Abstract
It is proved that ifκ is supercompact, there are at least (2• P κ(β)•)+normal ultrafilters overP k (β) and ifV=H.O.D. exactly (22• P κ(β)•) normal ultrafilters.
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References
Montague-Vaught,Natural models of set theories, Fund. Math.47 (1959), 219–242.
Myhill-D. Scott,Ordinal definability, to appear in the Proceedings of the U.C.L.A. Seminar of Set Theory, 1967.
W. N. Reinhardt and R. Solovay,Strong axioms of infinity and elementary embeddings, (to appear).
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This is a part of the author’s Ph.D. thesis prepared under the supervision of Professor Azriel Levy for whose help the author is grateful.
In 1966–67, Solovay proved Theorem 1 for the caseβ=κ without the condition of extendability. The same result, under a somewhat weaker assumption was proved by Namba in 1967–68. As noted by Solovay, his proof can be adapted to a generalβ (under weaker assumptions; if |P k (β) |=β it is only needed that ℵ is 2β-supercompact). Solovay’s result will be published in [3].
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Magidor, M. There are many normal ultrafiltres corresponding to a supercompact cardinal. Israel J. Math. 9, 186–192 (1971). https://doi.org/10.1007/BF02771583
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DOI: https://doi.org/10.1007/BF02771583