Rigid ideals
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2016
- DOI:
- 10.48550/arXiv.1606.00040
- arXiv:
- arXiv:1606.00040
- Bibcode:
- 2016arXiv160600040C
- Keywords:
-
- Mathematics - Logic;
- 03E55;
- 03E35
- E-Print:
- Israel J. Math. 224 (2018), no. 1, 343--366. MR3799759