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On elementary embeddings from an inner model to the universe

Published online by Cambridge University Press:  12 March 2014

J. Vickers  and
P. D. Welch
J. Vickers
Affiliation:
University of Bristol, Department of Mathematics, Bristol BS8 1TW, England
P. D. Welch
Affiliation:
Kobe University, Graduate School of Science & Technology, Rokko-Dai, Nada-Ku, Kobe 657, Japan, E-mail: p.welch@bristol.ac.uk
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Abstract

We consider the following question of Kunen:

Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j: M → V)

imply Con(ZFC + ∃ a measurable cardinal)?

We use core model theory to investigate consequences of the existence of such a j: M → 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 3 , September 2001 , pp. 1090 - 1116
Copyright
Copyright © Association for Symbolic Logic 2001

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References

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