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  1. Remarks on Gitik's model and symmetric extensions on products of the Lévy collapse.Amitayu Banerjee - 2020 - Mathematical Logic Quarterly 66 (3):259-279.
    We improve on results and constructions by Apter, Dimitriou, Gitik, Hayut, Karagila, and Koepke concerning large cardinals, ultrafilters, and cofinalities without the axiom of choice. In particular, we show the consistency of the following statements from certain assumptions: the first supercompact cardinal can be the first uncountable regular cardinal, all successors of regular cardinals are Ramsey, every sequence of stationary sets in is mutually stationary, an infinitary Chang conjecture holds for the cardinals, and all are singular. In each of the (...)
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  • Combinatorial properties and dependent choice in symmetric extensions based on Lévy collapse.Amitayu Banerjee - 2022 - Archive for Mathematical Logic 62 (3):369-399.
    We work with symmetric extensions based on Lévy collapse and extend a few results of Apter, Cody, and Koepke. We prove a conjecture of Dimitriou from her Ph.D. thesis. We also observe that if V is a model of $$\textsf {ZFC}$$ ZFC, then $$\textsf {DC}_{<\kappa }$$ DC < κ can be preserved in the symmetric extension of V in terms of symmetric system $$\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle $$ ⟨ P, G, F ⟩, if $${\mathbb {P}}$$ P is $$\kappa $$ (...)
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  • The consistency strength of choiceless failures of SCH.Arthur W. Apter & Peter Koepke - 2010 - Journal of Symbolic Logic 75 (3):1066-1080.
    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 $\aleph _{\omega}$ . Using symmetric collapses to $\aleph _{\omega}$ , $\aleph _{\omega _{1}}$ , (...)
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  • Making all cardinals almost Ramsey.Arthur W. Apter & Peter Koepke - 2008 - Archive for Mathematical Logic 47 (7-8):769-783.
    We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + ${\neg {\rm AC}_\omega}$ in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular cardinals, we can weaken this to an equiconsistency of the following theories: “ZFC + There is a proper class of regular almost (...)
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