I’m looking for reading material about the philosophical and mathematical issues that may result from using an axiom schema of replacement rather than an axiom of replacement. Among other things, it would be nice to be able to point to specific results along these lines, to help readers understand why presentations of the axiom schema of separation such as the one given in Wikipedia mention “definable” functions rather than just functions in the discussion. The Wikipedia article itself doesn’t seem to clarify why the modifier “definable” Is used.
I think it would help if such presentations included such discussions.
For this, I’m referring to the following statement as the Axiom of Replacement:
“The range of a function is a set.”
The Wikipedia article on the Axiom Schema of Replacement begins with
“In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF.”
From a pedagogical perspective, the following question arises immediately: “Why does the author of this passage include the term definable mapping rather than just the term mapping in this informal description of the axiom scheme? It raises the question then of how one might give an example of a function or mapping that isn’t definable, and how it would affect the meaning of the passage of the term “definable mapping” were reduced to just “mapping”.
The problem is that the principle
(where "function" means "set $f$ of ordered pairs such that for each $x$ there is at most one $y$ with $\langle x,y\rangle\in f$" as usual) is too weak to be useful. In particular, it already follows from the other axioms, specifically Powerset and Separation. The whole "target" of Replacement is relations which are not immediately sets.
For example, consider the transitive set $V_{\omega+\omega}$. This is a model of all the axioms of ZFC except Replacement. The issue is that, within $V_{\omega+\omega}$, we can define the relation $$R(\alpha,\beta):\equiv {}"\beta=\omega+\alpha"$$ but the class $\{\beta: \exists x\in\omega(R(x,\beta))\}$ is not a set in $V_{\omega+\omega}$.