In certain set definitions, it may be necessary to be able to assert an arbitrary number of quantifiers in your definitions. Sometimes, it might even be necessary to suggest that the number of quantifiers may range from one number to another. However, it is not really clear to me how that could be rigorously implied in our definition.
Part 1:
For instance take the definition of the product of n subsets $H_1, H_2 ..., H_n$ of a group G. I understand this definition can be done inductively, but that is besides the point. I'm merely showcasing this concept as an example for cases where substituting this definition with an inductive one might not be possible($\mathbb{*}$) . Let n be an arbitrary natural number
$H_1...H_n= \{x| \underbrace{(\exists h_1 \in H_1)(\exists h_2 \in H_2)...(\exists h_n \in H_n)}_{\text{n times}}(h_1..h_n = x) \}$
How does one specify the "..." within the definition above? We are currently able to define $H_1H_2$, $H_1H_2H_3$, $H_1H_2H_3H_4$ and so one, if we need to. But the general definition for any n seems impossible to me, currently.
part 2:
How do we specify a "changing number of quantifiers" in a definition? I'll alter the above definition to a new (and meaningless in Group theory) definition to showcase my point. I'll just call the set S:
$S := \{x| \underbrace{(\exists h_i \in H_i)(\exists h_{i+1} \in H_{i+1})...(\exists h_j \in H_j)}_{1 \leq i \leq j \leq n}(h_i..h_j = x) \}$
How can one describe such concept, precisely?
I'm very new to logic, but I'm aware the answers to these questions depend highly on what language we are working with. So, I suppose a better question would be to ask if there is a language in which answers exist to my above questions. And if so, what are those answers?
($\mathbb{*}$) I mentioned that an inductive solution to part 1's example is not of interest here, but I am curious if an inductive solution could be applied to any case in which part 1's (and part 2's) problem occurs. Also, if we do inductive definitions here, we are doing it on quantifiers; Is this something done within our metalanguage? Is it even allowed?
I'd be most thankful for your guidance.
First case:
$\exists n\geq 1\forall i\in\{1,\ldots,n\} \exists h_i \in H_i : \prod_{j=1}^n h_j = x$