Approaching a Bristol model

TL;DR

This paper provides an accessible overview, intuition, and new results on the Bristol model, an inner model of set theory that challenges constructibility assumptions and explores the structure of models of ZF.

Contribution

It offers a detailed guide and new insights into the Bristol model, including proofs that the Boolean Prime Ideal theorem fails and that the ground model is definable within its extensions.

Findings

Boolean Prime Ideal theorem fails in the Bristol model

Ground model is always definable in Bristol extensions

Sets in the Bristol model cannot be linearly ordered

Abstract

The Bristol model is an inner model of $L[c]$, where $c$ is a Cohen real, which is not constructible from a set. The idea was developed in 2011 in a workshop taking place in Bristol, but was only written in detail by the author in [8]. This paper is a guide for those who want to get a broader view of the construction. We try to provide more intuition that might serve as a jumping board for those interested in this construction and in odd models of $\mathsf{ZF}$. We also correct a few minor issues in the original paper, as well as prove new results. For example, that the Boolean Prime Ideal theorem fails in the Bristol model, as some sets cannot be linearly ordered, and the ground model is always definable in its Bristol extensions. In addition to this we include a discussion on Kinna--Wagner Principles, which we think may play an important role in understanding the generic multiverse in $\mathsf{ZF}$.