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.

We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to$\mathsf {DC}$-preserving symmetric submodels of forcing extensions. Hence,$\mathsf {ZF}+\mathsf {DC}$ not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in$\mathsf {ZF}$, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in$\mathsf {ZF}+\mathsf {DC}$ and$\mathsf {ZFC}$. Our results confirm$\mathsf {ZF} + \mathsf {DC}$ as a natural foundation for a significant portion of “classical mathematics” and provide support to the idea of this theory being also a natural foundation for a large part of set theory.

J. L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e., symmetric extensions. We also provide a new condition for preserving well ordered, and other particular type of choice, in the general settings of symmetric extensions.

The notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of $ZF$ between the ground model and the generic extension, and often the axiom of choice fails in these models. Symmetric extensions are generally used to prove choiceless consistency results. We develop a framework for iterating symmetric extensions in order to construct new models of $ZF$. We show how to obtain some well-known and lesser-known results using this framework. Specifically, we discuss Kinna–Wagner principles and obtain some results related to their failure.