ZFC - Prove existence of set {A_1, A_2, A_3, ...}

Question

Good morning,

I am currently taking a course in axiomatic set theory and I have encountered a problem in showing the existence of a certain infinite (countable) set.

Let $A_1$, $A_2$, $A_3$, ... be a countably infinite number of given sets. How can I prove the existence of the set $\{A_1,A_2,A_3,...\}$? Which axioms do I need to use?

I have already made some research, but I have not found an explicit explanation on how to prove the existence of the above set. Its existence is usually simply assumed but not formally proven. I just know that the existence of the finite set $\{A_1,A_2,...,A_{n-1},A_n\}$ can be shown using the pairing and union axioms.

Thanks for your help!

Answer 1

Replacement.

You have the set of natural numbers, and for each natural number $i$ you have a unique $A_i$. It is the axiom of replacement which lets you "replace" each $i\in\Bbb N$ with the corresponding $A_i$ and still have a set.