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Community Bot
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Suppose I have a transitive model $M$ of ZFC, and - in $M$ - $U$ is a measure on $\kappa$. Then the transitive collapse of the ultrapower of $M$ along $U$ is an inner model, $N\subset M$.

My question is:

Can we ever have $M$ be a class-generic extension of $N$?

Context: The set-forcing version of this question was answered by Douglas Ulrichwas answered by Douglas Ulrich, using a result of Hamkins-Kirmayer-Perlmutter; and Joel David Hamkins pointed out that the same result holds for many "nice" class forcings. However, the general class-forcing version remains open, and general class forcing is potentially so nasty that I thought this warranted its own question.

Suppose I have a transitive model $M$ of ZFC, and - in $M$ - $U$ is a measure on $\kappa$. Then the transitive collapse of the ultrapower of $M$ along $U$ is an inner model, $N\subset M$.

My question is:

Can we ever have $M$ be a class-generic extension of $N$?

Context: The set-forcing version of this question was answered by Douglas Ulrich, using a result of Hamkins-Kirmayer-Perlmutter; and Joel David Hamkins pointed out that the same result holds for many "nice" class forcings. However, the general class-forcing version remains open, and general class forcing is potentially so nasty that I thought this warranted its own question.

Suppose I have a transitive model $M$ of ZFC, and - in $M$ - $U$ is a measure on $\kappa$. Then the transitive collapse of the ultrapower of $M$ along $U$ is an inner model, $N\subset M$.

My question is:

Can we ever have $M$ be a class-generic extension of $N$?

Context: The set-forcing version of this question was answered by Douglas Ulrich, using a result of Hamkins-Kirmayer-Perlmutter; and Joel David Hamkins pointed out that the same result holds for many "nice" class forcings. However, the general class-forcing version remains open, and general class forcing is potentially so nasty that I thought this warranted its own question.

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Noah Schweber
Noah Schweber
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Can an ultrapower be undone by class forcing?

Suppose I have a transitive model $M$ of ZFC, and - in $M$ - $U$ is a measure on $\kappa$. Then the transitive collapse of the ultrapower of $M$ along $U$ is an inner model, $N\subset M$.

My question is:

Can we ever have $M$ be a class-generic extension of $N$?

Context: The set-forcing version of this question was answered by Douglas Ulrich, using a result of Hamkins-Kirmayer-Perlmutter; and Joel David Hamkins pointed out that the same result holds for many "nice" class forcings. However, the general class-forcing version remains open, and general class forcing is potentially so nasty that I thought this warranted its own question.