The ultrafilter characterization of huge cardinals

Mignone, Robert J.

doi:10.1090/S0002-9939-1984-0733411-6

The ultrafilter characterization of huge cardinals
HTML articles powered by AMS MathViewer

by Robert J. Mignone PDF

Proc. Amer. Math. Soc. 90 (1984), 585-590 Request permission

Abstract:

A huge cardinal can be characterized using ultrafilters. After an argument is made for a particular ultrafilter characterization, it is used to prove the existence of a measurable cardinal above the huge cardinal, and an ultrafilter over the set of all subsets of this measurable cardinal of size smaller than the huge cardinal. Finally, this last ultrafilter is disassembled intact by a process which often produces a different ultrafilter from the one started out with. An important point of this paper is given the existence of the particular ultrafilter characterization of a huge cardinal mentioned above these results are proved in Zermelo-Fraenkel set theory without the axiom of choice.

References

Many-times huge and supercompact cardinals

49

C. A. Di Prisco and J. Henle, On the compactness of $\aleph _{1}$ and $\aleph _{2}$, J. Symbolic Logic 43 (1978), no. 3, 394–401. MR 503778, DOI 10.2307/2273517
Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
Thomas J. Jech, Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic 5 (1972/73), 165–198. MR 325397, DOI 10.1016/0003-4843(73)90014-4
Telis K. Menas, A combinatorial property of $p_{k}\lambda$, J. Symbolic Logic 41 (1976), no. 1, 225–234. MR 409186, DOI 10.2307/2272962
Robert Mignone, Ultrafilters resulting from the axiom of determinateness, Proc. London Math. Soc. (3) 43 (1981), no. 3, 582–605. MR 635570, DOI 10.1112/plms/s3-43.3.582
Robert M. Solovay, William N. Reinhardt, and Akihiro Kanamori, Strong axioms of infinity and elementary embeddings, Ann. Math. Logic 13 (1978), no. 1, 73–116. MR 482431, DOI 10.1016/0003-4843(78)90031-1