## Abstract

We consider the following question of Kunen: Does Con(ZFC + $\exists$M a transitive inner model and a non-trivial elementary embedding j: M $\longrightarrow$ V) imply Con (ZFC + $\exists$ a measurable cardinal)? We use core model theory to investigate consequences of the existence of such a j : M $\rightarrow$ V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of "there exists a proper class of almost Ramsey cardinals". Conversely, if On is Ramsey, then such a j, M are definable. We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points.

Translated title of the contribution | On elementary embeddings from an inner model to the universe |
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Original language | English |

Pages (from-to) | 1090 - 1116 |

Number of pages | 27 |

Journal | Journal of Symbolic Logic |

Volume | 66 (3) |

Publication status | Published - Sep 2001 |