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 $κ$ is characterized by a Scott sentence $ϕ_M$, if $ϕ_M$ has a model of size $κ$, but no model of $κ^+$. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if $\aleph_β$ is characterized by a Scott sentence, then $2^{\aleph_{β+β_1}}$ is (homogeneously) characterized by a Scott sentence, for all $0<β_1<ω_1$. So, the answer to the above question is positive, except the case $β_1=0$ which remains open. As a consequence we derive that if $α\leβ$ and $\aleph_β$ is characterized by a Scott sentence, then $\aleph_{α+α_1}^{\aleph_{β+β_1}}$ is also characterized by a Scott sentence, for all $α_1<ω_1$ and $0<β_1<ω_1$. 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.