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DEPENDENT CHOICE, PROPERNESS, AND GENERIC ABSOLUTENESS

Part of: Set theory Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  02 July 2020

DAVID ASPERÓ  and
ASAF KARAGILA
DAVID ASPERÓ
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF EAST ANGLIA NORWICH NR4 7TJ, UKE-mail: d.aspero@uea.ac.ukE-mail: karagila@math.huji.ac.il
ASAF KARAGILA
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF EAST ANGLIA NORWICH NR4 7TJ, UKE-mail: d.aspero@uea.ac.ukE-mail: karagila@math.huji.ac.il
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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.

Keywords

Axiom of Choiceforcing axiomsproper forcingPFAgeneric absolutenessthe Chang model

MSC classification

Primary: 03E25: Axiom of choice and related propositions
Secondary: 03E55: Large cardinals 03E35: Consistency and independence results 03E57: Generic absoluteness and forcing axioms 03A05: Philosophical and critical
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 1 , March 2021 , pp. 225 - 249
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Copyright
© Association for Symbolic Logic, 2020

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