The axiom scheme of replacement

We saw that there are two kinds of functions in axiomatic set theory: class functions and set functions. A class function is a rule, defined by a first order formula. A set function is a function which is a set of pairs. The axiom scheme of replacement relates these two notions.

Axiom Scheme of Replacement: for any first order $\varphi (x,y,\stackrel{\_}{a})$, if $\forall x \exists y \varphi (x,y,\stackrel{\_}{a})$ and $b$ is a set then $\left\{⟨u,v⟩:u\in b\wedge \varphi (u,v,\stackrel{\_}{a})\right\}$ is a set.

This is a new general principle building sets, and another instance of Frege's comprehension axiom. It was added by Fraenkel after Zermelo proposed his earlier set of axioms. (Zermelo endorsed the addition - he said that he had overlooked it originally.)

As an application, we look at the idea of transitive sets.

To find examples of transitive sets we apply the recursion theorem on $\omega$.

Using replacement, $t=\left\{\text{TC}(x,n):n\in \omega \right\}$ is a set. We let $\text{TC}\left(x\right)$ be the set $\bigcup t$ and call this the transitive closure of $x$.