Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T22:49:38.579Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Dodd, A. J., The Core Model, Lecture Notes in Maths, vol. 61, London Mathematical Society, 1982.CrossRefGoogle Scholar
[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. 55–97.CrossRefGoogle Scholar
[3]Jensen, R. B., The Core Model for Measures of Order Zero, 1989, circulated manuscript.Google Scholar
[4]Jensen, R. B. and Vickers, J., A covering lemma part 2, manuscript.Google Scholar
[5]Jensen, R. B. and Vickers, J., Absoluteness of Jónsson Cardinals, 1997, preprint.Google Scholar
[6]Kanamori, A., The Higher Infinite, Perspectives in Mathematical Logic, Springer Verlag, Berlin, 1994.Google Scholar
[7]Miller, A., Some interesting problems, Set Theory of the Reals, (Proceedings of the Israeli Mathematical Conference), vol. 6, American Mathematical Society, 1993, pp. 645–654, http://www.math.wisc.edu/~miller.Google Scholar
[8]Steel, J. R., The Core Model Iterability Problem, Springer Lecture Notes in Maths, vol. 8, Springer, 1988.Google Scholar
[9]Vickers, J., A covering lemma part 1, 1990, manuscript.Google Scholar
[10]Welch, P. D., Countable unions of simple sets in the Core Model, this Journal, vol. 61 (1996), no. 1, pp. 293–312.Google Scholar
[11]Woodin, W. H, Σ12-absoluteness and supercompact cardinals, 05 1985, handwritten notes.Google Scholar