Question about definition by induction. I am reading some notes about naive set theory. I know the "definition by induction", but I can't apply it to the following cases directly. 
Suppose $A$ is a set, let $A^+$ be the set $A\cup \{A\}$. The we can inductively define a function $f$ whose domain is a natural number $n$, such that $f(0)=A$,  and $f(i^+)=f(i)^+$ whenever $i^+<n$.
My question is why does such a function (a set, why its range is a set) exist? 
A similar question is as following: 
Suppose we have a way $g$ to construct a sequence of sets $A_0$, $A_1$,... where $A_n$ is constructed from $A_{n-1}$. How does this process end or run out all the natural numbers? I mean we should have a set $Y$, and a function $h$ from the set of natural numbers to $Y$ such that $h(n)$ is exactly the set $A_n$ for each natural number $n$, right? 
Thank you!
 A: An example: Let $A=\emptyset$, that is, let  $A$ be empty. Then $$A^+=\emptyset \cup\{\emptyset\}=\{\emptyset\}$$ and 
$$(A^+)^+=A^+\cup\{A^+\}=\{\emptyset\}\cup\{\{\emptyset\}\}=\{\emptyset,\{\emptyset\}\}$$
and 
$$((A^+)^+)^+=(A^+)^+\cup \{(A^+)^+\}=\{\emptyset,\{\emptyset\}\}\cup\{\{\emptyset,\{\emptyset\}\}\}=$$$$=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\},$$
and so on.
With the $f$ notation:
$$f(0)=\emptyset, f(1)=\{\emptyset\}, f(2)=\{\emptyset,\{\emptyset\}\},\cdots$$
The question "why does such a function (a set, why its range is a set) exist?" is answered. I've shown at least one such function whose domain is the natural numbers and whose range is a set of well defined  sets.
As far as the second question: I am not sure if I understand that one correctly. I guess the answer is that it is not necessary in general that a closed formula exists when something is defined recursively. In the case our $f$ above such  a closed formula certainly exist in a clumzy form like "the $n^{th}$ expression starts with an opening curly bracket then, and then and then a certain number of opening and  closing curly bracket, blah blah blah..."
A: I find that in the book Set Theory by Thomas Jech, a more "general" theorem called "transfinite recursion" can answer all my questions, where he do not need to worry that "function" must be a set, ect.
Thanks again.
