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An Introduction to Recursively Saturated and Resplendent Models

Jon Barwise & John Schlipf

Journal of Symbolic Logic 47 (2):440-440 (1982)

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Model theory of the regularity and reflection schemes.Ali Enayat & Shahram Mohsenipour - 2008 - Archive for Mathematical Logic 47 (5):447-464.details This paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme and its strengthening REF ${(\mathcal{L})}$ (the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here ${\mathcal{L}}$ is a language with a distinguished linear order <, and REF ${(\mathcal {L})}$ consists of formulas of the form $$\exists x \forall y_{1} < x \ldots \forall y_{n} < x \varphi (y_{1},\ldots ,y_{n})\leftrightarrow \varphi^{ < x}(y_1, \ldots ,y_n),$$ where φ is an ${\mathcal{L}}$ -formula, φ (...) shrink) Download Export citation Bookmark 2 citations
Real closures of models of weak arithmetic.Emil Jeřábek & Leszek Aleksander Kołodziejczyk - 2013 - Archive for Mathematical Logic 52 (1):143-157.details D’Aquino et al. (J Symb Log 75(1):1–11, 2010) have recently shown that every real-closed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by IΔ0. We prove that the theorem holds if IΣ4 is replaced by weak subtheories of Buss’ bounded arithmetic: PV or $${\Sigma^b_1-IND^{\|x\|_k}}$$. It also holds for IΔ0 (and even its subtheory IE 2) under a rather mild assumption on cofinality. On the other hand, it (...) Download Export citation Bookmark 3 citations
Positive logics.Saharon Shelah & Jouko Väänänen - 2023 - Archive for Mathematical Logic 62 (1):207-223.details Lindström’s Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context (...) Download Export citation Bookmark
Condensable models of set theory.Ali Enayat - 2022 - Archive for Mathematical Logic 61 (3):299-315.details A model \ of ZF is said to be condensable if \\prec _{\mathbb {L}_{{\mathcal {M}}}} {\mathcal {M}}\) for some “ordinal” \, where \:=,\in )^{{\mathcal {M}}}\) and \ is the set of formulae of the infinitary logic \ that appear in the well-founded part of \. The work of Barwise and Schlipf in the 1970s revealed the fact that every countable recursively saturated model of ZF is cofinally condensable \prec _{\mathbb {L}_{{\mathcal {M}}}}{\mathcal {M}}\) for an unbounded collection of \). Moreover, it (...) Download Export citation Bookmark 2 citations
Foundations of recursive model theory.Terrence S. Millar - 1978 - Annals of Mathematical Logic 13 (1):45.details Download Export citation Bookmark 19 citations

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