Characterizing the powerset by a complete (Scott) sentence
Volume 222 / 2013
Fundamenta Mathematicae 222 (2013), 131-154
MSC: Primary 03C75, 03C30; Secondary 03C35, 03E10, 03E75.
DOI: 10.4064/fm222-2-2
Abstract
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.
