Zermelo–Fraenkel set theory

Two sets are equal (are the same set) if they have the same elements.

${\displaystyle \forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow x=y].}$

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 you are constructing set theory does not include equality "${\displaystyle =}$", ${\displaystyle x=y}$ may be defined as an abbreviation for the following formula:[6] ${\displaystyle \forall z[z\in x\Leftrightarrow z\in y]\land \forall w[x\in w\Leftrightarrow y\in w].}$

In this case, the axiom of extensionality can be reformulated as

${\displaystyle \forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow \forall w(x\in w\Leftrightarrow y\in w)],}$

which says that if ${\displaystyle x}$ and ${\displaystyle y}$ have the same elements, then they belong to the same sets.[7]

Every non-empty set ${\displaystyle x}$ contains a member ${\displaystyle y}$ such that ${\displaystyle x}$ and ${\displaystyle y}$ are disjoint sets.

${\displaystyle \forall x[\exists a(a\in x)\Rightarrow \exists y(y\in x\land \lnot \exists z(z\in y\land z\in x))].}$[8]

or in modern notation: ${\displaystyle \forall x\,(x\neq \varnothing \Rightarrow \exists y(y\in x\land y\cap x=\varnothing )).}$

This (along with the Axiom of Pairing) implies, for example, that no set is an element of itself and that every set has an ordinal rank.

Subsets are commonly constructed using set builder notation. For example, the even integers can be constructed as the subset of the integers ${\displaystyle \mathbb {Z} }$ satisfying the congruence modulo predicate ${\displaystyle x\equiv 0{\pmod {2}}}$:

${\displaystyle \{x\in \mathbb {Z} :x\equiv 0{\pmod {2}}\}.}$

In general, the subset of a set ${\displaystyle z}$ obeying a formula ${\displaystyle \varphi (x)}$ with one free variable ${\displaystyle x}$ may be written as:

${\displaystyle \{x\in z:\varphi (x)\}.}$

The axiom schema of specification states that this subset always exists (it is an axiom schema because there is one axiom for each ${\displaystyle \varphi }$). Formally, let ${\displaystyle \varphi }$ be any formula in the language of ZFC with all free variables among ${\displaystyle x,z,w_{1},\ldots ,w_{n}}$ (${\displaystyle y}$ is not free in ${\displaystyle \varphi }$). Then:

${\displaystyle \forall z\forall w_{1}\forall w_{2}\ldots \forall w_{n}\exists y\forall x[x\in y\Leftrightarrow ((x\in z)\land \varphi (x,w_{1},w_{2},...,w_{n},z))].}$

Note that the axiom schema of specification can only construct subsets and does not allow the construction of entities of the more general form:

${\displaystyle \{x:\varphi (x)\}.}$

This restriction is necessary to avoid Russell's paradox (let ${\displaystyle y=\{x:x\notin x\}}$ then ${\displaystyle y\in y\Leftrightarrow y\notin y}$) and its variants that accompany naive set theory with unrestricted comprehension.

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 ${\displaystyle \varnothing }$, once at least one set is known to exist (see above). One way to do this is to use a property ${\displaystyle \varphi }$ which no set has. For example, if ${\displaystyle w}$ is any existing set, the empty set can be constructed as

${\displaystyle \varnothing =\{u\in w\mid (u\in u)\land \lnot (u\in u)\}.}$

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 ${\displaystyle w}$). It is common to make a definitional extension that adds the symbol "${\displaystyle \varnothing }$" to the language of ZFC.

If ${\displaystyle x}$ and ${\displaystyle y}$ are sets, then there exists a set which contains ${\displaystyle x}$ and ${\displaystyle y}$ as elements, for example if x = {1,2} and y = {2,3} then z will be {{1,2},{2,3}}

${\displaystyle \forall x\forall y\exists z((x\in z)\land (y\in z)).}$

The axiom schema of specification must be used to reduce this to a set with exactly these two elements. The axiom of pairing 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.

The union over the elements of a set exists. For example, the union over the elements of the set ${\displaystyle \{\{1,2\},\{2,3\}\}}$ is ${\displaystyle \{1,2,3\}.}$

The axiom of union states that for any set of sets ${\displaystyle {\mathcal {F}}}$, there is a set ${\displaystyle A}$ containing every element that is a member of some member of ${\displaystyle {\mathcal {F}}}$:

${\displaystyle \forall {\mathcal {F}}\,\exists A\,\forall Y\,\forall x[(x\in Y\land Y\in {\mathcal {F}})\Rightarrow x\in A].}$

Although this formula doesn't directly assert the existence of ${\displaystyle \cup {\mathcal {F}}}$, the set ${\displaystyle \cup {\mathcal {F}}}$ can be constructed from ${\displaystyle A}$ in the above using the axiom schema of specification:

${\displaystyle \cup {\mathcal {F}}=\{x\in A:\exists Y(x\in Y\land Y\in {\mathcal {F}})\}.}$

The axiom schema of replacement asserts that the image of a set under any definable function will also fall inside a set.

Formally, let ${\displaystyle \varphi }$ be any formula in the language of ZFC whose free variables are among ${\displaystyle x,y,A,w_{1},\dotsc ,w_{n},}$ so that in particular ${\displaystyle B}$ is not free in ${\displaystyle \varphi }$. Then:

${\displaystyle \forall A\forall w_{1}\forall w_{2}\ldots \forall w_{n}{\bigl [}\forall x(x\in A\Rightarrow \exists !y\,\varphi )\Rightarrow \exists B\ \forall x{\bigl (}x\in A\Rightarrow \exists y(y\in B\land \varphi ){\bigr )}{\bigr ]}.}$

(The unique existential quantifier ${\displaystyle \exists$ !} denotes the existence of exactly one element such that it follows a given statement. For more, see uniqueness quantification.)

In other words, if the relation ${\displaystyle \varphi }$ represents a definable function ${\displaystyle f}$, ${\displaystyle A}$ represents its domain, and ${\displaystyle f(x)}$ is a set for every ${\displaystyle x\in A,}$ then the range of ${\displaystyle f}$ is a subset of some set ${\displaystyle B}$. The form stated here, in which ${\displaystyle B}$ may be larger than strictly necessary, is sometimes called the axiom schema of collection.

Let ${\displaystyle S(w)}$ abbreviate ${\displaystyle w\cup \{w\},}$ where ${\displaystyle w}$ is some set. (We can see that ${\displaystyle \{w\}}$ is a valid set by applying the Axiom of Pairing with ${\displaystyle x=y=w}$ so that the set z is ${\displaystyle \{w\}}$). Then there exists a set X such that the empty set ${\displaystyle \varnothing }$, defined axiomatically, is a member of X and, whenever a set y is a member of X then ${\displaystyle S(y)}$ is also a member of X.

${\displaystyle \exists X\left[\exists e(\forall z\,\neg (z\in e)\land e\in X)\land \forall y(y\in X\Rightarrow S(y)\in X)\right].}$

More colloquially, there exists a set X having infinitely many members. (It must be established, however, that these members are all different because if two elements are the same, the sequence will loop around in a finite cycle of sets. The axiom of regularity prevents this from happening.) 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 ${\displaystyle \mathbb {N} .}$

By definition, a set ${\displaystyle z}$ is a subset of a set ${\displaystyle x}$ if and only if every element of ${\displaystyle z}$ is also an element of ${\displaystyle x}$:

${\displaystyle (z\subseteq x)\Leftrightarrow (\forall q(q\in z\Rightarrow q\in x)).}$

The Axiom of Power Set states that for any set ${\displaystyle x}$, there is a set ${\displaystyle y}$ that contains every subset of ${\displaystyle x}$:

${\displaystyle \forall x\exists y\forall z[z\subseteq x\Rightarrow z\in y].}$

The axiom schema of specification is then used to define the power set ${\displaystyle {\mathcal {P}}(x)}$ as the subset of such a ${\displaystyle y}$ containing the subsets of ${\displaystyle x}$ exactly:

${\displaystyle {\mathcal {P}}(x)=\{z\in y:z\subseteq x\}.}$

Axioms 1–8 define ZF. Alternative forms of these axioms are often encountered, some of which are listed in Jech (2003). 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 ${\displaystyle x}$ whose existence is being asserted are just those sets which the axiom asserts ${\displaystyle x}$ must contain.

The following axiom is added to turn ZF into ZFC:

The last axiom, commonly known as the axiom of choice, is presented here as a property about well-orders, as in Kunen (1980). For any set ${\displaystyle X}$, there exists a binary relation ${\displaystyle R}$ which well-orders ${\displaystyle X}$. This means ${\displaystyle R}$ is a linear order on ${\displaystyle X}$ such that every nonempty subset of ${\displaystyle X}$ has a member which is minimal under ${\displaystyle R}$.

${\displaystyle \forall X\exists R(R\;{\mbox{well-orders}}\;X).}$

Given axioms 1 – 8, many statements are provably equivalent to axiom 9. The most common of these goes as follows. Let ${\displaystyle X}$ be a set whose members are all nonempty. Then there exists a function ${\displaystyle f}$ from ${\displaystyle X}$ to the union of the members of ${\displaystyle X}$, called a "choice function", such that for all ${\displaystyle Y\in X}$ one has ${\displaystyle f(Y)\in Y}$. A third version of the axiom, also equivalent, is Zorn's lemma.

Since the existence of a choice function when ${\displaystyle X}$ 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."

As noted earlier, proper classes (collections of mathematical objects defined by a property shared by their members which are too big to be sets) can only be treated indirectly in ZF (and thus ZFC). An alternative to proper classes while staying within ZF and ZFC is the virtual class notational construct introduced by Quine (1969), where the entire construct y ∈ { x | Fx } is simply defined as Fy.[11] This provides a simple notation for classes that can contain sets but need not themselves be sets, while not committing to the ontology of classes (because the notation can be syntactically converted to one that only uses sets). Quine's approach built on the earlier approach of Bernays & Fraenkel (1958). Virtual classes are also used in Levy (2002), Takeuti & Zaring (1982), and in the Metamath implementation of ZFC.

The axiom schemata of replacement and separation each contain infinitely many instances. 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 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.

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 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.

Abian & LaMacchia (1978) studied a subtheory of ZFC consisting of the axioms of extensionality, union, powerset, replacement, and choice. Using 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 models that satisfy each axiom of ZFC except the axiom of regularity, that axiom is independent of the other ZFC axioms.

If consistent, ZFC cannot prove the existence of the inaccessible cardinals that category theory requires. Huge sets of this nature are possible if ZF is augmented with Tarski's axiom.[12] Assuming that axiom turns the axioms of infinity, power set, and choice (7 – 9 above) into theorems.

Many important statements are independent of ZFC (see list of statements independent of ZFC). The independence is usually proved by forcing, whereby it is shown that every countable transitive model of ZFC (sometimes augmented with large cardinal axioms) 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 models, such as in the constructible universe. However, some statements that are true about constructible sets are not consistent with hypothesized large cardinal axioms.

Forcing proves that the following statements are independent of ZFC:

• Continuum hypothesis
• Diamond principle
• Suslin hypothesis
• Martin's axiom (which is not a ZFC axiom)
• Axiom of Constructibility (V=L) (which is also not a ZFC axiom).

Remarks:

• The consistency of V=L is provable by inner models but not forcing: every model of ZF can be trimmed to become a model of ZFC + V=L.
• The Diamond Principle implies the Continuum Hypothesis and the negation of the Suslin Hypothesis.
• Martin's axiom plus the negation of the Continuum Hypothesis implies the Suslin Hypothesis.
• The constructible universe satisfies the Generalized Continuum Hypothesis, the Diamond Principle, Martin's Axiom and the Kurepa Hypothesis.
• The failure of the Kurepa hypothesis is equiconsistent with the existence of a strongly inaccessible cardinal.

A variation on the method of 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.

Another method of proving independence results, one owing nothing to forcing, is based on 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.

The project to unify set theorists behind additional axioms to resolve the Continuum Hypothesis or other meta-mathematical ambiguities is sometimes known as "Gödel's program".[13] Mathematicians currently debate which axioms are the most plausible or "self-evident", which axioms are the most useful in various domains, and about to what degree usefulness should be traded off with plausibility; some "multiverse" set theorists argue that usefulness should be the sole ultimate criterion in which axioms to customarily adopt. One school of thought leans on expanding the "iterative" concept of a set to produce a set-theoretic universe with an interesting and complex but reasonably tractable structure by adopting forcing axioms; another school advocates for a tidier, less cluttered universe, perhaps focused on a "core" inner model.[14]