New Foundations

New Foundations

En théorie des ensembles, New Foundations (NF) est une théorie axiomatique découverte par Willard Van Orman Quine. En 1937 , il écrivit un article intitulé "New Foundations for Mathematical Logic" . En 1983, le mathématicien belge Marcel Crabbé a prouvé la consistance d'un système qu'il a appelé NFI .

Sommaire

Les prédicats et la hiérarchie d'appartenance

Les prédicats utilisés sont l'égalité et l'appartenance. La hiérarchie d'appartenance définit :

  • pour chaque nombre naturel n, les objets de type 'n+1 sont des ensembles d'éléments n
  • les ensembles de type n ont des éléments de type n-1.
  • Les objets reliés entre eux par une relation d'équivalence (prédicat d'identité) doivent être du même type.

x^{n} = y^{n}\! and x^{n} \in y^{n+1}

Les axiomes

Si \phi(x^n)\! est une formule , alors l'ensemble \{x^n \mid \phi(x^n)\}^{n+1}\! existe.
Dit autrement , étant donnée une formule \phi(x^n)\!, la formule \exists A^{n+1} \forall x^n [x^n \in A^{n+1} \leftrightarrow \phi(x^n)] est un axiome où A^{n+1}\! represente l'ensemble \{x^n \mid \phi(x^n)\}^{n+1}\!.


Références

Bibliographie

  • Crabbé, Marcel, 1982, On the consistency of an impredicative fragment of Quine's NF, The Journal of Symbolic Logic 47: 131-136.
  • Holmes, Randall, 1998. Elementary Set Theory with a Universal Set. Academia-Bruylant. The publisher has graciously consented to permit diffusion of this introduction to NFU via the web. Copyright is reserved.
  • Jensen, R. B., 1969, "On the Consistency of a Slight(?) Modification of Quine's NF," Synthese 19: 250-63. With discussion by Quine.
  • Quine, W. V., 1980, "New Foundations for Mathematical Logic" in From a Logical Point of View, 2nd ed., revised. Harvard Univ. Press: 80-101. The definitive version of where it all began, namely Quine's 1937 paper in the American Mathematical Monthly.

Lien interne

Liens externes


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