# The Bristol Model: an abyss called a Cohen real

## TL;DR

This paper constructs a new ZF model between L and L[c] for a Cohen real c, providing insights into models of ZF, the HOD Conjecture, and intermediate models, while also addressing questions about Kinna–Wagner Principles.

## Contribution

It introduces a novel ZF model between L and L[c] that is not of the form L(x), advancing understanding of models of ZF and related conjectures.

## Key findings

- Constructed a ZF model between L and L[c] for a Cohen real.
- Provided a positive answer to Grigorieff's question on intermediate models.
- Demonstrated the failure of Kinna–Wagner Principles in ZF.

## Abstract

We construct a model $M$ of ZF which lies between $L$ and $L[c]$ for a Cohen real $c$ and does not have the form $L(x)$ for any set $x$. This is loosely based on the unwritten work done in a Bristol workshop about Woodin's HOD Conjecture in 2011. The construction given here allows for a finer analysis of the needed assumptions on the ground models, thus taking us one step closer to understanding models of ZF, and the HOD Conjecture and its relatives. This model also provides a positive answer to a question of Grigorieff about intermediate models of ZF, and we use it to show the failure of Kinna--Wagner Principles in ZF.