Universal Graphs at $\aleph_{\omega_1+1}$ and Set-theoretic Geology

TL;DR

This thesis constructs a model with a jointly universal family of graphs at a large cardinal and explores foundational aspects of set-theoretic geology, including mantle and ground models, using advanced forcing techniques.

Contribution

It introduces a method to create models with universal graphs at $eth_{ ext{omega}_1+1}$ and advances understanding of set-theoretic geology, especially regarding mantle and ground models.

Findings

Constructed a model with a jointly universal family of graphs of size $eth_{ ext{omega}_1+2}$.

Showed that certain class forcing iterations produce a universe that is its own mantle.

Established conditions under which different generic extensions share a common ground.

Abstract

This thesis consists of two parts: the construction of a jointly universal family of graphs, and then an exploration of set-theoretic geology. Firstly we shall construct a model in which $2^{\aleph_{\omega_1}}=2^{\aleph_{\omega_1+1}}=\aleph_{\omega_1+3}$ but there is a jointly universal family of size $\aleph_{\omega_1+2}$ of graphs on $\aleph_{\omega_1+1}$. We take a supercompact cardinal $\kappa$ and will use Radin forcing with interleaved collapses to change $\kappa$ into $\aleph_{\omega_1}$. Prior to the Radin forcing we perform a preparatory iteration to add functions from $\kappa^+$ into Radin names for what will become members of the jointly universal family on $\kappa^+$. The same technique can be used with any uncountable cardinal in place of $\omega_1$. Secondly we explore various topics in set-theoretic geology. We begin by showing that a class Easton support iteration of $\mathrm{Add}(\kappa,1)$ at $\kappa$ regular results in a universe that is its own generic mantle. We then consider set forcings $\mathbb{P}$, $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{S}$ with respective generics $G$, $H$, $I$ and $J$ such that $V[G][I]=V[H][J]$ and show that $V[G]$ and $V[H]$ must have a shared ground via $(|\mathbb{R}|+|\mathbb{S}|)^+$-cc forcing. This allows a similar analysis of the related situation when $\mathbb{P}$ is replaced by a class iteration and $V[H]$ by a generic ground of $V[G]$. We conclude with a simple characterisation of the mantle of a class forcing extension, and an investigation of the possibilities for a version of the intermediate model theorem that applies to class forcing.