There are many equivalent formulations of the ZFC axioms. The following particular axiom set is from . The axioms in order below are expressed in a mixture of
first-order logic and high-level abbreviations. Axioms 1–8 form ZF, while the axiom 9 turns ZF into ZFC. Following , we use the equivalent
well-ordering theorem in place of the
axiom of choice for axiom 9. All formulations of ZFC imply that at least one set exists. Kunen includes an axiom that directly asserts the existence of a set, although he notes that he does so only "for emphasis". 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 existsusually expressed as the assertion that something is identical to itself, \exists 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 asserts that an
infinite set exists. This implies that
a set exists, and so, once again, it is superfluous to include an axiom asserting as much.
Axiom of extensionality - 1 Two sets are equal (are the same set) if they have the same elements. \forall x \forall y [\forall z (z \in x \Leftrightarrow z \in y) \Rightarrow x = y]. The converse of this axiom follows from the substitution property of
equality. ZFC is constructed in first-order logic. Some formulations of first-order logic include identity; others do not. If the variety of first-order logic in which one is constructing set theory does not include equality "=", x=y may be defined as an abbreviation for the following formula: \forall z [z \in x \Leftrightarrow z \in y] \land \forall w [x \in w \Leftrightarrow y \in w]. In this case, the axiom of extensionality can be reformulated as \forall x \forall y [\forall z (z \in x \Leftrightarrow z \in y) \Rightarrow \forall w (x \in w \Leftrightarrow y \in w)], which says that if x and y have the same elements, then they belong to the same sets.
Axiom of regularity (also called the axiom of foundation) - 2 Every non-empty set x contains a member y such that x and y are
disjoint sets. \forall x [(\exists a ( a \in x)) \Rightarrow \exists y ( y \in x \land \lnot \exists z (z \in y \land z \in x))]. or in modern notation: \forall x\,(x \neq \varnothing \Rightarrow \exists y (y \in x \land y \cap x = \varnothing)). With the axioms of pairing and union, this implies that no set is an element of itself. With the axioms of infinity, replacement, and union, this implies that every set has an
ordinal rank.
Axiom schema of specification (or of separation, or of restricted comprehension) - 3 Subsets are commonly constructed using
set builder notation. For example, the even integers can be constructed as the subset of the integers \mathbb{Z} satisfying the
congruence modulo predicate x \equiv 0 \pmod 2: \{x \in \mathbb{Z} : x \equiv 0 \pmod 2\}. In general, the subset of a set z obeying a formula \varphi(x) with one free variable x may be written as: \{x \in z : \varphi(x)\}. The axiom schema of specification states that this subset always exists (it is an
axiom schema because there is one axiom for each \varphi). Formally, let \varphi be any formula in the language of ZFC with all free variables among x,z,w_{1},\ldots,w_{n} (y is not free in \varphi). Then: \forall z \forall w_1 \forall w_2\ldots \forall w_n \exists y \forall x [x \in y \Leftrightarrow (( x \in z )\land \varphi(x,w_1,w_2,...,w_n,z) )]. Note that the axiom schema of specification can only construct subsets and does not allow the construction of entities of the more general form: \{x : \varphi(x)\}. This restriction is necessary to avoid
Russell's paradox (let y=\{x:x\notin x\} then y \in y \Leftrightarrow y \notin y) and its variants that accompany naive set theory with
unrestricted comprehension (since under this restriction y only refers to sets '''
within z''' that don't belong to themselves, and y \in z has
not been established, even though y \subseteq z is the case, so y stands in a separate position from which it can't refer to or comprehend itself; therefore, in a certain sense, this axiom schema is saying that in order to build a y on the basis of a formula \varphi(x), we need to previously restrict the sets y will regard within a set z that leaves y outside so y can't refer to itself; or, in other words, sets shouldn't refer to themselves). In some other axiomatizations of ZF, this axiom is redundant in that it follows from the
axiom schema of replacement and the
axiom of the empty set. On the other hand, the axiom schema of specification can be used to prove the existence of the
empty set, denoted \varnothing, once at least one set is known to exist. One way to do this is to use a property \varphi which no set has. For example, if w is any existing set, the empty set can be constructed as \varnothing = \{u \in w \mid (u \notin w) \} . 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 "\varnothing" to the language of ZFC.
Axiom of pairing - 4 If x and y are sets, then there exists a set which contains x and y as elements; for example, if x = \{1,2\} and y = \{2,3\}, then z might be \{\{1,2\},\{2,3\}\}. \forall x \forall y \exists z ((x \in z) \land (y \in z)). The axiom schema of specification must be used to reduce this to a set with exactly these two elements.
Axiom of union - 5 The
union over the elements of a set exists. For example, the union over the elements of the set \{\{1,2\},\{2,3\}\} is \{1,2,3\}. The axiom of union states that for any set of sets \mathcal{F}, there is a set A containing every element that is a member of some member of \mathcal{F}: \forall \mathcal{F} \,\exists A \, \forall Y\, \forall x [(x \in Y \land Y \in \mathcal{F}) \Rightarrow x \in A]. Although this formula doesn't directly assert the existence of \cup \mathcal{F}, the set \cup \mathcal{F} can be constructed from A in the above using the axiom schema of specification: \cup \mathcal{F}=\{x\in A : \exists Y (x \in Y \land Y \in \mathcal{F})\}.
Axiom schema of replacement - 6 The axiom schema of replacement asserts that the
image of a set under any definable
function will also fall inside a set. Formally, let \varphi be any
formula in the language of ZFC whose
free variables are among x, y, A, w_1, \dotsc, w_n, so that in particular B is not free in \varphi. Then: \forall A\forall w_1 \forall w_2\ldots \forall w_n \bigl[\forall x ( x\in A \Rightarrow \exists! y\,\varphi ) \Rightarrow \exists B \ \forall x \bigl(x\in A \Rightarrow \exists y (y\in B \land \varphi)\bigr)\bigr]. (The
unique existential quantifier \exists! denotes the existence of exactly one element such that it follows a given statement.) In other words, if the relation \varphi represents a definable function f, A represents its
domain, and f(x) is a set for every x \in A, then the
range of f is a subset of some set B. The form stated here, in which B may be larger than strictly necessary, is sometimes called the
axiom schema of collection.
Axiom of infinity - 7 Let S(w) abbreviate w \cup \{w\}, where w is some set. (We can see that \{w\} is a valid set by applying the axiom of pairing with x = y = w so that the set is \{w\}). Then there exists a set such that the empty set \varnothing, defined axiomatically, is a member of and, whenever a set is a member of then S(y) is also a member of . \exists X \left [\exists e (\forall z \, \neg (z \in e) \land e \in X) \land \forall y (y \in X \Rightarrow S(y) \in X)\right]. or in modern notation: \exists X \left [\varnothing \in X \land \forall y (y \in X \Rightarrow S(y) \in X)\right]. More colloquially, there exists a set having infinitely many members. These members are built by repeatedly applying the operation S starting from the empty set. Each result of this construction is distinct from the previous ones, so the process does not loop or repeat. The minimal set satisfying the axiom of infinity is the
von Neumann ordinal , which can also be thought of as the set of
natural numbers \mathbb{N}. (Note that the well-foundedness of \mathbb{N} does not require the axiom of regularity; it follows naturally from the structure of the construction.)
Axiom of power set - 8 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: (z \subseteq x) \Leftrightarrow ( \forall q (q \in z \Rightarrow q \in x)). The Axiom of power set states that for any set x, there is a set y that contains every subset of x: \forall x \exists y \forall z (z \subseteq x \Rightarrow z \in y). The axiom schema of specification is then used to define the
power set \mathcal{P}(x) as the subset of such a y containing the subsets of x exactly: \mathcal{P}(x) = \{ z \in y: z \subseteq x \}. Axioms
1–8 define ZF. Alternative forms of these axioms are often encountered, some of which are listed in . Some ZF axiomatizations include an axiom asserting that the
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. The following axiom is added to turn ZF into ZFC:
Axiom of well-ordering (choice) - 9 The last axiom, commonly known as the
axiom of choice, is presented here as a property about
well-orders, as in . For any set X, there exists a
binary relation R which
well-orders X. This means R is a
linear order on X such that every nonempty
subset of X has a
least element under the order R. \forall X \exists R ( R \;\mbox{well-orders}\; X). Given axioms 1–8, many statements are equivalent to axiom 9. The most common of these goes as follows. Let X be a set whose members are all nonempty. Then there exists a function f from X to the union of the members of X, called a "
choice function", such that for all Y\in X one has f(Y)\in Y. A third version of the axiom, also equivalent, is
Zorn's lemma. 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 sets. AC is characterized as
nonconstructive because it asserts the existence of a choice function but says nothing about how this choice function is to be "constructed". == Motivation via the cumulative hierarchy ==