Notre Dame Journal of Formal Logic

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

Ioannis Souldatos

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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.

Article information

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

Dates
First available in Project Euclid: 7 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1415382954

Digital Object Identifier
doi:10.1215/00294527-2798727

Mathematical Reviews number (MathSciNet)
MR3276410

Zentralblatt MATH identifier
1338.03072

Subjects
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

Keywords
infinitary logic Scott sentence complete sentence characterizable cardinals

Citation

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. https://projecteuclid.org/euclid.ndjfl/1415382954


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References

  • [1] Baumgartner, J. E., “The Hanf number for complete $L_{\omega_{1},\omega}$ sentences (without GCH),” Journal of Symbolic Logic, vol. 39 (1974), pp. 575–78.
  • [2] Hanf, W., “Incompactness in languages with infinitely long expressions,” Fundamenta Mathematicae, vol. 53 (1963/1964), pp. 309–24.
  • [3] Hjorth, G., “Knight’s model, its automorphism group, and characterizing the uncountable cardinals,” Journal of Mathematical Logic, vol. 2 (2002), pp. 113–44.
  • [4] Keisler, H. J., Model Theory for Infinitary Logic: Logic with Countable Conjunctions and Finite Quantifiers, vol. 62 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1971.
  • [5] Knight, J. F., “A complete $L_{\omega_{1}\omega}$-sentence characterizing $\aleph_{1}$,” Journal of Symbolic Logic, vol. 42 (1977), pp. 59–62.
  • [6] Malitz, J., “The Hanf number for complete $L_{\omega_{1}},_{\omega}$ sentences,” pp. 166–81 in The Syntax and Semantics of Infinitary Languages (Los Angeles, 1967), vol. 72 of Lecture Notes in Mathematics, Springer, New York, 1968.
  • [7] Scott, D., “Logic with denumerably long formulas and finite strings of quantifiers,” pp. 329–41 in Theory of Models (Berkeley, 1963), North-Holland, Amsterdam, 1965.
  • [8] Shelah, S., “Borel sets with large squares,” Fundamenta Mathematicae, vol. 159 (1999), pp. 1–50.
  • [9] Souldatos, I., “Linear orderings and powers of characterizable cardinals,” Annals of Pure and Applied Logic, vol. 163 (2012), pp. 225–37.