Sequential and distributive forcings without choice

@article{Karagila2021SequentialAD,
  title={Sequential and distributive forcings without choice},
  author={Asaf Karagila and Jonathan Schilhan},
  journal={Canadian Mathematical Bulletin},
  year={2021}
}
. In the Zermelo–Fraenkel set theory with the Axiom of Choice a forcing notion is “ κ -distributive” if and only if it is “ κ -sequential”. We show that without the Axiom of Choice this equivalence fails, even if we include a weak form of the Axiom of Choice, the Principle of Dependent Choice for κ . Still, the equivalence may still hold along with very strong failures of the Axiom of Choice, assuming the consistency of large cardinal axioms. We also prove that while a κ -distributive forcing… 
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Forcing over choiceless models and generic absoluteness

. We develop a toolbox for forcing over arbitrary models of set theory without the axiom of choice. In particular, we introduce a variant of the countable chain condition and prove an iteration…

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