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AN AXIOMATIC APPROACH TO FORCING IN A GENERAL SETTING

Part of: Set theory Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  04 April 2022

RODRIGO A. FREIRE  and
PETER HOLY
RODRIGO A. FREIRE
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF BRASÍLIABRASÍLIA, BRAZILE-mail: rodrigofreire@unb.br
PETER HOLY
Affiliation:
UNIVERSITY OF UDINEUDINE, ITALYE-mail: pholy@math.uni-bonn.de
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Abstract

The technique of forcing is almost ubiquitous in set theory, and it seems to be based on technicalities like the concepts of genericity, forcing names and their evaluations, and on the recursively defined forcing predicates, the definition of which is particularly intricate for the basic case of atomic first order formulas. In his [3], the first author has provided an axiomatic framework for set forcing over models of $\mathrm {ZFC}$ that is a collection of guiding principles for extensions over which one still has control from the ground model, and has shown that these axiomatics necessarily lead to the usual concepts of genericity and of forcing extensions, and also that one can infer from them the usual recursive definition of forcing predicates. In this paper, we present a more general such approach, covering both class forcing and set forcing, over various base theories, and we provide additional details regarding the formal setting that was outlined in [3].

Keywords

forcingclass forcingaxiomatic framework for forcing

MSC classification

Primary: 03E40: Other aspects of forcing and Boolean-valued models
Secondary: 03E70: Nonclassical and second-order set theories 03A05: Philosophical and critical
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 28 , Issue 3 , September 2022 , pp. 427 - 450
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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