L’« axiome du choix simple » est le principe selon lequel on peut choisir un élément dans tout ensemble non vide. Cet « autre axiome du choix » a une histoire paradoxale et riche, dont la première partie de cet article recherche les traces et repère les enjeux. Apparaissent comme décisifs le statut de la théorie des ensembles dans les mathématiques intuitionnistes, mais aussi la tension croissante entre technicisation de la logique et réflexion épistémologique des mathématiciens. La deuxième partie...

In set theory without the Axiom of Choice ZF, we prove that for every commutative field $𝕂$, the following statement ${𝐃}_{𝕂}$: “On every non null $𝕂$-vector space, there exists a non null linear form” implies the existence of a “$𝕂$-linear extender” on every vector subspace of a $𝕂$-vector space. This solves a question raised in Morillon M., Linear forms and axioms of choice, Comment. Math. Univ. Carolin. 50 (2009), no. 3, 421-431. In the second part of the paper, we generalize our results in the case of spherically...

We work in set-theory without choice ZF. Given a commutative field $𝕂$, we consider the statement $𝐃\left(𝕂\right)$: “On every non null $𝕂$-vector space there exists a non-null linear form.” We investigate various statements which are equivalent to $𝐃\left(𝕂\right)$ in ZF. Denoting by ${\mathbb{Z}}_2$ the two-element field, we deduce that $𝐃\left({\mathbb{Z}}_2\right)$ implies the axiom of choice for pairs. We also deduce that $𝐃\left(\mathbb{Q}\right)$ implies the axiom of choice for linearly ordered sets isomorphic with $\mathbb{Z}$.