Zermelo–Fraenkel set theory

In <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, Zermelo–Fraenkel set theory, named after mathematicians <a href="/facts/Ernst_Zermelo/uvW8gDq0">Ernst Zermelo</a> and <a href="/facts/Abraham_Fraenkel/Hc8IBfmY">Abraham Fraenkel</a>, is an <a href="/facts/Axiomatic_system/dwMz70d5">axiomatic system</a> that was proposed in the early twentieth century in order to formulate a <a href="/facts/Theory_of_sets/1ptfsvUP">theory of sets</a> free of paradoxes such as <a href="/facts/Russell%27s_paradox/XupqWGYt">Russell's paradox</a>. Today, Zermelo–Fraenkel set theory, with the historically controversial <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a> (AC) included, is the standard form of <a href="/facts/Axiomatic_set_theory/1ptfsvUP">axiomatic set theory</a> and as such is the most common <a href="/facts/Foundations_of_mathematics/yjpbbUkt">foundation of mathematics</a>. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.
Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a <a href="/facts/Hereditary_set/edFwmXa8">hereditary</a> <a href="/facts/Well-founded/zyRRyJ4M">well-founded</a> <a href="/facts/Set_(mathematics)/BfucfHMq">set</a>, so that all entities in the <a href="/facts/Universe_of_discourse/T1WRQxx6">universe of discourse</a> are such sets. Thus the <a href="/facts/Axiom/mLjP14Mc">axioms</a> of Zermelo–Fraenkel set theory refer only to <a href="/facts/Pure_set/edFwmXa8">pure sets</a> and prevent its <a href="/facts/Model_theory/vLWVV8sQ">models</a> from containing <a href="/facts/Urelement/YLbuKp37">urelements</a> (elements that are not themselves sets). Furthermore, <a href="/facts/Proper_class/F5EmDd9b">proper classes</a> (collections of <a href="/facts/Mathematical_object/Juhyow9M">mathematical objects</a> defined by a property shared by their members where the collections are too big to be sets) can only be treated indirectly. Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a <a href="/facts/Universal_set/QLFgSfox">universal set</a> (a set containing all sets) nor for <a href="/facts/Unrestricted_comprehension/FVgjEBfT">unrestricted comprehension</a>, thereby avoiding Russell's paradox. <a href="/facts/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory/5wLhAIMz">Von Neumann–Bernays–Gödel set theory</a> (NBG) is a commonly used <a href="/facts/Conservative_extension/bqjq9Q2X">conservative extension</a> of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes.
There are many equivalent formulations of the axioms of Zermelo–Fraenkel set theory. Most of the axioms state the existence of particular sets defined from other sets. For example, the <a href="/facts/Axiom_of_pairing/yva0UafC">axiom of pairing</a> says that given any two sets 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 b
 
 
 {\displaystyle b}
 
 there is a new set 
 
 
 
 {
 a
 ,
 b
 }
 
 
 {\displaystyle \{a,b\}}
 
 containing exactly 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 b
 
 
 {\displaystyle b}
 
. Other axioms describe properties of set membership. A goal of the axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the <a href="/facts/Von_Neumann_universe/EhkJAxXc">von Neumann universe</a> (also known as the cumulative hierarchy).
The <a href="/facts/Metamathematics/yeIm8vqC">metamathematics</a> of Zermelo–Fraenkel set theory has been extensively studied. Landmark results in this area established the <a href="/facts/Independence_(mathematical_logic)/xW9tCrGB">logical independence</a> of the axiom of choice from the remaining Zermelo-Fraenkel axioms and of the <a href="/facts/Continuum_hypothesis/R0yGqHND">continuum hypothesis</a> from ZFC. The <a href="/facts/Consistency/IHkdnA9M">consistency</a> of a theory such as ZFC cannot be proved within the theory itself, as shown by <a href="/facts/G%C3%B6del%27s_second_incompleteness_theorem/du8PRu8g">Gödel's second incompleteness theorem</a>.

Zermelo–Fraenkel set theory open-in-new

Zermelo–Fraenkel set theory