## Notre Dame Journal of Formal Logic

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

Ioannis Souldatos

#### 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 $\phi$, in the sense that $\phi$ characterizes $\kappa$, if $\phi$ has a model of size $\kappa$ but no models of size $\kappa^{+}$.

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 $\aleph_{\alpha}$ is characterized by a Scott sentence, at least one of $\aleph_{\alpha}$, $\aleph_{\alpha+1}$, or $(\aleph_{\alpha+1},\aleph_{\alpha})$ 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

#### 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

#### 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.