I noticed that quantifiers are defined in a way resembling induction: we define the quantifier for the empty set, for singletons and then for the union of sets.

The problem is, such a method doesn’t work for infinite sets (for an example of why, consider the fact that finite intersections of open sets are open, but arbitrary intersections are not necessarily so. In topology, a similar statement about the intersection of two open sets is taken as an axiom, but many infinite intersections can easily be shown to be singletons and hence not open). Yet, in the lectures, we saw examples of both infinite sums (over the naturals), universal quantifiers (over the rationals), existential quantifiers (over the naturals) and even products (over the naturals), none of which can be handled by the definitions given, at least according to my understanding.

Are we going to be given different definitions, or am I making a mistake here?