementary extensions.