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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.

MSC:

03E45 Inner models, including constructibility, ordinal definability, and core models
03E15 Descriptive set theory
03E55 Large cardinals
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[1] DOI: 10.1090/S0894-0347-1994-1224594-7 · doi:10.1090/S0894-0347-1994-1224594-7
[2] The core model for non-overlapping extender sequences (1991)
[3] The core model (1982)
[4] Core models with more Woodin cardinals 67 pp 1197– (2002) · Zbl 1012.03055
[5] DOI: 10.1016/S0168-0072(96)00032-2 · Zbl 0868.03021 · doi:10.1016/S0168-0072(96)00032-2
[6] The core model iterability problem (1996) · Zbl 0864.03035
[7] Annals of Pure and Applied Logic 116 pp 207– (2002)
[8] DOI: 10.1016/S0168-0072(01)00103-8 · Zbl 1012.03054 · doi:10.1016/S0168-0072(01)00103-8
[9] Fine structure and iteration trees (1994) · Zbl 0805.03042
[10] Logic Colloquium ’01 20 pp 386– (2005)
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