Comptes Rendus
no. 6
Logique
An axiomatic approach to forcing and generic extensions
[Une approche axiomatique du forcing et des extensions génériques]
Rodrigo A. Freire1
1 Department of Philosophy, University of Brasília, Brasil
Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 757-775.

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.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.97
Rodrigo A. Freire 1

1 Department of Philosophy, University of Brasília, Brasil
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2020__358_6_757_0,
     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},
}
TY  - JOUR
AU  - Rodrigo A. Freire
TI  - An axiomatic approach to forcing and generic extensions
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 757
EP  - 775
VL  - 358
IS  - 6
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.97
LA  - en
ID  - CRMATH_2020__358_6_757_0
ER  - 
%0 Journal Article
%A Rodrigo A. Freire
%T An axiomatic approach to forcing and generic extensions
%J Comptes Rendus. Mathématique
%D 2020
%P 757-775
%V 358
%N 6
%I Académie des sciences, Paris
%R 10.5802/crmath.97
%G en
%F CRMATH_2020__358_6_757_0
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/

[1] Thomas Jech Set theory, Springer Monographs in Mathematics, Springer, 2003 | Zbl

[2] Kenneth Kunen Set theory: An introduction to independence proofs, Studies in Logic and the Foundations of Mathematics, 102, North-Holland, 1980 | Zbl

[3] Azriel Levy Definability in axiomatic set theory. I, Logic, Methodology and Philosophy of Science: Proceedings of the 1964 International Congress (Studies in Logic and the Foundations of Mathematics), North-Holland, 1964

[4] Joseph R. Shoenfield Mathematical logic, Addison-Wesley Publishing Group, 1967 | MR | Zbl

[5] Joseph R. Shoenfield Unramified forcing, Proceedings of Symposia in Pure Mathematics, 13, American Mathematical Society, 1971 | Zbl

[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

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Type-definable groups in C-minimal structures

Fares Maalouf

C. R. Math (2010)

3D Ising model: a view from the conformal bootstrap island

Slava Rychkov

C. R. Phys (2020)

The complexity to compute the Euler characteristic of complex varieties

Peter Bürgisser; Felipe Cucker; Martin Lotz

C. R. Math (2004)