An ultrafilter with property 𝜎

Shioya, Masahiro

doi:10.1090/S0002-9939-05-08161-X

An ultrafilter with property $\sigma$
HTML articles powered by AMS MathViewer

by Masahiro Shioya

Proc. Amer. Math. Soc. 134 (2006), 1819-1821

DOI: https://doi.org/10.1090/S0002-9939-05-08161-X

Published electronically: October 28, 2005

PDF | Request permission

Abstract:

Let $\kappa$ be a $\lambda$-supercompact cardinal. We show that $\mathcal {P}_\kappa \lambda$ carries a normal ultrafilter with a property introduced by Menas. With it we give a transparent proof of Kamo’s theorem that $\mathcal {P}_\kappa \lambda$ carries a normal ultrafilter with the partition property.

References

Shizuo Kamo, Partition properties on $\scr P_{\kappa }\lambda$, J. Math. Soc. Japan 54 (2002), no. 1, 123–133. MR 1864930, DOI 10.2969/jmsj/1191593957
Shizuo Kamo, Normal ultrafilters without the partition property, Sūrikaisekikenkyūsho K\B{o}kyūroku 1202 (2001), 1–6. Axiomatic set theory (Japanese) (Kyoto, 2000). MR 1855546
Akihiro Kanamori, The higher infinite, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Large cardinals in set theory from their beginnings. MR 1994835
Kenneth Kunen and Donald H. Pelletier, On a combinatorial property of Menas related to the partition property for measures on supercompact cardinals, J. Symbolic Logic 48 (1983), no. 2, 475–481. MR 704100, DOI 10.2307/2273563
Telis K. Menas, On strong compactness and supercompactness, Ann. Math. Logic 7 (1974/75), 327–359. MR 357121, DOI 10.1016/0003-4843(75)90009-1
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 M. Solovay, Strongly compact cardinals and the GCH, Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1974, pp. 365–372. MR 0379200