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On elementary embeddings from an inner model to the universe

  • J. Vickers (a1) and P. D. Welch (a2)
Abstract

We consider the following question of Kunen:

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

imply Con(ZFC + ∃ a measurable cardinal)?

We use core model theory to investigate consequences of the existence of such a j: MV. 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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References
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[1]Dodd, A. J., The Core Model, Lecture Notes in Maths, vol. 61, London Mathematical Society, 1982.
[2]Donder, H-D., Jensen, R. B., and Koppelberg, B. J, Some applictions of the Core Model, Set Theory and Model Theory, Springer Lecture Notes in Maths, vol. 872, 1981, pp. 5597.
[3]Jensen, R. B., The Core Model for Measures of Order Zero, 1989, circulated manuscript.
[4]Jensen, R. B. and Vickers, J., A covering lemma part 2, manuscript.
[5]Jensen, R. B. and Vickers, J., Absoluteness of Jónsson Cardinals, 1997, preprint.
[6]Kanamori, A., The Higher Infinite, Perspectives in Mathematical Logic, Springer Verlag, Berlin, 1994.
[7]Miller, A., Some interesting problems, Set Theory of the Reals, (Proceedings of the Israeli Mathematical Conference), vol. 6, American Mathematical Society, 1993, pp. 645654, http://www.math.wisc.edu/~miller.
[8]Steel, J. R., The Core Model Iterability Problem, Springer Lecture Notes in Maths, vol. 8, Springer, 1988.
[9]Vickers, J., A covering lemma part 1, 1990, manuscript.
[10]Welch, P. D., Countable unions of simple sets in the Core Model, this Journal, vol. 61 (1996), no. 1, pp. 293312.
[11]Woodin, W. H, Σ12-absoluteness and supercompact cardinals, 05 1985, handwritten notes.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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