Let N be a transitive model of ZFC such that
ω N ⊂ N and 𝒫(ℝ) ⊂ N.
Assume that both V and N satisfy “the core model K exists.”
Then KN is an iterate of K, i.e.,
there exists an iteration tree 𝒯 on K
such that 𝒯 has successor length and
ℳ𝒯∞ = KN. Moreover,
if there exists an elementary embedding π : V → N
then the iteration map associated to the main branch of
𝒯 equals π ↾ K. (This answers
a question
of W. H. Woodin, M. Gitik, and others.)
The hypothesis that 𝒫(ℝ) ⊂ N is not
needed if there does not exist a transitive model of ZFC with
infinitely many Woodin cardinals.