Remarques sur l'expression de la g\'en\'eralit\'e en math\'ematiques

TL;DR

This paper explores the expression of generality in mathematics by applying the Löwenheim-Skolem theorem to Zermelo's axioms and analyzing an example from Gauss's Disquisitiones Arithmeticae.

Contribution

It introduces a condition for expressing generality in mathematics based on model theory and set theory, illustrated through historical and mathematical examples.

Findings

Löwenheim-Skolem theorem applied to Zermelo's axioms yields conditions for generality.

An example from Gauss's Disquisitiones Arithmeticae illustrates the expression problem.

Sets play a crucial role in the expression of mathematical generality.

Abstract

This paper gives a condition of the expression of generality in mathematics from the application of L\"owenheim-Skolem theorem to Zermelo's axioms. It gives an example of an "expression problem" from Gauss's Disquisitiones Arithmeticae and caracterizes the used of sets in it.---L'article d\'egage une condition de l'expression de la g\'en\'eralit\'e en math\'ematiques \`a partir de l'application du th\'eor\`eme de L\"owenheim-Skolem aux axiomes de Zermelo. Il donne un exemple de "probl\`eme d'expression" \`a partir des Disquisitiones Arithmeticae de Gauss, d\'egageant ainsi une condition du recours aux ensembles.