Kripke–Platek set theory

is a first-order axiomatic set theory that is weaker than ZF. It uses the language of set theory, with infinitely many set variables, denoted by letters, and two set connectives \(=\), \(\in\).

Axioms
Under the name there are two axiomatic systems - a system without axiom of infinity (abbreviated KP) and one with axiom of infinity (abbreviated KPω, but some authors use KP).

Axiom of Extensionality
\[\forall x\forall y(\forall z(z\in x\leftrightarrow z\in y)\rightarrow x=y)\] Given sets \(x\), \(y\), if they have exactly the same elements, then they are equal.

Axiom of Empty Set
\[\exists x\forall y(y\notin x)\] There exists some set. In fact, there is a set which contains no members.

Axiom of Pairing
\[\forall x\forall y\exists z\forall w(w\in z\leftrightarrow(w=x\lor w=y))\] Given set \(x\), \(y\), there exists set \(z=\{x,y\}\).

Axiom of Union
\[\forall x\exists y\forall z(z\in y\leftrightarrow\exists w(z\in w\land w\in x))\] For every set \(x\), there exists set \(y\) whose members are exactly all the members of the members of \(x\), denoted \(y=\bigcup x\) or \(y=\bigcup_{z\in x}z\).

Axiom Schema of Foundation
For every formula \(\phi(x,p_1,\cdots,p_n)\) or abbreviated \(\phi(x,\overrightarrow{p})\), the following is an instance of the axiom schema: \[\forall\overrightarrow{p}(\exists x\phi(x,\overrightarrow{p})\rightarrow\exists x(\phi(x,\overrightarrow{p})\land\neg\exists y(y\in x\land\phi(y,\overrightarrow{p}))))\] Suppose that a given property \(\phi\) is true for some set \(x\). Then there is a \(\in\)-minimal set for which \(\phi\) is true.

Axiom Schema of \(\Sigma_0\)-Separation
A \(\Sigma_0\) formula is a formula in which every occurence of quantifier is bounded, namely in the form \(\forall x\in y\phi\) (i.e. \(\forall x(x\in y\rightarrow\phi)\)) or \(\exists x\in y\phi\) (i.e. \(\exists x(x\in y\land\phi)\)).

For every \(\Sigma_0\) formula \(\phi(x,\overrightarrow{p})\), the following is an instance of the axiom schema: \[\forall\overrightarrow{p}\forall x\exists y\forall z(z\in y\leftrightarrow(z\in x\land\phi(z,\overrightarrow{p})))\] Given set \(x\), there exists set \(y=\{z\in x|\phi(z,\overrightarrow{p})\}\).

Axiom Schema of \(\Sigma_0\)-Collection
For every \(\Sigma_0\) formula \(\phi(x,y,\overrightarrow{p})\), the following is an instance of the axiom schema: \[\forall\overrightarrow{p}\forall a(\forall x\in a\exists y\phi(x,y,\overrightarrow{p})\rightarrow\exists b\forall x\in a\exists y\in b\phi(x,y,\overrightarrow{p}))\] Given set \(a\), if for all its member \(x\) there is some \(y\) such that \((x,y)\) satisfies a given \(\Sigma_0\)-property, then there exists set \(b\) such that such \(y\) can be found in \(b\).

Axiom of Infinity
\[\exists x(\varnothing\in x\land\forall y\in x(y\cup\{y\}\in x))\] where \(\varnothing\in x\) is shorthand for \(\exists y\in x\forall z(\neg z\in y)\), and \(y\cup\{y\}\in x\) is shorthand for \(\exists z\in x\forall w(w\in z\leftrightarrow(w\in y\lor w=y))\). There exists a set containing infinitely many elements, and it follows from this axiom that empty set exists.