Notre Dame Journal of Formal Logic

Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence

Ioannis Souldatos

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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.

Article information

Notre Dame J. Formal Logic, Volume 55, Number 4 (2014), 533-551.

First available in Project Euclid: 7 November 2014

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Zentralblatt MATH identifier

Primary: 03C75: Other infinitary logic 03C30: Other model constructions
Secondary: 03C35: Categoricity and completeness of theories 03E10: Ordinal and cardinal numbers 03E75: Applications of set theory

infinitary logic Scott sentence complete sentence characterizable cardinals


Souldatos, Ioannis. Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence. Notre Dame J. Formal Logic 55 (2014), no. 4, 533--551. doi:10.1215/00294527-2798727.

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