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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!

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  • $\begingroup$ 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? $\endgroup$ – DanielWainfleet Apr 3 at 19:55
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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.

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  • $\begingroup$ Thanks, that is an interesting idea. But what can I do if I do not have the set of natural numbers yet? $\endgroup$ – mathfreak Mar 23 at 10:49
  • $\begingroup$ @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. $\endgroup$ – Arthur Mar 23 at 11:09

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