Zermelo–Fraenkel set theory

Zermelo–Fraenkel set theory with the Axiom of Choice or ZFC is a popular first-order axiomatic set theory.

Axioms
All lowercase letters denote sets.

Axiom of Extensionality
Two sets with the same elements are equal.

\[\forall x: \forall y: [\forall z: (z \in x \Leftrightarrow z \in y) \Rightarrow (x = y)]\]

Axiom of Pairing
Given two sets, there is a third one containing only those two.

\[\forall x: \forall y: \exists z: \forall w: [w \in z \Rightarrow (w = x \vee w = y)]\]

Axiom of Regularity
Every nonempty set is disjoint with one of its members.

\[\forall x: [\exists a: (a \in x) \Rightarrow \exists y: (y \in x \wedge \not \exists z: (z \in y \wedge z \in x))]\]

Axiom Schema of Specification
Sets may be constructed by filtering a larger set with a formula.

The following is an axiom for every formula \(\phi\) with two free variables:

\[\forall x: \forall w: \exists y: \forall z: (z \in y \Leftrightarrow (z \in x \wedge \phi(u, w)))\]

Axiom of Power Set
Every set has a power set.

\[\forall x: \exists y: \forall z: (z \in y \Leftrightarrow \forall w: (w \in z \Rightarrow w \in x))\]

Axiom of Infinity
There is an infinite set.

\[\exists x: (\empty \in x \wedge \forall y: (y \in x \Rightarrow (y \cup \{y\}) \in x))\]

Axiom of Union
The union over a set's members exists.

\[\forall x: \exists y: \forall z: (z \in y \Leftrightarrow \exists w: (w \in x \wedge z \in w))\]

Axiom Schema of Replacement
We can create an image of a set over a function and a codomain.

The following is an axiom for every formula \(\phi\) with three free variables:

\[\forall p: \left[ \forall x: \forall y: \forall z: [\phi(x, y, p) \wedge \phi(x, z, p) \Rightarrow] \Rightarrow \forall x \exists y \forall z [z \in y \Leftrightarrow \exists w (w \in x \Rightarrow \phi(x, y, p))] \right]\]