Axiom Schema of Specification

Last updated Nov 1, 2022

# Axiom Schema

Let $k \in \mathbb{N}$ and let $\phi$ be a Formula with Free Variables among $x, t_{1}, \dots, t_{k}, c$. Then the Sentence $$\forall t_{1} \cdots \forall t_{k} \forall c \exists B \forall x [x \in B \Leftrightarrow (x \in c \wedge \phi(x, c, t_{1}, \dots, t_{k}))]$$ is an Axiom in ZF-Theory.

1. This Axiom effectively allows us to take any subset of a Set using a finite Formula.
2. This is often known as the “Subset Axiom”.
3. If $A$ is a Set and $\phi$ is a Formula with Free Variables among $x, t_{1}, \dots, t_{k}, A$, then we often write our desired subset as ${x \in A : \phi(x, A, t_{1}, \dots, t_{k})}$.
4. If we instead want to take a definable subset of $\mathcal{P}(A)$, we can instead write ${x \subset A : \phi(x, \mathcal{P}(A), A, t_{1}, \dots, t_{k})}$, where $A$ can be treated as one of the extra Free Variables of $\phi$.