## Notre Dame Journal of Formal Logic

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

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

#### 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 $\varphi $, in the sense that $\varphi $ characterizes $\kappa $, if $\varphi $ 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

**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