An axiomatic approach to forcing and generic extensions

Rodrigo A. Freire

doi:10.5802/crmath.97

Logic

An axiomatic approach to forcing and generic extensions

Rodrigo A. Freire ¹

¹ Department of Philosophy, University of Brasília, Brasil

Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 757-775.

Abstract
Résumé

This paper provides a conceptual analysis of forcing and generic extensions. Our goal is to give general axioms for the concept of standard forcing-generic extension and to show that the usual (poset) constructions are unified and explained as realizations of this concept. According to our approach, the basic idea behind forcing and generic extensions is that the latter are uniform adjunctions which are ground-controlled by forcing, and forcing is nothing more than that ground-control. As a result of our axiomatization of this idea, the usual definitions of forcing and genericity are derived.

Cet article présente une analyse conceptuelle du forcing et des extensions génériques. Notre objectif est de donner des axiomes généraux pour le concept d’extension forcing-générique standard, et de montrer que les constructions habituelles sont unifiées et expliquées comme étant des réalisations de ce concept. Selon notre approche, l’idée-clé sous-tendant le forcing et les extensions génériques est que ces dernières sont des adjonctions uniformes qui sont contrôlées par le forcing, ainsi le forcing n’est rien de plus que ce contrôle. Comme conséquence de notre axiomatisation de cette idée, on dérive les définitions habituelles du forcing et de la généricité.

Received: 2020-04-10
Revised: 2020-07-17
Accepted: 2020-07-17
Published online: 2020-10-02

DOI: 10.5802/crmath.97

Author's affiliations:

Rodrigo A. Freire ¹

¹ Department of Philosophy, University of Brasília, Brasil

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Rodrigo A. Freire},
     title = {An axiomatic approach to forcing and generic extensions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {757--775},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {6},
     year = {2020},
     doi = {10.5802/crmath.97},
     language = {en},
}

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Rodrigo A. Freire. An axiomatic approach to forcing and generic extensions. Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 757-775. doi : 10.5802/crmath.97. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.97/

References
Cited by

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[6] Matteo Viale The proper forcing axiom and the singular cardinal hypothesis, J. Symb. Log., Volume 71 (2006) no. 2, pp. 473-479 | DOI | MR | Zbl

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