1 results
Sequential and distributive forcings without choice
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 66 / Issue 3 / September 2023
- Published online by Cambridge University Press:
- 29 December 2022, pp. 868-880
- Print publication:
- September 2023
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- Article
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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.
