Model theory of the regularity and reflection schemes (Q938239)

A well-known theorem of Keisler characterizes first-order theories containing the regularity scheme (or, equivalently, the collection scheme), in terms of existence of elementary end extensions and \(\kappa\)-like models for regular uncountable cardinals \(\kappa\). Enayat and Mohsenipour prove an analogous result for theories containing the reflection scheme. A linearly ordered \(L\)-structure \(M=(A,<,\dots)\) is a model of the reflection scheme REF\((L)\) if \(A\) has no last element and and for every \(L\)-formula \(\varphi(y_1,\dots,y_n)\) there is a \(c\in A\) such that for all \(a_1,\dots,a_n<c\), \(M\models \varphi(a_1,\dots,a_n)\) iff \((\{x\in A: x<c\},<,\dots)\models \varphi(a_1,dots,a_n)\). The main theorem of the paper gives several conditions equivalent to \(T\vdash \text{REF}(L)\), among them: some model of \(T\) has an elementary end extension with a first new element, and \(T\) has an \(\omega_1\)-like model that continuously embeds \(\omega_1\). The authors develop model theory of REF\((L)\) proving a number of results generalizing analogous theorems from model theory of PA and ZF. In particular, they prove a variant of Gaifman's splitting theorem and a theorem characterizing tallness in terms of existence of cofinal recursively saturated elementary end extensions. The paper concludes with a list of attractive open problems.