Editing Ernst Zermelo (section)

==Research in set theory==

In 1900, in the Paris conference of the [[International Congress of Mathematicians]], [[David Hilbert]] challenged the mathematical community with his famous [[Hilbert's problems]], a list of 23 unsolved fundamental questions which mathematicians should attack during the coming century. The first of these, a problem of [[set theory]], was the [[continuum hypothesis]] introduced by [[Georg Cantor|Cantor]] in 1878, and in the course of its statement Hilbert also mentioned the need to prove the [[well-ordering theorem]].

Zermelo began to work on the problems of set theory under Hilbert's influence and in 1902 published his first work concerning the addition of [[Cardinal numbers|transfinite cardinals]]. By that time he had also discovered the so-called [[Russell paradox]]. In 1904, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he proved the [[well-ordering theorem]] (''every set can be well ordered''). This result brought fame to Zermelo, who was appointed Professor in Göttingen, in 1905. His proof of the [[well-ordering theorem]], based on the powerset axiom and the [[axiom of choice]], was not accepted by all mathematicians, mostly because the axiom of choice was a paradigm of non-constructive mathematics. In 1908, Zermelo succeeded in producing an improved proof making use of Dedekind's notion of the "chain" of a set, which became more widely accepted; this was mainly because that same year he also offered an [[axiomatization]] of set theory.

Zermelo began to axiomatize set theory in 1905; in 1908, he published his results despite his failure to prove the consistency of his axiomatic system. See the article on [[Zermelo set theory]] for an outline of this paper, together with the original axioms, and the original numbering.

In 1922, [[Abraham Fraenkel]] and [[Thoralf Skolem]] independently improved Zermelo's axiom system. The resulting system, now called [[Zermelo–Fraenkel set theory|Zermelo–Fraenkel axioms]] (ZF), is now the most commonly used system for [[axiomatic set theory]].