A note on a question of Woodin

• Autorzy: Apter Arthur W.

Identyfikatory

DOI

10.4064/ba230918-26-10

• Języki publikacji: EN

Abstrakty

EN

A question of Woodin from the 1980s asks, assuming there is no inner model of ZFC with a strong cardinal, if it is possible for there to be a model M of ZFC such that M ⊨ “2אω > אω+2 and 2אn = אn+1 for every n < ω”, together with the existence of an inner model N∗ ⊆ M of ZFC such that for the γ, δ satisfying γ = (אω) and δ = (אω+3)M,N∗ ⊨ “γ is measurable and 2γ ≥ δ”. We show that this is the case for a choiceless version of Woodin’s question, where we assume AC fails in M but holds in N∗. We also prove analogous results for אω1 and אω2 . The methods used allow for equiconsistencies in certain cases.

Słowa kluczowe

EN

Singular Cardinal Hypothesis (SCH); core model; symmetric inner model; equiconsistency

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2023
• Tom: Vol. 71, no 2
• Strony: 115--122
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Apter Arthur W.

• Department of Mathematics, Baruch College of CUNY, New York, NY 10010, USA
• The CUNY Graduate Center, Mathematics, New York, NY 10016, USA

• https://orcid.org/0000-0002-7091-3628

Bibliografia

• [1] A. Apter, On weak square, approachability, the tree property, and failures of SCH in a choiceless context, Math. Logic Quart. 66 (2020), 115-120.
• [2] A. Apter and P. Koepke, The consistency strength of אω and אω1being Rowbottom cardinals without the Axiom of Choice, Arch. Math. Logic 45 (2006), 721-737.
• [3] A. Apter and P. Koepke, The consistency strength of choiceless failures of SCH, J. Symbolic Logic 75 (2010), 1066-1080.
• [4] M. Gitik, Blowing up power of a singular cardinal—wider gaps, Ann. Pure Appl.Logic 116 (2002), 1-38.
• [5] M. Gitik, On measurable cardinals violating the continuum hypothesis, Ann. Pure Appl. Logic 63 (1993), 227-240.
• [6] M. Gitik, The negation of the Singular Cardinal Hypothesis from o(κ) = κ++, Ann.Pure Appl. Logic 43 (1989), 209-234.
• [7] M. Gitik and W. Mitchell, Indiscernible sequences for extenders, and the singular cardinal hypothesis, Ann. Pure Appl. Logic 82 (1996), 273-316.
• [8] M. Magidor, Changing cofinality of cardinals, Fund. Math. 99 (1978), 61-71.[9] S. Shelah, Cardinal Arithmetic, Oxford Logic Guides 29, Oxford Univ. Press, New York, 1994.
• [10] J. Silver, On the singular cardinals problem, in: Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974), Vol. 1, Canad. Math. Congr., Montreal, Quebec, 1975, 265-268.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024)