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Strong ultrapowers and long core models

Published online by Cambridge University Press:  12 March 2014

James Cummings
James Cummings*
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755, E-mail: [email protected]
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Extract

In his paper [7] Steel asked whether there can exist a normal measure U on a cardinal κ such that

We use Reverse Easton forcing to show that this is consistent from a P2κ hypermeasure; we also show that the result is sharp, using the core model for nonoverlapping coherent extender sequences.

The proof uses forcing technology due to Woodin.

In this section we collect some facts that are useful in the forcing constructions of the next section. None of them are due to us, and we are unsure to whom they should be attributed for the most part. We give sketchy proofs; the reader who wants to see more details is referred to [2].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 1 , March 1993 , pp. 240 - 248
Copyright
Copyright © Association for Symbolic Logic 1993

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References

REFERENCES

[1]Baumgartner, J. E., Iterated forcing, Surveys in set theory, Cambridge University Press, London and New York, 1983, pp. 1–55.Google Scholar
[2]Cummings, J., A model in which GCH holds at successors but fails at limits, Transactions of the American Mathematical Society, vol. 329 (1992), pp. 1–39.Google Scholar
[3]Gitik, M., The negation of SCH from o(κ) = κ ++, Annals of Pure and Applied Logic, vol. 43(1989), pp. 209–234.CrossRefGoogle Scholar
[4]Gitik, M., On measurable cardinals violating GCH, to appear.Google Scholar
[5]Koepke, P., An introduction to extenders and core models for extender sequences, Logic Colloquium '87, North-Holland, Amsterdam, 1989, pp. 137–182.Google Scholar
[6]Solovay, Reinhardt, and Kanamori, , Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978).CrossRefGoogle Scholar
[7]Steel, J., The wellfoundedness of the Mitchell order, to appear.Google Scholar