Journal of Symbolic Logic

The consistency strength of choiceless failures of SCH

Arthur W. Apter and Peter Koepke

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis (SCH) in the setting of the Zermelo-Fraenkel axiom system ZF without the Axiom of Choice (AC). By the new notion of parallel Prikry forcing that we introduce, we obtain surjective failures of SCH using only one measurable cardinal, including a surjective failure of Shelah's pcf theorem about the size of the power set of ℵω. Using symmetric collapses to ℵω, ℵω₁, or ℵω₂, we show that injective failures at ℵω, ℵω₁, or ℵω₂ can have relatively mild consistency strengths in terms of Mitchell orders of measurable cardinals. Injective failures of both the aforementioned theorem of Shelah and Silver's theorem that GCH cannot first fail at a singular strong limit cardinal of uncountable cofinality are also obtained. Lower bounds are shown by core model techniques and methods due to Gitik and Mitchell.

Article information

Source
J. Symbolic Logic, Volume 75, Issue 3 (2010), 1066-1080.

Dates
First available in Project Euclid: 9 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1278682215

Digital Object Identifier
doi:10.2178/jsl/1278682215

Mathematical Reviews number (MathSciNet)
MR2723782

Zentralblatt MATH identifier
1202.03056

Keywords
Singular Cardinals Hypothesis (SCH) core model parallel Prikry forcing symmetric inner model

Citation

Apter, Arthur W.; Koepke, Peter. The consistency strength of choiceless failures of SCH. J. Symbolic Logic 75 (2010), no. 3, 1066--1080. doi:10.2178/jsl/1278682215. https://projecteuclid.org/euclid.jsl/1278682215


Export citation