Dependent Choice, Properness, and Generic Absoluteness

Abstract

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.

$I$-regularity, determinacy, and $\infty$-Borel sets of reals

“…Asperó and Karagila [1] modified the definition of hereditary sets H(κ) in such a way that it can be defined in ZF without using the Axiom of Choice while ensuring some basic facts on H(κ), and that it is equivalent to the standard definition of H(κ) under ZFC. Using this modified definition of H(κ), they developed the basic theory of proper forcings under ZF + DC.…”

“…Using this modified definition of H(κ), they developed the basic theory of proper forcings under ZF + DC. Definition 2.2 (Asperó and Karagila [1]).…”

“…As pointed by Asperó and Karagila [1], if we do not assume DC, the notion of properness becomes obscure (or every forcing would become proper) given that DC is equivalent to having countable elementary substrctures of V α for any infinite ordinal α.…”

“…Then every set of reals is I-regular. If we do not assume DC, the notion of properness becomes obscure as pointed out by Asperó and Karagila [1]. Using the notion of strong properness in Definition 2.3 similar to the one introduced by Bagaria and Bosch [2, Definition 5], we will prove the following theorem: Theorem 1.5.…”

“…After the above results on AD and Solovay models, many other regularity properties for sets of reals have been investigated. Ikegami [3] developed a general framework of regularity properties by introducing the notion of strongly arboreal forcings and assigning a regularity property called P-measurability to each strongly arboreal forcing P. Many of the regularity properties are equivalent to P-measurability for some P and he has shown the general equivalence among P-measurability for ∆ 1 2 -sets of reals, the generic absoluteness for Σ 1 3 -statements via P, and a transcendence property over L[x] for all reals x if P is proper and simply definable.…”