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The axiom asserts the existence of a set containing the empty set and closed under the successor operation $x\mapsto x\cup \{x\}$:In Zermelo–Fraenkel set theory, this supplies a set from which the set of natural numbers is constructed.

Every proper lattice ideal of a Boolean algebra extends to a prime lattice ideal. More generally, an ideal disjoint from a lattice filter extends to a prime ideal still disjoint from that filter. This Boolean case of the Stone prime filter theorem is equivalent to the ultrafilter lemma over Zermelo–Fraenkel set theory and is weaker than the axiom of choice.

A set $X$ is Dedekind-finite when no injective function $\mathbb{N}\to X$ exists. In classical Zermelo–Fraenkel set theory this is equivalent to $X$ not being in bijection with a proper subset: an injective non-surjective self-map generates distinct iterates from an element outside its image, and a countably infinite subset permits a shift fixing its complement. A finite set is Dedekind-finite. Without the axiom of choice, some infinite sets can also be Dedekind-finite.

The Axiom of foundation states that every nonempty set $A$ has a member $a$ with $a\cap A=\varnothing$. The principle of epsilon induction states that a formula $\varphi$ satisfying $\forall x[(\forall y\in x\varphi (y))\Rightarrow \varphi (x)]$ holds for every set $x$.

Assume Axiom of foundation and suppose a progressive formula has a counterexample $a$. In the transitive closure of $\{a\}$, use axiom schema of separation to collect the counterexamples into a nonempty set $B$. Foundation supplies $b\in B$ having no member in $B$. Since the closure is transitive, every member of $b$ satisfies $\varphi$, contradicting progressiveness. Conversely, if a nonempty set $A$ had no foundation-minimal member, every $x\in A$ would have a member $y\in x\cap A$. The formula $x\notin A$ is then progressive: if every member of $x$ lies outside $A$, $x$ cannot belong to $A$. Epsilon induction would make $A$ empty, a contradiction. This proves equivalence in the presence of the other Zermelo–Fraenkel set theory.

To obtain the cumulative hierarchy, perform epsilon induction on the assertion that $x$ belongs to some $V_{\alpha }$. If this holds for each member $y$, Axiom schema of replacement collects their least such ordinals into a set. Choose an ordinal $\beta$ above them all. Then $x\subseteq V_{\beta }$ and $x\in V_{\beta +1}$. Thus every set occurs in the hierarchy. Use the rank of a set convention $\mathrm{rk}(x)={\sup }_{y\in x}(\mathrm{rk}(y)+1)$, equivalently the least $\alpha$ with $x\subseteq V_{\alpha }$.

Set theory studies sets, their set elements, and the constructions and sizes possible for them. It supplies a shared vocabulary for mathematics: a function connects two sets, a binary relation specifies which pairs of objects are related, and a probability space starts with a set of possible outcomes. Giving these objects a common language lets a construction from one subject be used in another without repeating its underlying bookkeeping.

A particularly beautiful question is how to compare sizes without first assigning a number. Two sets have the same cardinality when a bijection pairs every member of one with exactly one member of the other. For finite sets, this agrees with ordinary counting. For infinite sets, it produces surprises: $n\mapsto 2n$ bijects the natural numbers with the nonnegative even integers, and the Cantor pairing function bijects the Cartesian product ${\mathbb{N}}^2$ with $\mathbb{N}$. A proper subset can have the same cardinality as its containing set.

This does not make all infinite sets equally large. Cantor's diagonal argument shows that all infinite binary sequences cannot be listed by the natural numbers. Cantor's theorem gives the wider principle: the power set of any set has strictly greater cardinality. The contrast between an explicit enumeration and an impossibility argument makes the meaning of infinity much sharper than a picture of something merely continuing forever.

Set formation itself requires care. Russell's paradox shows why not every condition defines a set of all objects satisfying it. Zermelo–Fraenkel set theory instead uses axioms specifying permitted constructions, including set comprehension within a previously given set. The restrictions are useful: they let us form the collections ordinary arguments need while blocking the unrestricted construction that produces the paradox.

ZFC is Zermelo–Fraenkel set theory together with the axiom of choice.