zbMATH — the first resource for mathematics

Some new upper bounds in consistency strength for certain choiceless large cardinal patterns. (English) Zbl 0755.03028
Assuming the existence of an almost huge cardinal, Gitik and the author have proved the existence of various models of \(\text{ZF}+\neg\text{AC}_ \omega\) in which large classes of cardinals are singular (e.g. a model of ZF in which all uncountable nonmeasurable cardinals are singular). Using the method of extenders, the author improves these results by weakening the assumption in consistency strength.

03E60 Determinacy principles
03E55 Large cardinals
03E35 Consistency and independence results
Full Text: DOI
[1] [An] Andretta, A.: Handwritten notes on extenders
[2] [Ap1] Apter, A.: On a problem inspired by determinacy. Isr. J. Math.61, 256-270 (1988) · Zbl 0661.03040
[3] [Ap2] Apter, A.: Some results on consecutive large cardinals. II. Applications of Radin forcing. Isr. J. Math.52, 273-292 (1985) · Zbl 0603.03016
[4] [C1] Cummings, J.: A model in which GCH holds at successors but fails at limits. Trans. Am. Math. Soc. (to appear) · Zbl 0758.03022
[5] [C2] Cummings, J.: Forcing and the arithmetic of small cardinals (to appear)
[6] [FW] Foreman, M., Woodin, H.: The GCH can fail everywhere. Ann. Math.133, 1-36 (1991) · Zbl 0718.03040
[7] [G] Gitik, M.: Regular cardinals in models of ZF. Trans. Am. Math. Soc.290, 41-68 (1985) · Zbl 0589.03033
[8] [GS] Gitik, M., Shelah, S.: On certain indestructibility of strong cardinals and a question of Hajnal. Arch. Math. Logic28, 35-42 (1989) · Zbl 0663.03041
[9] [Ma1] Magidor, M.: On the singular cardinals problem. I. Isr. J. Math.28, 1-31 (1977) · Zbl 0364.02040
[10] [Ma2] Magidor, M.: On the singular cardinals problem. II. Ann. Math.106, 517-549 (1977) · Zbl 0365.02057
[11] [Mi1] Mitchell, W.: Sets constructed from sequences of measures revisited. J. Symb. Logic48, 600-609 (1983) · Zbl 0527.03032
[12] [Mi2] Mitchell, W.: The core model for sequences of measures. I. Proc. Cambridge Philos. Soc.95, 229-260 (1984) · Zbl 0539.03030
[13] [MS1] Martin, D.A., Steel, J.: A proof of projective determinacy. J. Am. Math. Soc.2, 71-125 (1989) · Zbl 0668.03021
[14] [MS2] Martin, D.A., Steel, J.: Projective determinacy. Proc. Natl. Acad. Sci. USA85, 6582-6586 (1988) · Zbl 0656.03036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.