Force to change large cardinal strength

TL;DR

This dissertation explores how to alter large cardinal properties using forcing, providing new definitions and methods to change the degrees of inaccessibility, Mahlo, measurable, supercompact, and strongly compact cardinals.

Contribution

It introduces new forcing techniques to modify large cardinal degrees and ranks, extending the hyper-inaccessible hierarchy and demonstrating the manipulability of large cardinal properties.

Findings

Forcing can preserve or destroy specific large cardinal degrees.

It is possible to assign any inaccessible degree to a cardinal via forcing.

Forcing can reduce Mitchell rank of measurable and supercompact cardinals.

Abstract

This dissertation includes many theorems which show how to change large cardinal properties with forcing. I consider in detail the degrees of inaccessible cardinals (an analogue of the classical degrees of Mahlo cardinals) and provide new large cardinal definitions for degrees of inaccessible cardinals extending the hyper-inaccessible hierarchy. I showed that for every cardinal $\kappa$, and ordinal $\alpha$, if $\kappa$ is $\alpha$-inaccerssible, then there is a $\mathbb{P}$ forcing that $\kappa$ which preserves that $\alpha$-inaccessible but destorys that $\kappa$ is $(\alpha+1)$-inaccessible. I also consider Mahlo cardinals and degrees of Mahlo cardinals. I showed that for every cardinal $\kappa$, and ordinal $\alpha$, there is a notion of forcing $\mathbb{P}$ such that $\kappa$ is still $\alpha$-Mahlo in the extension, but $\kappa$ is no longer $(\alpha +1)$-Mahlo. I also show that a cardinal $\kappa$ which is Mahlo in the ground model can have every possible inaccessible degree in the forcing extension, but no longer be Mahlo there. The thesis includes a collection of results which give forcing notions which change large cardinal strength from weakly compact to weakly measurable, including some earlier work by others that fit this theme. I consider in detail measurable cardinals and Mitchell rank. I show how to change a class of measurable cardinals by forcing to an extension where all measurable cardinals above some fixed ordinal $\alpha$ have Mitchell rank below $\alpha.$ Finally, I consider supercompact cardinals, and a few theorems about strongly compact cardinals. Here, I show how to change the Mitchell rank for supercompactness for a class of cardinals.