ZF cento anni fa

Lolli, Gabriele

Matematica, cultura e società, Série 1, Tome 7 (2022) no. 1, pp. 5-33

Résumé

One hundred years ago the search for an axiom system for the new theory of sets was completed, almost fifty years after the first Cantor's discovery in 1874 of the existence of two different actual infinities, the numerabile and the continuum. To Zermelo's 1908 axioms, in 1922, Thoralf Skolem and Abraham Fraenkel independently added the axiom of replacement, that guarantees a sufficient extension for the universe of sets; on this basis the theory could purport in the twentieth century to be a framework of all of known mathematics. It became clear after the fact that the difficulty to conceive the new axiom depended on the necessity of expressing in the set theoretic language a concept of application that was more general than that of a function defined as a set of ordered pairs. While logicians as Skolem and John von Neumann were quick to see an answer, Fraenkel had to be helped by the latter. But the upshot suggests that pure set theory could be not enough to grasp the concept of function.