Iterates of the core model

Abstract

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 K^N is an iterate of K, i.e., there exists an iteration tree 𝒯 on K such that 𝒯 has successor length and ℳ^𝒯_∞ = K^N. 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.