I've seen in various sources such as this one describe the axiom schema of replacement involving a function:
If $F$ is a function, then for any $X$ there exists a set $Y=F[X]=\{ F(x):x \in X \}$.
My questions are really elementary, so please bare with me.
- What exactly is the domain and codomain of $F$? Shouldn't that be specified?
- In the axiom schema, we talk about any set $X$. Does this mean the domain of $F$ is the entire universe of ZFC? How can any function possibly be defined on every possible element in ZFC? Is there an example that can help me grasp this?
Best Answer
Let's look at an example. Consider the following "function" (see below): $$F:x\mapsto \{x\}.$$ This makes sense for arbitrary $x$. The corresponding instance of Replacement says that for each $A$ the class $\{\{x\}: x\in A\}$ is again a set. Note that $F$ isn't limited to $A$ in any sense, but when we use Replacement we do have to restrict attention to a "starting set."
Now - and this gets to the scare quotes around the word "function" above - $F$ isn't, properly speaking, a function. In set theory a function is a set of ordered pairs, and the class $$\{(x,\{x\}): x\mbox{ is a set}\}$$ is a proper class. To be precise, Replacement is cast in terms of formulas which we think of as defining functions. That said, talking about functions may be intuitively helpful.