Dependent choice, properness, and generic absoluteness

Aspero, David and Karagila, Asaf (2020) Dependent choice, properness, and generic absoluteness. The Review of Symbolic Logic. pp. 1-23. ISSN 1755-0203

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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 DC-preserving symmetric submodels of forcing extensions. Hence, ZF+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 ZF, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in ZF+DC and ZFC. Our results confirm ZF+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.

Item Type: Article
Faculty \ School: Faculty of Science > School of Mathematics
Related URLs:
Depositing User: LivePure Connector
Date Deposited: 03 Apr 2020 00:41
Last Modified: 09 Jul 2020 23:54
URI: https://ueaeprints.uea.ac.uk/id/eprint/74693
DOI: 10.1017/S1755020320000143

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