i added an answer and then realized it was a complete nonsense, i was getting at the following question.
Suppose kappa is an inaccessible cardinal. Is there a transitive model M of ZFC such that
- M has height kappa
- for some stationary in kappa set S, for every alpha in S, M computes alpha^+ correctly,
- for some stationary set D, for every alpha in D, alpha is inaccessible in M
- there are no inacessibles below kappa.
One can show that if the answer is yes then there is an inner model with a Woodin cardinal.
Assume M and N are as in the statement. Assume there is no inner model with a woodin. Then clause 2 implies that K^M and K must coiterate (K is the core model). This means that K has stationary set of inaccessibles, so by Trevor's trick V must also have inaccessibles. So we must have covering fails in N which then implies that K doesn't exist.
there is one subtle point here, we need to see that K^M is iterable in V but this can be done by some absoluteness trick.
maybe one can use stationary tower to produce such an M?