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 , if and is a set then 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.
A set is transitive if for all if and then .
To find examples of transitive sets we apply the recursion theorem on .
We define by and .
Using replacement, is a set. We let be the set and call this the transitive closure of .
(a) The transitive closure of is transitive. (b) If and is transitive then .