Characterizing the powerset by a complete (Scott) sentence

TL;DR

This paper investigates whether the class of cardinals characterized by Scott sentences is closed under the powerset operation, providing partial positive results and implications for the richness of such cardinals under ZFC.

Contribution

It proves that if a cardinal is characterized by a Scott sentence, then certain powerset cardinals are also characterized, expanding understanding of the structure of characterizable cardinals.

Findings

Characterizable cardinals are closed under powerset operation for certain cases.

The class of characterizable cardinals is richer than previously known under ZFC.

Open questions remain about the case when the exponent is zero.

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.