# Conceptual question about the Axiom of Replacement and functions that map between "different looking" elements.

For the following question, there is likely terminology that I am not privy to that should be used. I apologize in advanced for this, as I am asking a question about a territory of math I am not overly familiar with.

In Tao's Analysis I, the Axiom of Replacement is stated as follows:

Let $$A$$ be a set. For any object $$x \in A$$, and any object $$y$$, suppose we have a statement $$P(x,y)$$ pertaining to $$x$$ and $$y$$, such that for each $$x \in A$$ there is at most one $$y$$ for which $$P(x,y)$$ is true. Then there exists a set $$\{y:P(x,y)$$ is true for some $$x \in A \}$$, such that for any object $$z$$, $$z\in\{y:P(x,y)$$ is true for some $$x \in A\} \iff P(x,z)$$ is true for some $$x \in A$$.

For quite some time, I thought this axiom was restricted to depicting scenarios where functions would map between "common objects". For example, consider the function $$f: \mathbb R \to \mathbb R$$ where $$f$$ is defined as $$x \mapsto 2x$$, i.e. $$f(x)=2x$$.

Or, alternatively, consider the function $$f: \mathbb R \to \mathbb Z$$ where $$f$$ is defined as $$x \mapsto \lfloor x \rfloor$$.

In either case, the idea that I thought was being captured (and therefore in agreement with my understanding of what was implied by the Axiom of Replacement) was that a number was being mapped to a number. i.e. "common objects/things/items" were being mapped to one another.

However, I recently encountered an exercise that required me to employ the Axiom of Replacement in ways that I have not previously explored.

The exercise, ($$3.5.1$$), asks the reader to demonstrate that the Cartesian product $$X \times Y$$ is a set. In order to carry this out, I first created a function of the following form:

$$f_x(y) = \Big\{\{x\},\{x,y\}\Big\}$$

i.e. this function takes any object $$y$$ in set $$Y$$ and maps it to the following different object: $$\Big\{\{x\},\{x,y\}\Big\}$$... where $$x$$, an element of $$X$$, is fixed

After this, I created a set invoking the Axiom of Replacement:

$$Y_x = \bigg\{\Big\{\{x\},\{x,y\}\Big\}: \forall y \in Y, f_x(y)=\Big\{\{x\},\{x,y\}\Big\}\bigg\}$$

Now, I basically made arbitrarily many of these functions...e.g. :

$$f_{x'}(y) = \Big\{\{x'\},\{x',y\}\Big\}$$

$$f_{x''}(y) = \Big\{\{x''\},\{x'',y\}\Big\}$$

ect ect...where $$x'$$, $$x''$$, etc are all different elements of $$X$$...resulting in all of the following different sets:

$$Y_{x'} = \bigg\{\Big\{\{x'\},\{x',y\}\Big\}: \forall y \in Y, f_{x'}(y)=\Big\{\{x'\},\{x',y\}\Big\}\bigg\}$$

$$Y_{x''} = \bigg\{\Big\{\{x''\},\{x'',y\}\Big\}: \forall y \in Y, f_{x''}(y)=\Big\{\{x''\},\{x'',y\}\Big\}\bigg\}$$ etc etc

I lingered on this for a bit, recognizing that $$y$$ and $$\{\{x'\},\{x',y\}\}$$ are, somehow, not "the same thing"...but continued on nonetheless.

Finally, I made the function $$g(x) = Y_x$$, which is even more bizarre to me, because this is effectively saying:

$$g: x\mapsto \bigg\{\Big\{\{x\},\{x,a\}\Big\},\Big\{\{x\},\{x,b\}\Big\}, \Big\{\{x\},\{x,c\}\Big\}...\bigg\}$$ where $$a,b,c,...$$ etc are all elements of $$Y$$.

Invoking the Axiom of Replacement one last time:

$$Z=\{Y_x: \forall x \in X, g(x) = Y_x\}$$. Following this with the Axiom of Union, I successfully generate the set of all ordered pairs (as defined through the Kuratowski notation), thus demonstrating that the Cartesian product is, in fact, a set.

The idea of what a function can do, and what the Axiom of Replacement entails, seems significantly richer than I initially imagined. In the above example, it is evident to me that I am mapping (for a lack of a better word), lower-tiered objects to higher-tiered objects. For example, in function $$g$$, it seems clear to me that an element $$g(x)$$ has significantly "more structure" than the element $$x$$ that maps to it. (Perhaps a better way of phrasing it is that $$g(x)$$ sits higher on some sort of set construction hierarchy than the element $$x$$)

So, really, my question is fairly simple. Is there a name for what I am observing? And, further, is there any significance to this concept?

Long comment

The Axiom does not "use" a function in the universe of sets (i.e. a set) but a formula $$P(x,y)$$ of the language of the theory of sets that "behaves like a function", i.e. such that it satisfies the condition

"that for each $$x ∈ A$$ there is at most one $$y$$ for which $$P(x,y)$$ holds."

The key points of the Axiom is the set $$A$$ above.

Per se, a "functional" formula $$P(x,y)$$ does not license us to assert that a corresponding set exists.

What does it mean ?

Consider the collection of pairs $$\{ (x,y) \mid P(x,y) \text { holds, for some } x \}$$.

If such a collection is a set, clearly it will be a function. But thus, we can use it to carve out the collection of all first "coordinates" of the pairs it contains.

Clearly, such a collection is the "universe" $$V$$ itself, and we know that in $$\mathsf {ZFC}$$ the universe is not a set.

Thus, neither the collection above is a set.

What the Axiom of Replacement asserts is that, for a set $$A$$ whatever, there is a set $$Y$$ consisting of all those $$y$$ such that $$P(x,y)$$ holds for some $$x \in A$$.

But then, all pairs $$(x,y)$$ are elements of $$A \times Y$$ and thus, by Separation, the collection of pairs $$\{ (x,y) \in A \times Y \mid P(x,y) \text { holds } \}$$ is a set.

Because the formula $$P(x,y)$$ "behaves like a function", that set is indeed a function.