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.
"The consistency strength of choiceless failures of SCH." J. Symbolic Logic 75 (3) 1066 - 1080, September 2010. https://doi.org/10.2178/jsl/1278682215