Editing Mathematical logic (section)

==== Set theory and paradoxes ====
[[Ernst Zermelo]] gave a proof that [[Well-ordering theorem|every set could be well-ordered]], a result [[Georg Cantor]] had been unable to obtain.{{sfnp|Zermelo|1904}} To achieve the proof, Zermelo introduced the [[axiom of choice]], which drew heated debate and research among mathematicians and the pioneers of set theory. The immediate criticism of the method led Zermelo to publish a second exposition of his result, directly addressing criticisms of his proof.{{sfnp|Zermelo|1908a}} This paper led to the general acceptance of the axiom of choice in the mathematics community.

Skepticism about the axiom of choice was reinforced by recently discovered paradoxes in [[naive set theory]]. [[Cesare Burali-Forti]]{{sfnp|Burali-Forti|1897}} was the first to state a paradox: the [[Burali-Forti paradox]] shows that the collection of all [[ordinal number]]s cannot form a set. Very soon thereafter, [[Bertrand Russell]] discovered [[Russell's paradox]] in 1901, and [[Jules Richard (mathematician)|Jules Richard]]  discovered [[Richard's paradox]].{{sfnp|Richard|1905}}

Zermelo provided the first set of axioms for set theory.{{sfnp|Zermelo|1908b}} These axioms, together with the additional [[axiom of replacement]] proposed by [[Abraham Fraenkel]], are now called [[Zermelo–Fraenkel set theory]] (ZF). Zermelo's axioms incorporated the principle of [[limitation of size]] to avoid Russell's paradox.

In 1910, the first volume of ''[[Principia Mathematica]]'' by Russell and [[Alfred North Whitehead]] was published. This seminal work developed the theory of functions and cardinality in a completely formal framework of [[type theory]], which Russell and Whitehead developed in an effort to avoid the paradoxes. ''Principia Mathematica'' is considered one of the most influential works of the 20th century, although the framework of type theory did not prove popular as a foundational theory for mathematics.{{sfnp|Ferreirós|2001|p=445}}

Fraenkel{{sfnp|Fraenkel|1922}} proved that the axiom of choice cannot be proved from the axioms of Zermelo's set theory with [[urelements]]. Later work by [[Paul Cohen]]{{sfnp|Cohen|1966}} showed that the addition of urelements is not needed, and the axiom of choice is unprovable in ZF. Cohen's proof developed the method of [[forcing (mathematics)|forcing]], which is now an important tool for establishing [[independence result]]s in set theory.<ref>See also {{harvnb|Cohen|2008}}.</ref>