Archive for Mathematical Logic 60 (1-2):17-47 (2021)

Abstract
This note concerns the model theoretic properties of logics extending the first-order logic with monadic second-order variables equipped with the stationarity quantifier. The eight variations of the strong downward Löwenheim–Skolem Theorem down to <ℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<\aleph _2$$\end{document} for this logic with the interpretation of second-order variables as countable subsets of the structures are classified into four principles. The strongest of these four is shown to be equivalent to the conjunction of CH and the Diagonal Reflection Principle for internally clubness of S. Cox. We show that a further strengthening of this SDLS and its variations follow from the Game Reflection Principle of B. König and its generalizations.
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DOI 10.1007/s00153-020-00730-x
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References found in this work BETA

Barwise: Abstract Model Theory and Generalized Quantifiers.Vaananen Jouko - 2004 - Bulletin of Symbolic Logic 10 (1):37-53.
Semistationary and Stationary Reflection.Hiroshi Sakai - 2008 - Journal of Symbolic Logic 73 (1):181-192.
Barwise: Abstract Model Theory and Generalized Quantifiers.Jouko Va An Anen - 2004 - Bulletin of Symbolic Logic 10 (1):37-53.
Generic Compactness Reformulated.Bernhard König - 2004 - Archive for Mathematical Logic 43 (3):311-326.

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