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| {{redirects|ZFC}}
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| In mathematics, '''Zermelo–Fraenkel set theory with the [[axiom of choice]]''', named after mathematicians [[Ernst Zermelo]] and [[Abraham Fraenkel]] and commonly abbreviated '''ZFC''', is one of several [[axiomatic system]]s that were proposed in the early twentieth century to formulate a [[theory of sets]] without the paradoxes of [[naive set theory]] such as [[Russell's paradox]]. Specifically, ZFC does not allow [[unrestricted comprehension]]. Today ZFC is the standard form of [[axiomatic set theory]] and as such is the most common [[foundations of mathematics|foundation of mathematics]].
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| ZFC is intended to formalize a single primitive notion, that of a [[hereditary set|hereditary]] [[well-founded]] [[Set (mathematics)|set]], so that all [[entities]] in the [[universe of discourse]] are such sets. Thus the axioms of ZFC refer only to sets, not to [[urelement]]s (elements of sets which are not themselves sets) or [[class (set theory)|class]]es (collections of mathematical objects defined by a property shared by their members). The axioms of ZFC prevent its [[Model theory|model]]s from containing urelements, and [[proper class]]es can only be treated indirectly.
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| Formally, ZFC is a [[First-order logic#Many-sorted logic|one-sorted theory]] in [[first-order logic]]. The [[signature (mathematical logic)|signature]] has equality and a single primitive [[binary relation]], [[set membership]], which is usually denoted ∈. The formula ''a'' ∈ ''b'' means that the set ''a'' is a member of the set ''b'' (which is also read, "''a'' is an element of ''b''" or "''a'' is in ''b''").
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| There are many equivalent formulations of the ZFC [[axiom]]s. Most of the ZFC axioms state the existence of particular sets defined from other sets. For example, the [[axiom of pairing]] says that given any two sets ''a'' and ''b'' there is a new set {''a'', ''b''} containing exactly ''a'' and ''b''. Other axioms describe properties of set membership. A goal of the ZFC axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the [[von Neumann universe]] (also known as the cumulative hierarchy).
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| The [[metamathematics]] of ZFC has been extensively studied. Landmark results in this area established the independence of the [[continuum hypothesis]] from ZFC, and of the [[axiom of choice]] from the remaining ZFC axioms.
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| ==History==
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| In 1908, [[Ernst Zermelo]] proposed the first [[axiomatic set theory]], [[Zermelo set theory]]. However, as first pointed out by [[Abraham Fraenkel]] in a 1921 letter to Zermelo, this theory was incapable of proving the existence of certain sets and [[cardinal number]]s whose existence was taken for granted by most set theorists of the time, notably, the cardinal number ℵ<sub>ω</sub> and, where Z<sub>0</sub> is any infinite set and ℘ is the power set operation, the set {Z<sub>0</sub>, ℘(Z<sub>0</sub>), ℘(℘(Z<sub>0</sub>)),...} (Ebbinghaus 2007, p. 136). Moreover, one of Zermelo's axioms invoked a concept, that of a "definite" property, whose operational meaning was not clear. In 1922, Fraenkel and [[Thoralf Skolem]] independently proposed operationalizing a "definite" property as one that could be formulated as a [[first-order logic|first order theory]] whose [[atomic formula]]s were limited to set membership and identity. They also independently proposed replacing the [[axiom schema of specification]] with the [[axiom schema of replacement]]. Appending this schema, as well as the [[axiom of regularity]] (first proposed by [[Dimitry Mirimanoff]] in 1917), to Zermelo set theory yields the theory denoted by '''ZF'''. Adding to ZF either the [[axiom of choice]] (AC) or a statement that is equivalent to it yields ZFC.
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| == The axioms ==
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| There are many equivalent formulations of the ZFC axioms; for a rich but somewhat dated discussion of this fact, see Fraenkel ''et al.'' (1973). The following particular axiom set is from Kunen (1980). The axioms per se are expressed in the symbolism of [[first order logic]]. The associated English prose is only intended to aid the intuition.
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| <!-- **NOTE TO EDITORS** The symbolic axioms are taken from Kunen, and should not be changed without discussion on the talk page. Making up a new set of axioms is original research. The English descriptions are new here and can be freely changed. The choice of Kunen instead of Jech or some other author was arbitrary, and is open for discussion on the talk page. -->
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| All formulations of ZFC imply that at least one set exists. Kunen includes an axiom that directly asserts the existence of a set, in addition to the axioms given below (although he notes that he does so only “for emphasis” (''ibid.'', p. 10)). Its omission here can be justified in two ways. First, in the standard semantics of first-order logic in which ZFC is typically formalized, the [[domain of discourse]] must be nonempty. Hence, it is a logical theorem of first-order logic that something exists — usually expressed as the assertion that something is identical to itself, ∃x(x=x). Consequently, it is a theorem of every first-order theory that something exists. However, as noted above, because in the intended semantics of ZFC there are only sets, the interpretation of this logical theorem in the context of ZFC is that some ''set'' exists. Hence, there is no need for a separate axiom asserting that a set exists. Second, however, even if ZFC is formulated in so-called [[free logic]], in which it is not provable from logic alone that something exists, the axiom of infinity (below) asserts that an ''infinite'' set exists. This obviously implies that ''a'' set exists and so, once again, it is superfluous to include an axiom asserting as much.
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| === 1. Axiom of extensionality ===
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| {{Main| Axiom of extensionality}}
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| Two sets are equal (are the same set) if they have the same elements.
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| :<math>\forall x \forall y [ \forall z (z \in x \Leftrightarrow z \in y) \Rightarrow x = y].</math>
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| The converse of this axiom follows from the substitution property of [[equality (mathematics)|equality]]. If the background logic does not include equality "=", ''x''=''y'' may be defined as an abbreviation for the following formula (Hatcher 1982, p. 138, def. 1):
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| :<math>\forall z [ z \in x \Leftrightarrow z \in y] \land \forall w [x \in w \Leftrightarrow y \in w].</math>
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| In this case, the axiom of extensionality can be reformulated as
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| :<math>\forall x \forall y [ \forall z (z \in x \Leftrightarrow z \in y) \Rightarrow \forall w (x \in w \Leftrightarrow y \in w) ],</math>
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| which says that if ''x'' and ''y'' have the same elements, then they belong to the same sets (Fraenkel ''et al.'' 1973).
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| === 2. Axiom of regularity (also called the ''Axiom of foundation'') ===
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| {{Main|Axiom of regularity}}
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| Every non-empty set ''x'' contains a member ''y'' such that ''x'' and ''y'' are [[disjoint sets]].
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| :<math>\forall x [ \exists a ( a \in x) \Rightarrow \exists y ( y \in x \land \lnot \exists z (z \in y \land z \in x))].</math>
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| This implies, for example, that every set has an [[ordinal number|ordinal]] [[rank (set theory)|rank]].
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| ===3. Axiom schema of specification (also called the axiom schema of ''separation'' or of ''restricted comprehension'')===
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| {{Main|Axiom schema of specification}}
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| Subsets are commonly constructed using [[set builder notation]]. For example, the even integers can be constructed as the subset of the integers <math>\mathbb{Z}</math> satisfying the predicate <math>x \equiv 0 \pmod 2</math>:
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| :<math>\{x \in \mathbb{Z} : x \equiv 0 \pmod 2\}.</math>
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| In general, the subset of a set ''z'' obeying a formula <math>\phi</math>(''x'') with one free variable ''x'' may be written as:
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| :<math>\{x \in z : \phi(x)\}.</math>
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| The axiom schema of specification states that this subset always exists (it is an [[axiom schema|axiom ''schema'']] because there is one axiom for each <math>\phi</math>). Formally, let <math>\phi\!</math> be any formula in the language of ZFC with all free variables among <math>x,z,w_1,\ldots,w_n\!</math> (''y'' is ''not'' free in <math>\phi\!</math>). Then:
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| :<math>\forall z \forall w_1 \forall w_2\ldots \forall w_n \exists y \forall x [x \in y \Leftrightarrow ( x \in z \land \phi )].</math>
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| Note that the axiom schema of specification can only construct subsets, and does not allow the construction of sets of the more general form:
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| :<math>\{x : \phi(x)\}.</math>
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| This restriction is necessary to avoid [[Russell's paradox]] and its variants.
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| In some other axiomatizations of ZF, this axiom is redundant in that it follows from the [[axiom schema of replacement]].
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| The axiom of specification can be used to prove the existence of the [[empty set]], denoted <math>\varnothing</math>, once at least one set is known to exist (see above). One way to do this is to use a property <math>\phi\!</math> which no set has. For example, if ''w'' is any existing set, the empty set can be constructed as
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| :<math>\varnothing = \{u \in w \mid (u \in u) \land \lnot (u \in u) \}</math>.
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| Thus the [[axiom of the empty set]] is implied by the nine axioms presented here. The axiom of extensionality implies the empty set is unique (does not depend on ''w''). It is common to make a [[definitional extension]] that adds the symbol <math>\varnothing</math> to the language of ZFC.
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| ===4. Axiom of pairing===
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| {{Main|Axiom of pairing}}
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| If ''x'' and ''y'' are sets, then there exists a set which contains ''x'' and ''y'' as elements.
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| :<math>\forall x \forall y \exist z (x \in z \land y \in z).</math>
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| The axiom schema of specification must be used to reduce this to a set with exactly these two elements. This axiom is part of Z, but is redundant in ZF because it follows from the axiom schema of replacement, if we are given a set with at least two elements. The existence of a set with at least two elements is assured by either the [[axiom of infinity]], or by the axiom schema of specification and the [[axiom of the power set]] applied twice to any set.
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| === 5. Axiom of union ===
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| {{Main|Axiom of union}}
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| The union over the elements of a set exists. For example, the union over the elements of the set <math>\{\{1,2\},\{2,3\}\}</math> is <math>\{1,2,3\}</math>.
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| Formally, for any set <math>\mathcal{F}</math> there is a set ''A'' containing every element that is a member of some member of <math>\mathcal{F}</math>:
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| :<math>\forall \mathcal{F} \,\exists A \, \forall Y\, \forall x [(x \in Y \land Y \in \mathcal{F}) \Rightarrow x \in A].</math>
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| [[File:Codomain2_A_B.SVG|thumb|Axiom schema of replacement: the image of the domain set ''A'' under the definable function ''f'' (i.e. the range of ''f'') falls inside a set ''B''.]]
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| === 6. Axiom schema of replacement ===
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| {{Main|Axiom schema of replacement}}
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| The axiom schema of replacement asserts that the [[image (mathematics)|image]] of a set under any definable function will also fall inside a set.
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| Formally, let <math>\phi \!</math> be any [[Well-formed formula|formula]] in the language of ZFC whose [[free variable]]s are among <math>x,y,A,w_1,\ldots,w_n \!</math>, so that in particular <math>B</math> is not free in <math>\phi \!</math>. Then:
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| :<math>\forall A\forall w_1 \forall w_2\ldots \forall w_n \bigl[ \forall x ( x\in A \Rightarrow \exists! y\,\phi ) \Rightarrow \exists B \ \forall x \bigl(x\in A \Rightarrow \exists y (y\in B \land \phi)\bigr)\bigr].</math>
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| In other words, if the relation <math>\phi \!</math> represents a definable function ''f'', <math>A</math> represents its [[domain of a function|domain]], and ''f''(''x'') is a set for every ''x'' in that domain, then the [[range (mathematics)|range]] of ''f'' is a subset of some set <math>B</math>. The form stated here, in which <math>B</math> may be larger than strictly necessary, is sometimes called the [[Axiom schema of replacement#Axiom schema of collection|axiom schema of collection]].
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| === 7. Axiom of infinity ===
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| {{Main|Axiom of infinity}}
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| Let <math>S(w)\!</math> abbreviate <math> w \cup \{w\} \!</math>, where <math> w \!</math> is some set (We can see that <math>\{w\}</math> is a valid set by applying the Axiom of Pairing with <math> x=y=w \!</math> so that the set <math>z\!</math> is <math>\{w\} \!</math>). Then there exists a set ''X'' such that the empty set <math>\varnothing</math> is a member of ''X'' and, whenever a set ''y'' is a member of ''X'', then <math>S(y)\!</math> is also a member of ''X''.
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| :<math>\exist X \left [\varnothing \in X \and \forall y (y \in X \Rightarrow S(y) \in X)\right ].</math>
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| More colloquially, there exists a set ''X'' having infinitely many members. The minimal set ''X'' satisfying the axiom of infinity is the [[von Neumann ordinal]] ω, which can also be thought of as the set of [[natural numbers]] <math>\mathbb{N}</math>.
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| === 8. Axiom of power set ===
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| {{Main|Axiom of power set}}
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| By definition a set ''z'' is a [[subset]] of a set ''x'' if and only if every element of ''z'' is also an element of ''x'':
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| :<math>(z \subseteq x) \Leftrightarrow ( \forall q (q \in z \Rightarrow q \in x)).</math>
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| The Axiom of Power Set states that for any set ''x'', there is a set ''y'' that contains every subset of ''x'':
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| :<math>\forall x \exists y \forall z [z \subseteq x \Rightarrow z \in y].</math>
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| The axiom schema of specification is then used to define the [[power set]] ''P(x)'' as the subset of such a ''y'' containing the subsets of ''x'' exactly:
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| :<math>P(x) = \{ z \in y: z \subseteq x \}</math>
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| Axioms '''1–8''' define ZF. Alternative forms of these axioms are often encountered, some of which are listed in Jech (2003). Some ZF axiomatizations include an axiom asserting that the [[axiom of empty set|empty set exists]]. The axioms of pairing, union, replacement, and power set are often stated so that the members of the set ''x'' whose existence is being asserted are just those sets which the axiom asserts ''x'' must contain.
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| The following axiom is added to turn ZF into ZFC:
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| === 9. Well-ordering theorem ===
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| {{Main|Well-ordering theorem}}
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| For any set ''X'', there is a [[binary relation]] ''R'' which [[wellordering|well-orders]] ''X''. This means ''R'' is a [[linear order]] on ''X'' such that every nonempty [[subset]] of ''X'' has a member which is minimal under ''R''.
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| :<math>\forall X \exists R ( R \;\mbox{well-orders}\; X).</math>
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| Given axioms '''1–8''', there are many statements {{not a typo|provably}} equivalent to axiom '''9''', the best known of which is the [[axiom of choice]] (AC), which goes as follows. Let ''X'' be a set whose members are all non-empty. Then there exists a function ''f'' from ''X'' to the union of the members of ''X'', called a "[[choice function]]", such that for all {{math|''Y'' ∈ ''X''}} one has {{math|''f''(''Y'') ∈ ''Y''}}. Since the existence of a choice function when ''X'' is a [[finite set]] is easily proved from axioms '''1–8''', AC only matters for certain [[infinite set]]s. AC is characterized as [[constructive mathematics|nonconstructive]] because it asserts the existence of a choice set but says nothing about how the choice set is to be "constructed." Much research has sought to characterize the definability (or lack thereof) of certain sets whose existence AC asserts.
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| == Motivation via the cumulative hierarchy ==
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| One motivation for the ZFC axioms is the [[cumulative hierarchy]] of sets introduced by [[John von Neumann]] (Shoenfield 1977, sec. 2). In this viewpoint, the universe of set theory is built up in stages, with one stage for each [[ordinal number]]. At stage 0 there are no sets yet. At each following stage, a set is added to the universe if all of its elements have been added at previous stages. Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2; see Hinman (2005, p. 467). The collection of all sets that are obtained in this way, over all the stages, is known as V. The sets in V can be arranged into a hierarchy by assigning to each set the first stage at which that set was added to V.
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| It is provable that a set is in V if and only if the set is [[pure set|pure]] and [[well-founded set|well-founded]]; and provable that V satisfies all the axioms of ZFC, if the class of ordinals has appropriate reflection properties. For example, suppose that a set ''x'' is added at stage α, which means that every element of ''x'' was added at a stage earlier than α. Then every subset of ''x'' is also added at stage α, because all elements of any subset of ''x'' were also added before stage α. This means that any subset of ''x'' which the axiom of separation can construct is added at stage α, and that the powerset of ''x'' will be added at the next stage after α. For a complete argument that V satisfies ZFC see Shoenfield (1977).
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| The picture of the universe of sets stratified into the cumulative hierarchy is characteristic of ZFC and related axiomatic set theories such as [[Von Neumann–Bernays–Gödel set theory]] (often called NBG) and [[Morse–Kelley set theory]]. The cumulative hierarchy is not compatible with other set theories such as [[New Foundations]].
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| It is possible to change the definition of ''V'' so that at each stage, instead of adding all the subsets of the union of the previous stages, subsets are only added if they are definable in a certain sense. This results in a more "narrow" hierarchy which gives the [[constructible universe]] ''L'', which also satisfies all the axioms of ZFC, including the axiom of choice. It is independent from the ZFC axioms whether ''V'' = ''L''. Although the structure of ''L'' is more regular and well behaved than that of ''V'', few mathematicians argue that ''V'' = ''L'' should be added to ZFC as an additional axiom.
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| ==Metamathematics==
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| The axiom schemata of replacement and separation each contain infinitely many instances. [[Richard Montague|Montague]] (1961) included a result first proved in his 1957 Ph.D. thesis: if ZFC is consistent, it is impossible to axiomatize ZFC using only finitely many axioms. On the other hand, [[Von Neumann–Bernays–Gödel set theory]] (NBG) can be finitely axiomatized. The ontology of NBG includes [[class (set theory)|proper classes]] as well as sets; a set is any class that can be a member of another class. NBG and ZFC are equivalent set theories in the sense that any [[theorem]] not mentioning classes and provable in one theory can be proved in the other.
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| [[Gödel's second incompleteness theorem]] says that a recursively axiomatizable system that can interpret [[Robinson arithmetic]] can prove its own consistency only if it is inconsistent. Moreover, Robinson arithmetic can be interpreted in [[general set theory]], a small fragment of ZFC. Hence the [[consistency proof|consistency]] of ZFC cannot be proved within ZFC itself (unless it is actually inconsistent). Thus, to the extent that ZFC is identified with ordinary mathematics, the consistency of ZFC cannot be demonstrated in ordinary mathematics. The consistency of ZFC does follow from the existence of a weakly [[inaccessible cardinal]], which is unprovable in ZFC if ZFC is consistent. Nevertheless, it is deemed unlikely that ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC were inconsistent, that fact would have been uncovered by now. This much is certain — ZFC is immune to the classic paradoxes of [[naive set theory]]: [[Russell's paradox]], the [[Burali-Forti paradox]], and [[Cantor's paradox]].
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| Abian and LaMacchia (1978) studied a [[subtheory]] of ZFC consisting of the axioms of extensionality, union, powerset, replacement, and choice. Using [[model theory|models]], they proved this subtheory consistent, and proved that each of the axioms of extensionality, replacement, and power set is independent of the four remaining axioms of this subtheory. If this subtheory is augmented with the axiom of infinity, each of the axioms of union, choice, and infinity is independent of the five remaining axioms. Because there are [[non-well-founded set theory|non-well-founded]] models that satisfy each axiom of ZFC except the axiom of regularity, that axiom is independent of the other ZFC axioms.
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| If consistent, ZFC cannot prove the existence of the [[inaccessible cardinal]]s that [[category theory]] requires. Huge sets of this nature are possible if ZF is augmented with [[Tarski–Grothendieck set theory|Tarski's axiom]] (Tarski 1939). Assuming that axiom turns the axioms of [[axiom of infinity|infinity]], [[axiom of power set|power set]], and [[axiom of choice|choice]] ('''7''' − '''9''' above) into theorems.
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| ===Independence in ZFC===
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| Many important statements are [[Logical independence|independent]] of ZFC (see [[list of statements undecidable in ZFC]]). The independence is usually proved by [[forcing (mathematics)|forcing]], whereby it is shown that every countable transitive [[model theory|model]] of ZFC (sometimes augmented with [[large cardinal axiom]]s) can be expanded to satisfy the statement in question. A different expansion is then shown to satisfy the negation of the statement. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can be proven to hold in particular [[inner model]]s, such as in the [[constructible universe]]. However, some statements that are true about constructible sets are not consistent with hypothesized large cardinal axioms.
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| Forcing proves that the following statements are independent of ZFC:
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| *[[Continuum hypothesis]]
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| *[[Diamondsuit|Diamond principle]]
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| *[[Suslin's problem|Suslin hypothesis]]
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| *[[Martin's axiom]] (which is not a ZFC axiom)
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| *[[Axiom of constructibility|Axiom of Constructibility (V=L)]] (which is also not a ZFC axiom).
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| Remarks:
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| *The consistency of V=L is provable by [[inner model]]s but not forcing: every model of ZF can be trimmed to become a model of ZFC + V=L.
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| *The Diamond Principle implies the Continuum Hypothesis and the negation of the Suslin Hypothesis.
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| *Martin's axiom plus the negation of the Continuum Hypothesis implies the Suslin Hypothesis.
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| *The [[constructible universe]] satisfies the [[Generalized Continuum Hypothesis]], the Diamond Principle, Martin's Axiom and the Kurepa Hypothesis.
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| *The failure of the [[Kurepa tree|Kurepa hypothesis]] is equiconsistent with the existence of a [[strongly inaccessible cardinal]].
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| A variation on the method of [[forcing (mathematics)|forcing]] can also be used to demonstrate the consistency and unprovability of the [[axiom of choice]], i.e., that the axiom of choice is independent of ZF. The consistency of choice can be (relatively) easily verified by proving that the inner model L satisfies choice. (Thus every model of ZF contains a submodel of ZFC, so that Con(ZF) implies Con(ZFC).) Since forcing preserves choice, we cannot directly produce a model contradicting choice from a model satisfying choice. However, we can use forcing to create a model which contains a suitable submodel, namely one satisfying ZF but not C.
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| Another method of proving independence results, one owing nothing to forcing, is based on [[Gödel's incompleteness theorem|Gödel's second incompleteness theorem]]. This approach employs the statement whose independence is being examined, to prove the existence of a set model of ZFC, in which case Con(ZFC) is true. Since ZFC satisfies the conditions of Gödel's second theorem, the consistency of ZFC is unprovable in ZFC (provided that ZFC is, in fact, consistent). Hence no statement allowing such a proof can be proved in ZFC. This method can prove that the existence of [[large cardinals]] is not provable in ZFC, but cannot prove that assuming such cardinals, given ZFC, is free of contradiction.
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| ==Criticisms==
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| :''For criticism of set theory in general, see [[Set_theory#Objections_to_set_theory_as_a_foundation_for_mathematics|Objections to set theory]]''
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| ZFC has been criticized both for being excessively strong and for being excessively weak, as well as for its failure to capture objects such as proper classes and the [[universal set]].
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| Many mathematical theorems can be proven in much weaker systems than ZFC, such as [[Peano arithmetic]] and [[second order arithmetic]] (as explored by the program of [[reverse mathematics]]). [[Saunders Mac Lane]] and [[Solomon Feferman]] have both made this point. Some of "mainstream mathematics" (mathematics not directly connected with axiomatic set theory) is beyond Peano arithmetic and second order arithmetic, but still, all such mathematics can be carried out in ZC ([[Zermelo set theory]] with choice), another theory weaker than ZFC. Much of the power of ZFC, including the axiom of regularity and the axiom schema of replacement, is included primarily to facilitate the study of the set theory itself.
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| On the other hand, among [[axiomatic set theories]], ZFC is comparatively weak. Unlike [[New Foundations]], ZFC does not admit the existence of a universal set. Hence the [[universe]] of sets under ZFC is not closed under the elementary operations of the [[algebra of sets]]. Unlike [[von Neumann–Bernays–Gödel set theory]] and [[Morse–Kelley set theory]] (MK), ZFC does not admit the existence of [[proper class]]es. These [[ontology|ontological]] restrictions are required for ZFC to avoid [[Russell's paradox]], but critics argue these restrictions make the ZFC axioms fail to capture the informal concept of ''set''. A further comparative weakness of ZFC is that the [[axiom of choice]] included in ZFC is weaker than the [[axiom of global choice]] included in MK.
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| There are numerous [[list of statements undecidable in ZFC|mathematical statements undecidable in ZFC]]. These include the [[continuum hypothesis]], the [[Whitehead problem]], and the [[Moore space (topology)|Normal Moore space conjecture]]. Some of these conjectures are provable with the addition of axioms such as [[Martin's axiom]], [[large cardinal axiom]]s to ZFC. Some others are decided in ZF+AD where AD is the [[axiom of determinacy]], a strong supposition incompatible with choice. One attraction of [[large cardinal axiom]]s is that they enable many results from ZF+AD to be established in ZFC adjoined by some large cardinal axiom (see [[projective determinacy]]). The [[Mizar system]] has adopted [[Tarski–Grothendieck set theory]] instead of ZFC so that proofs involving [[Grothendieck universe]]s (encountered in category theory and algebraic geometry) can be formalized.
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| == See also ==
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| * [[Foundation of mathematics]]
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| * [[Inner model]]
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| * [[Large cardinal axiom]]
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| Related [[axiomatic set theories]]:
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| *[[Morse–Kelley set theory]]
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| *[[Von Neumann–Bernays–Gödel set theory]]
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| *[[Tarski–Grothendieck set theory]]
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| *[[Constructive set theory]]
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| *[[Internal set theory]]
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| ==References==
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| *[[Alexander Abian]], 1965. ''The Theory of Sets and Transfinite Arithmetic''. W B Saunders.
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| *-------- and LaMacchia, Samuel, 1978, "[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ndjfl/1093888220 On the Consistency and Independence of Some Set-Theoretical Axioms,]" ''Notre Dame Journal of Formal Logic'' 19: 155-58.
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| *[[Keith Devlin]], 1996 (1984). ''The Joy of Sets''. [[Springer Science+Business Media|Springer]].
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| *[[Heinz-Dieter Ebbinghaus]], 2007. ''Ernst Zermelo: An Approach to His Life and Work''. Springer. ISBN 978-3-540-49551-2.
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| *[[Abraham Fraenkel]], [[Yehoshua Bar-Hillel]], and [[Azriel Levy]], 1973 (1958). ''Foundations of Set Theory''. [[North-Holland Publishing Company|North-Holland]]. Fraenkel's final word on ZF and ZFC.
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| *Hatcher, William, 1982 (1968). ''The Logical Foundations of Mathematics''. [[Pergamon Press]].
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| * Peter Hinman, 2005, ''Fundamentals of Mathematical Logic'', [[A K Peters]]. ISBN 978-1-56881-262-5
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| *[[Thomas Jech]], 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. ISBN 3-540-44085-2.
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| *[[Kenneth Kunen]], 1980. ''Set Theory: An Introduction to Independence Proofs''. [[Elsevier]]. ISBN 0-444-86839-9.
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| *[[Richard Montague]], 1961, "Semantic closure and non-finite axiomatizability" in ''Infinistic Methods''. London: Pergamon Press: 45–69.
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| *[[Patrick Suppes]], 1972 (1960). ''Axiomatic Set Theory''. Dover reprint. Perhaps the best exposition of ZFC before the independence of AC and the Continuum hypothesis, and the emergence of large cardinals. Includes many theorems.
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| *[[Gaisi Takeuti]] and Zaring, W M, 1971. ''Introduction to Axiomatic Set Theory''. [[Springer-Verlag]].
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| *[[Alfred Tarski]], 1939, "On well-ordered subsets of any set,", ''[[Fundamenta Mathematicae]]'' 32: 176-83.
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| *Tiles, Mary, 2004 (1989). ''The Philosophy of Set Theory''. Dover reprint. Weak on metatheory; the author is not a mathematician.
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| *Tourlakis, George, 2003. ''Lectures in Logic and Set Theory, Vol. 2''. [[Cambridge University Press]].
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| *[[Jean van Heijenoort]], 1967. ''From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931''. [[Harvard University Press]]. Includes annotated English translations of the classic articles by [[Zermelo]], [[Fraenkel]], and [[Skolem]] bearing on '''ZFC'''.
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| *{{citation|authorlink=Ernst Zermelo|first=Ernst|last= Zermelo|year=1908|title=Untersuchungen über die Grundlagen der Mengenlehre I|journal=[[Mathematische Annalen]] |volume=65|pages= 261–281|doi= 10.1007/BF01449999|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0065&DMDID=DMDLOG_0018&L=1}} English translation in *{{citation|authorlink=Jean van Heijenoort|first=Jean van|last= Heijenoort |year=1967 |title= From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 |series=Source Books in the History of the Sciences |chapter=Investigations in the foundations of set theory|publisher=Harvard University Press|pages=199–215|isbn= 978-0-674-32449-7}}
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| *{{Citation | last1=Zermelo | first1=Ernst | author1-link=Ernst Zermelo | title= Über Grenzzahlen und Mengenbereiche | url=http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=16 | year=1930 | journal=[[Fundamenta Mathematicae]] | issn=0016-2736 | volume=16 | pages=29–47}}
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| ==External links==
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| * {{springer|title=ZFC|id=p/z130100}}
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| * [[Stanford Encyclopedia of Philosophy]] articles by [[Thomas Jech]]:
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| ** [http://plato.stanford.edu/entries/set-theory/ Set Theory];
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| ** [http://plato.stanford.edu/entries/set-theory/ZF.html Axioms of Zermelo–Fraenkel Set Theory].
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| * [http://us.metamath.org/mpegif/mmset.html#staxioms Metamath version of the ZFC axioms] — A concise and nonredundant axiomatization. The background [[first order logic]] is defined especially to facilitate machine verification of proofs.
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| ** A [http://us.metamath.org/mpegif/axsep.html derivation] in [[Metamath]] of a version of the separation schema from a version of the replacement schema.
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| * {{planetmath reference|id=317|title=Zermelo-Fraenkel Axioms}}
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| * {{MathWorld |title=Zermelo-Fraenkel Set Theory |id=Zermelo-FraenkelSetTheory }}
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| {{Set theory}}
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| {{DEFAULTSORT:Zermelo-Fraenkel Set Theory}}
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| [[Category:Systems of set theory]]
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| [[Category:Z notation]]
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