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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

1. M has height kappa
2. for some stationary in kappa set S, for every alpha in S, M computes alpha^+ correctly,
3. for some stationary set D, for every alpha in D, alpha is inaccessible in M
4. there are no inacessibles below kappa.

One can show that if the answer is yes then there is an inner model with a proper class of measurables.

Assume M is as in the statement. Assume there is no inner model with a proper class of measurables. 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 V which then implies that K doesn't exist.

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

1. M has height kappa

2. for some stationary in kappa set S, for every alpha in S, M computes alpha^+ correctly,

3. for some stationary set D, for every alpha in D, alpha is inaccessible in M

4. there are no inacessibles below kappa.

   One can show that if the answer is yes then there is an inner model with a proper class of measurables.

Assume M is as in the statement. Assume there is no inner model with a proper class of measurables. 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 V which then implies that K doesn't exist.

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

1. M has height kappa
2. for some stationary in kappa set S, for every alpha in S, M computes alpha^+ correctly,
3. for some stationary set D, for every alpha in D, alpha is inaccessible in M
4. 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 Is 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 V 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?

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

1. M has height kappa
2. for some stationary in kappa set S, for every alpha in S, M computes alpha^+ correctly,
3. for some stationary set D, for every alpha in D, alpha is inaccessible in M
4. there are no inacessibles below kappa.

One can show that if the answer is yes then there is an inner model with a proper class of measurables.

Assume M is as in the statement. Assume there is no inner model with a proper class of measurables. 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 V which then implies that K doesn't exist.

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

1. M has height kappa
2. for some stationary in kappa set S, for every alpha in S, M computes alpha^+ correctly,
3. for some stationary set D, for every alpha in D, alpha is inaccessible in M
4. 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 V 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?

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

1. M has height kappa
2. for some stationary in kappa set S, for every alpha in S, M computes alpha^+ correctly,
3. for some stationary set D, for every alpha in D, alpha is inaccessible in M
4. 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 Is 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 V 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?