  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 $V[c]$.
