Sequential and distributive forcings without choice

TL;DR

This paper investigates the relationship between $oldsymbol{ ext{kappa}}$-distributive and $oldsymbol{ ext{kappa}}$-sequential forcing notions in set theory without the Axiom of Choice, revealing conditions under which their equivalence fails or holds.

Contribution

It demonstrates that the equivalence between $oldsymbol{ ext{kappa}}$-distributive and $oldsymbol{ ext{kappa}}$-sequential forcing notions fails without the Axiom of Choice and introduces conditions for preserving Dependent Choice.

Findings

Equivalence fails without the Axiom of Choice, even with weak forms like Dependent Choice.

A $oldsymbol{ ext{kappa}}$-distributive forcing can violate Dependent Choice but preserve Choice for families of size $oldsymbol{ ext{kappa}}$.

A $oldsymbol{ ext{kappa}}$-sequential forcing can violate Countable Choice.

Abstract

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 while 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.