# Intermediate submodels which do not satisfy AC

The following is known:

Theorem. Suppose $V[G]$ is a generic extension of $V$ by a set forcing, and let $N$ be a model of $ZFC$ with $V\subseteq N\subseteq V[G].$ Then $N$ is a generic extension of $V$ by a set forcing, in particular $N=V[A],$ for some set of ordinals.

It seems that the above theorem is not true if $N$ does not satisfy $AC$. In fact the following abstract is given in a talk by James Cummings (see http://settheory.mathtalks.org/cmu-math-logic-seminar-tues-11-september/):

If $c$ is Cohen-generic over $L$, then there is a transitive class model $M$ of $ZF$ intermediate between $L$ and $L[c]$ which is not of the form $L(A)$ for any $A.$

Does anyone know a proof of this fact?

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The Bristol model, I see. Have you contacted James or Menachem? Asaf Karagila was working on an alternative approach for a while. –  Andres Caicedo Feb 13 at 6:31
@AsafKaragila: Dear Asaf, have you completed your notes on Bristol model? –  Mohammad Golshani Nov 13 at 5:24

To my knowledge there is no written proof of this fact. I have all the available notes, which include a very very scattered description of $V_{\omega+1}$ and $V_{\omega+2}$ of this model $M$, and a single lemma which is used to proceed through successor of singular cardinals.