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Woodin cardinals, Shelah cardinals, and the Mitchell-Steel core model

Author(s): Ernest Schimmerling
Journal: Proc. Amer. Math. Soc. 130 (2002), 3385-3391.
MSC (1991): Primary 03E45, 03E55
Posted: March 18, 2002
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Abstract: Theorem 4 is a characterization of Woodin cardinals in terms of Skolem hulls and Mostowski collapses. We define weakly hyper-Woodin cardinals and hyper-Woodin cardinals. Theorem 5 is a covering theorem for the Mitchell-Steel core model, which is constructed using total background extenders. Roughly, Theorem 5 states that this core model correctly computes successors of hyper-Woodin cardinals. Within the large cardinal hierarchy, in increasing order we have: measurable Woodin, weakly hyper-Woodin, Shelah, hyper-Woodin, and superstrong cardinals. (The comparison of Shelah versus hyper-Woodin is due to James Cummings.)

References:

1.: A. Andretta, I. Neeman and J.R. Steel, The domestic levels of are iterable, to appear in Israel J. Math.
2.: A. Kanamori, The higher infinite, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1994. MR 96k:03125
3.: D.A. Martin and J.R. Steel, A proof of projective determinacy, J. Amer. Math. Soc. 2 (1989) 71-125. MR 89m:03042
4.: D.A. Martin and J.R. Steel, Iteration trees, J. Amer. Math. Soc. 7 (1994) 1-73. MR 94f:03062
5.: W.J. Mitchell and J.R. Steel, Fine structure and iteration trees, Lecture Notes in Logic, 3, Springer-Verlag, Berlin, 1994. MR 95m:03099
6.: I. Neeman, Inner models in the region of a Woodin limit of Woodin cardinals, preprint.
7.: J.R. Steel, The core model iterability problem, Lecture Notes in Logic, 8, Springer-Verlag, Berlin, 1996. MR 99k:03043

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Additional Information:

Ernest Schimmerling
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213-3890
Email: eschimme@andrew.cmu.edu

DOI: 10.1090/S0002-9939-02-06455-9
PII: S 0002-9939(02)06455-9
Keywords: Large cardinals, core models
Received by editor(s): May 16, 2001
Received by editor(s) in revised form: June 14, 2001
Posted: March 18, 2002
Additional Notes: This research was supported by NSF DMS-0088948.
Communicated by: Carl G. Jockusch, Jr.
Copyright of article: Copyright 2002, American Mathematical Society


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