Characterizing the powerset by a complete (Scott) sentence

Ioannis Souldatos¹

¹ Mathematics Department University of Detroit Mercy 4001 W. McNichols Detroit, MI 48221, U.S.A.

Fundamenta Mathematicae, Tome 222 (2013) no. 2, pp. 131-154.

Résumé

This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing previous work of the author. A cardinal $\kappa $ is characterized by a Scott sentence $\phi _ {\mathcal {M}}$ if $\phi _ {\mathcal {M}}$ has a model of size $\kappa $, but no model of size $\kappa ^+$. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if $ {\aleph _{\beta }}$ is characterized by a Scott sentence, then $2^{ {\aleph _{\beta +\beta _1}}}$ is (homogeneously) characterized by a Scott sentence, for all $0\beta _1 {\omega _1}$. So, the answer to the above question is positive, except the case $\beta _1=0$ which remains open. As a consequence we derive that if $\alpha \le \beta $ and $ {\aleph _{\beta }}$ is characterized by a Scott sentence, then $ {\aleph _{\alpha +\alpha _1}}^{ {\aleph _{\beta +\beta _1}}}$ is (homogeneously) characterized by a Scott sentence, for all $\alpha _1 {\omega _1}$ and $0\beta _1 {\omega _1}$. Hence, depending on the model of ZFC, we see that the class of characterizable and homogeneously characterizable cardinals is much richer than previously known. Several open questions are mentioned at the end.

Détail

DOI : 10.4064/fm222-2-2

Keywords: paper part study cardinals characterizable scott sentence continuing previous work author cardinal kappa characterized scott sentence phi mathcal phi mathcal has model size kappa model size kappa main question paper following characterizable cardinals closed under powerset operation prove aleph beta characterized scott sentence aleph beta beta homogeneously characterized scott sentence beta omega answer above question positive except beta which remains consequence derive alpha beta aleph beta characterized scott sentence aleph alpha alpha aleph beta beta homogeneously characterized scott sentence alpha omega beta omega hence depending model zfc see class characterizable homogeneously characterizable cardinals much richer previously known several questions mentioned end