Axiom schema of replacement

Remarks

Example: let $\varphi(x, y)$ say "$x$ is a natural number and $y$ is the set $\omega + x$". For $A = \omega$ the hypothesis holds, so replacement gives the set $\set{\omega + n : n \in \omega}$, whose union is the ordinal $\omega + \omega$. Non-example: the set $V_{\omega + \omega}$ of all sets of rank below $\omega + \omega$ satisfies all axioms of Zermelo set theory, but not replacement. In it, the instance above fails: each $\omega + n$ lies in $V_{\omega + \omega}$, but the set $\set{\omega + n : n \in \omega}$ has rank $\omega + \omega$ and does not. Replacement implies the separation scheme, given the other axioms of ZF. It is used to justify definitions by transfinite recursion on the ordinals.