Characterizing the powerset by a complete (Scott) sentence

Abstract

This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing the work from http://arxiv.org/abs/1007.2426v1. A cardinal $\kappa$ is characterized by a Scott sentence $\phi_M$, if $\phi_M$ has a model of size $\kappa$, but no model of $\kappa^+$. 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 also characterized by a Scott sentence, for all $\alpha_1<\omega_1$ and $0<\beta_1<\omega_1$. Whence, 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 also mentioned at the end.