This might be a request for further philosophical clarification, rather than a mathematical question. You know, we are working with two kind of quantifiers $\forall x$ and $\exists x$. Somehow, we decided this language is enough for us. I can see that. For example, I can express further things like "There exists more than three elements in this set that satisfies predicate P". For example, I can try to express this as
$$\exists x, y, z, w .P(x) \land P(y) \land P(z) \land P(w)$$
I think this captures my initial statement. If I can find 4 elements that P is true for all of them, then I can say there exists more than three elements in this set that satisfies P. You see though, I needed to use 4 auxiliary variables to express this and if I wanted to say "there are more than 100 elements", I could not find enough symbols to identify my variables.
In a way, what I am trying to say is, yes two quantifiers are powerful to express whole of these kind of things but they are not nice to use when things get ugly. It is, as if, predicate logic is a designed language just like programming languages and somewhere along the way designers decided 2 quantifiers are enough to express mathematics.
My question is, why we have 2 quantifiers? Who decided that it is best? Why we don't have further syntax to express other notions? For example, I could explain "there exists more than 100 elements in the set that satisfies predicate P" as $$\exists (> 100)x.P(x)$$
why we don't have something like this? I can see that mathematics is the collective work of many geniuses and there is definitely a reason why my above suggestion is not used in mathematics. But what is the reason for things being as they are?
Edit: There are clarifications and suggestions for further notation in the comments. But what I am really asking is, what is the "design rationale" you should have in mind when deciding on available quantifiers for your logic? There are logics beside predicate logic and we might devise new ones in the future. How mathematicians tackle this problem of making available "just enough" quantifiers? For example, do they seek minimal number of quantifiers that makes their logic complete, or some similar things?