Geodesic
Sequential and distributive forcings without choice
Canadian mathematical bulletin, Tome 66 (2023) no. 3, pp. 868-880

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DOI

In the Zermelo–Fraenkel set theory with the Axiom of Choice, a forcing notion is “$\kappa $-distributive” if and only if it is “$\kappa $-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 $\kappa $. 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 although a $\kappa $-distributive forcing notion may violate Dependent Choice, it must preserve the Axiom of Choice for families of size $\kappa $. On the other hand, a $\kappa $-sequential can violate the Axiom of Choice for countable families. We also provide a condition of “quasiproperness” which is sufficient for the preservation of Dependent Choice, and is also necessary if the forcing notion is sequential.
DOI : 10.4153/S0008439522000753
Mots-clés : Forcing, Axiom of Choice, distributive forcing, sequential forcing 
Karagila, Asaf; Schilhan, Jonathan. Sequential and distributive forcings without choice. Canadian mathematical bulletin, Tome 66 (2023) no. 3, pp. 868-880. doi: 10.4153/S0008439522000753
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