Journal of Symbolic Logic

On Elementary Embeddings from an Inner Model to the Universe

J. Vickers and P. D. Welch

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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.

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J. Symbolic Logic, Volume 66, Issue 3 (2001), 1090-1116.

First available in Project Euclid: 6 July 2007

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Vickers, J.; Welch, P. D. On Elementary Embeddings from an Inner Model to the Universe. J. Symbolic Logic 66 (2001), no. 3, 1090--1116.

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