Approaching a Bristol model
Asaf Karagila
TL;DR
The Bristol model provides a canonical choiceless intermediate between $V$ and $V[c]$ by coding a long sequence through symmetric iterations over a Cohen real. The paper distills three complementary viewpoints (Nature, Cause, Accident) and develops a detailed decoding apparatus to realize the model, establishing that the Bristol model satisfies $ZF$, fails the Boolean Prime Ideal theorem, and defies being of the form $L(x)$. It also investigates Kinna–Wagner Principles, ground-model definability, and the extent to which constructibility assumptions can be weakened, while outlining the multiverse implications and a suite of open questions. The work highlights how intricate symmetry, permutation, and scale-decoding mechanisms interact to shape a rich, largely undefinable landscape of intermediate $ZF$-models, with broad relevance to the study of choice principles and the generic multiverse.
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 $\mathsf{ZF}$.