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

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!

• In the formal Language Of Set Theory (a.k. a. Lost) we have a very limited alphabet and syntax. How would you say "Let $A_1,A_2,..$ be a countably infinite number of sets" in Lost? – DanielWainfleet Apr 3 at 19:55

## 1 Answer

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.

• Thanks, that is an interesting idea. But what can I do if I do not have the set of natural numbers yet? – mathfreak Mar 23 at 10:49
• @mathfreak In that case you don't have all of ZF, as the existence of the natural numbers (or the corresponding ordinal $\omega$) is one of the axioms. – Arthur Mar 23 at 11:09