The consistency strength of choiceless failures of SCH

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 . Using symmetric collapses to 2000 Mathematics Subject Classifications: 03E25, 03E35, 03E45, 03E55.

• Type: Journal
• Year: 2010
• Where: JSYML
• Authors: Arthur W. Apter, Peter Koepke
• Keywords: Exact Consistency Strengths, JSYML 2010, Singular Cardinals Hypothesis, Surjective Failure