In this paper we investigate some properties of forcing which can be considered “nice” in the context of singularizing regular cardinals to have an uncountable cofinality. We show that such forcing which changes cofinality of a regular cardinal, cannot be too nice and must cause some “damage” to the structure of cardinals and stationary sets. As a consequence there is no analogue to the Prikry forcing, in terms of “nice” properties, when changing cofinalities to be uncountable.

We construct a model [Formula: see text] of [Formula: see text] which lies between [Formula: see text] and [Formula: see text] for a Cohen real [Formula: see text] and does not have the form [Formula: see text] for any set [Formula: see text]. This is loosely based on the unwritten work done in a Bristol workshop about Woodin’s HOD Conjecture in 2011. The construction given here allows for a finer analysis of the needed assumptions on the ground models, thus taking (...) us one step closer to understanding models of [Formula: see text], and the HOD Conjecture and its relatives. This model also provides a positive answer to a question of Grigorieff about intermediate models of [Formula: see text], and we use it to show the failure of Kinna–Wagner Principles in [Formula: see text]. (shrink)

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. (shrink)

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.