Sandwiching the consistency strength of two global choiceless cardinal patterns

• Autorzy: Apter A. W.
• Języki publikacji: EN

Abstrakty

EN

We provide upper and lower bounds in consistency strength for the theories "ZF + ¬ACω + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω" and "ZF + ¬ACω + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω 1". In particular, our models for both of these theories satisfy "ZF + ¬ACω + κ is singular iff κ is either an uncountable limit cardinal or the successor of an uncountable limit cardinal".

Słowa kluczowe

EN

supercompact cardinal; supercompact Radin forcing; Radin sequence of measures; symmetric inner model

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2009
• Tom: Vol. 57, no 3-4
• Strony: 189--197
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Apter A. W.

• Department of Mathematics, Baruch College of CUNY, New York, NY 10010, U.S.A.

Bibliografia

• [1] A. Apter, A cardinal pattern inspired by AD, Math. Logic Quart. 42 (1996), 211-218.
• [2] A. Apter, How many normal measures can אω1+1 carry?, Math. Logic Quart. 56 (2010), to appear.
• [3] A. Apter, On the class of measurable cardinals without the axiom of choice, Israel J. Math. 79 (1992), 367-379.
• [4] A. Apter, Some new upper bounds in consistency strength for certain choiceless large cardinal patterns, Arch. Math. Logic 31 (1992), 201-205.
• [5] A. Apter, Some results on consecutive large cardinals II: Applications of Radin forcing, Israel J. Math. 52 (1985), 273-292.
• [6] A. Apter and P. Koepke, Making all cardinals almost Ramsey, Arch. Math. Logic 47 (2008), 769-783.
• [7] D. Busche and R. Schindler, The strength of choiceless patterns of singular and weakly compact cardinals, Ann. Pure Appl. Logic 159 (2009), 198-248.
• [8] J. Cummings and W. H. Woodin, Generalised Prikry forcings, circulated manuscript.
• [9] M. Foreman and W. H. Woodin, The GCH can fail everywhere, Ann. of Math. 133 (1991), 1-36.
• [10] M. Gitik, Prikry-type forcings, forthcoming article in the Handbook of Set Theory.
• [11] M. Gitik, Regular cardinals in models of ZF, Trans. Amer. Math. Soc. 290 (1985), 41-68.
• [12] L. Radin, Adding closed cofinal sequences to large cardinals, Ann. Math. Logic 23 (1982), 263-283.