Editing Foundations of mathematics (section)

===Infinite sets===
Before the second half of the 19th century, [[infinity]] was a philosophical concept that did not belong to mathematics. However, with the rise of [[infinitesimal calculus]], mathematicians became accustomed  to infinity, mainly through [[potential infinity]], that is, as the result of an endless process, such as the definition of an [[infinite sequence]], an [[infinite series]] or a [[limit (mathematics)|limit]]. The possibility of an [[actual infinity]] was the subject of many philosophical disputes.

[[Set (mathematics)|Set]]s, and more specially [[infinite set]]s were not considered as a mathematical concept; in particular, there was no fixed term for them. A dramatic change arose with the work of [[Georg Cantor]] who was the first mathematician to systematically study infinite sets. In particular, he introduced [[cardinal number]]s that measure the size of infinite sets, and [[ordinal number]]s that, roughly speaking, allow one to continue to count after having reach infinity. One of his major results is the discovery that there are strictly more real numbers than natural numbers (the cardinal of the [[continuum (set theory)|continuum]] of the real numbers is greater than that of the natural numbers).

These results were rejected by many mathematicians and philosophers, and led to debates that are a part of the [[#Foundational crisis of mathematics|foundational crisis of mathematics]].

The crisis was amplified with the [[Russel's paradox]] that asserts that the phrase "the set of all sets" is self-contradictory. This condradiction introduced a doubt on the [[consistency]] of all mathematics.

With the introduction of the [[Zermelo–Fraenkel set theory]] ({{circa|1925}}) and its adoption by the mathematical community, the doubt about the consistency was essentially removed, although consistency of set theory cannot be proved because of [[Gödel's incompleteness theorem]].