Notre Dame Journal of Formal Logic

Consecutive Singular Cardinals and the Continuum Function

Arthur W. Apter and Brent Cody

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Abstract

We show that from a supercompact cardinal κ, there is a forcing extension V[G] that has a symmetric inner model N in which ZF+¬AC holds, κ and κ+ are both singular, and the continuum function at κ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of κ of length equal to any predetermined ordinal. We also show that the above situation can be collapsed to obtain a model of ZF+¬ACω in which either (1) 1 and 2 are both singular and the continuum function at 1 can be precisely controlled, or (2) ω and ω+1 are both singular and the continuum function at ω can be precisely controlled. Additionally, we discuss a result in which we separate the lengths of sequences of distinct subsets of consecutive singular cardinals κ and κ+ in a model of ZF. Some open questions concerning the continuum function in models of ZF with consecutive singular cardinals are posed.

Article information

Source
Notre Dame J. Formal Logic, Volume 54, Number 2 (2013), 125-136.

Dates
First available in Project Euclid: 21 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1361454970

Digital Object Identifier
doi:10.1215/00294527-1960434

Mathematical Reviews number (MathSciNet)
MR3028791

Zentralblatt MATH identifier
1284.03235

Subjects
Primary: 03E25: Axiom of choice and related propositions
Secondary: 03E35: Consistency and independence results 03E45: Inner models, including constructibility, ordinal definability, and core models 03E55: Large cardinals

Keywords
supercompact cardinal supercompact Prikry forcing GCH symmetric inner model

Citation

Apter, Arthur W.; Cody, Brent. Consecutive Singular Cardinals and the Continuum Function. Notre Dame J. Formal Logic 54 (2013), no. 2, 125--136. doi:10.1215/00294527-1960434. https://projecteuclid.org/euclid.ndjfl/1361454970


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