Iterates of the core model. (English) Zbl 1109.03063

Summary: Let \(N\) be a transitive model of ZFC such that \(^\omega N \subset N\) and \({\mathcal P}(\mathbb{R})\subset N\). 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 \({\mathcal T}\) on \(K\) such that \({\mathcal T}\) has successor length and \({\mathcal M}^{\mathcal T}_\infty=K^N\). Moreover, if there exists an elementary embedding \(\pi:V\to N\) then the iteration map associated to the main branch of \({\mathcal T}\) equals \(\pi \upharpoonright K\). (This answers a question of W. H. Woodin, M. Gitik, and others.) The hypothesis that \({\mathcal P}(\mathbb{R})\subset N\) is not needed if there does not exist a transitive model of ZFC with infinitely many Woodin cardinals.


03E45 Inner models, including constructibility, ordinal definability, and core models
03E15 Descriptive set theory
03E55 Large cardinals
Full Text: DOI


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