Intermediate models with deep failure of choice

TL;DR

This paper demonstrates that for any ZF model and Cohen-generic real extension, there exist numerous intermediate models that do not arise from sets and lack the axiom of choice, resolving a longstanding question.

Contribution

It proves the existence of intermediate ZF models with unbounded Kinna-Wagner degrees in any Cohen-generic extension, fully answering Grigorieff's question.

Findings

Existence of intermediate models in any Cohen-generic extension

Intermediate models have unbounded Kinna-Wagner degrees

Many intermediate models do not satisfy the axiom of choice

Abstract

The following question was asked by Grigorieff: Suppose $V$ is a ZFC model and $V[G]$ is a set-generic extension of $V$. Can there be a ZF model $N$ so that $V\subset N \subset V[G]$ yet $N$ is not equal to $V(A)$ for any set $A\in V[G]$? The first such model was constructed by Karagila. This is the so-called \emph{Bristol model}, an intermediate model between $L$ and $L[c]$ where $c$ is a Cohen-generic real over $L$. Karagila further proves that the Kinna-Wager degree is unbounded in this model. We prove that such an intermediate extension can be found in a Cohen-generic extension of \emph{any} ground model, fully resolving Grigorieff's question. That is, let $V$ be \emph{any} ZF model and $c$ a Cohen-generic real over $V$. We prove that there is an intermediate ZF-model $V\subset N \subset V[c]$ so that $N$ is not equal to $V(A)$ for any set $A\in V[c]$, the Kinna-Wagner degree of $N$ is unbounded and, in particular, no set forcing in $N$ forces the axiom of choice. Therefore, there are class many different intermediate models of ZF between $V$ and $V[c]$.