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2013 Consecutive Singular Cardinals and the Continuum Function
Arthur W. Apter, Brent Cody
Notre Dame J. Formal Logic 54(2): 125-136 (2013). DOI: 10.1215/00294527-1960434

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.

Citation

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Arthur W. Apter. Brent Cody. "Consecutive Singular Cardinals and the Continuum Function." Notre Dame J. Formal Logic 54 (2) 125 - 136, 2013. https://doi.org/10.1215/00294527-1960434

Information

Published: 2013
First available in Project Euclid: 21 February 2013

zbMATH: 1284.03235
MathSciNet: MR3028791
Digital Object Identifier: 10.1215/00294527-1960434

Subjects:
Primary: 03E25
Secondary: 03E35, 03E45, 03E55

Rights: Copyright © 2013 University of Notre Dame

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Vol.54 • No. 2 • 2013
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