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2014 Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence
Ioannis Souldatos
Notre Dame J. Formal Logic 55(4): 533-551 (2014). DOI: 10.1215/00294527-2798727

Abstract

This is the first part of a study on cardinals that are characterizable by Scott sentences. Building on previous work of Hjorth, Malitz, and Baumgartner, we study which cardinals are characterizable by a Scott sentence ϕ, in the sense that ϕ characterizes κ, if ϕ has a model of size κ but no models of size κ+.

We show that the set of cardinals that are characterized by a Scott sentence is closed under successors, countable unions, and countable products (see Theorems 3.3 and 4.6 and Corollary 4.8). We also prove that if α is characterized by a Scott sentence, at least one of α, α+1, or (α+1,α) is homogeneously characterizable (see Definitions 1.3 and 1.4 and Theorem 3.19). Based on an argument of Shelah, we give counterexamples that characterizable cardinals are not closed under predecessors or cofinalities.

Citation

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Ioannis Souldatos. "Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence." Notre Dame J. Formal Logic 55 (4) 533 - 551, 2014. https://doi.org/10.1215/00294527-2798727

Information

Published: 2014
First available in Project Euclid: 7 November 2014

zbMATH: 1338.03072
MathSciNet: MR3276410
Digital Object Identifier: 10.1215/00294527-2798727

Subjects:
Primary: 03C30, 03C75
Secondary: 03C35, 03E10, 03E75

Rights: Copyright © 2014 University of Notre Dame

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Vol.55 • No. 4 • 2014
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