Rigid ideals

TL;DR

This paper explores the consistency of rigid ideals with various properties on successor cardinals, linking their existence to large cardinal assumptions and the failure or holding of GCH.

Contribution

It establishes the relative consistency of rigid saturated and presaturated ideals on successor cardinals under large cardinal assumptions, extending the understanding of ideal rigidity and GCH.

Findings

Existence of rigid saturated μ-minimal ideals implies GCH fails.

Rigid saturated ideals on μ+ are consistent with GCH given almost-huge cardinals.

Rigid presaturated ideals on ω₁ are consistent with CH assuming almost-huge cardinals.

Abstract

An ideal $I$ on a cardinal $\kappa$ is called \emph{rigid} if all automorphisms of $P(\kappa)/I$ are trivial. An ideal is called \emph{$\mu$-minimal} if whenever $G\subseteq P(\kappa)/I$ is generic and $X\in P(\mu)^{V[G]}\setminus V$, it follows that $V[X]=V[G]$. We prove that the existence of a rigid saturated $\mu$-minimal ideal on $\mu^+$, where $\mu$ is a regular cardinal, is consistent relative to the existence of large cardinals. The existence of such an ideal implies that GCH fails. However, we show that the existence of a rigid saturated ideal on $\mu^+$, where $\mu$ is an \emph{uncountable} regular cardinal, is consistent with GCH relative to the existence of an almost-huge cardinal. Addressing the case $\mu=\omega$, we show that the existence of a rigid \emph{presaturated} ideal on $\omega_1$ is consistent with CH relative to the existence of an almost-huge cardinal. The existence of a \emph{precipitous} rigid ideal on $\mu^+$ where $\mu$ is an uncountable regular cardinal is equiconsistent with the existence of a measurable cardinal.