Editing Mathematical logic (section)

== Set theory ==
{{Main|Set theory}}
'''[[Set theory]]''' is the study of [[Set (mathematics)|sets]], which are abstract collections of objects. Many of the basic notions, such as ordinal and cardinal numbers, were developed informally by Cantor before formal axiomatizations of set theory were developed. The [[Zermelo set theory|first such axiomatization]], due to Zermelo,{{sfnp|Zermelo|1908b}} was extended slightly to become [[Zermelo–Fraenkel set theory]] (ZF), which is now the most widely used foundational theory for mathematics.

Other formalizations of set theory have been proposed, including [[von Neumann–Bernays–Gödel set theory]] (NBG), [[Morse–Kelley set theory]] (MK), and [[New Foundations]] (NF).  Of these, ZF, NBG, and MK are similar in describing a [[cumulative hierarchy]] of sets. New Foundations takes a different approach; it allows objects such as the set of all sets at the cost of restrictions on its set-existence axioms. The system of [[Kripke–Platek set theory]] is closely related to generalized recursion theory.

Two famous statements in set theory are the [[axiom of choice]] and the [[continuum hypothesis]]. The axiom of choice, first stated by Zermelo,{{sfnp|Zermelo|1904}} was proved independent of ZF by Fraenkel,{{sfnp|Fraenkel|1922}} but has come to be widely accepted by mathematicians.  It states that given a collection of nonempty sets there is a single set ''C'' that contains exactly one element from each set in the collection. The set ''C'' is said to "choose" one element from each set in the collection. While the ability to make such a choice is considered obvious by some, since each set in the collection is nonempty, the lack of a general, concrete rule by which the choice can be made renders the axiom nonconstructive. [[Stefan Banach]] and [[Alfred Tarski]] showed that the axiom of choice can be used to decompose a solid ball into a finite number of pieces which can then be rearranged, with no scaling, to make two solid balls of the original size.{{sfnp|Banach|Tarski|1924}} This theorem, known as the [[Banach–Tarski paradox]], is one of many counterintuitive results of the axiom of choice.

The continuum hypothesis, first proposed as a conjecture by Cantor, was listed by [[David Hilbert]] as one of his 23 problems in 1900. Gödel showed that the continuum hypothesis cannot be disproven from the axioms of Zermelo–Fraenkel set theory (with or without the axiom of choice), by developing the [[constructible universe]] of set theory in which the continuum hypothesis must hold. In 1963, [[Paul Cohen]] showed that the continuum hypothesis cannot be proven from the axioms of Zermelo–Fraenkel set theory.{{sfnp|Cohen|1966}} This independence result did not completely settle Hilbert's question, however, as it is possible that new axioms for set theory could resolve the hypothesis. Recent work along these lines has been conducted by [[W. Hugh Woodin]], although its importance is not yet clear.{{sfnp|Woodin|2001}}

Contemporary research in set theory includes the study of [[large cardinal]]s and [[determinacy]].  Large cardinals are [[cardinal numbers]] with particular properties so strong that the existence of such cardinals cannot be proved in ZFC. The existence of the smallest large cardinal typically studied, an [[inaccessible cardinal]], already implies the consistency of ZFC.  Despite the fact that large cardinals have extremely high [[cardinality]], their existence has many ramifications for the structure of the real line.  ''Determinacy'' refers to the possible existence of winning strategies for certain two-player games (the games are said to be ''determined''). The existence of these strategies implies structural properties of the real line and other [[Polish space]]s.