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Model theory of the regularity and reflection schemes. (English) Zbl 1149.03026
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.

##### MSC:
 03C64 Model theory of ordered structures; o-minimality 03C62 Models of arithmetic and set theory 03C80 Logic with extra quantifiers and operators
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