Set Theory in the Foundation of Math; Internal Classes and External Sets

Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Each such set is described by a single set-theoretic formula with parameters unrelated to other formulas. Exotic expressions involving sets related to formulas of unbounded quantifier depth appear mostly in esoteric or foundational studies. Recognizing the internal-to-math (formula-specified) and external (parameter-based) aspects of math objects greatly simplifies foundations. I postulate external sets (not internally specified, constituting the domain of variables) to be hereditarily countable and independent of formula-defined classes, i.e. with finite algorithmic information about them. Variables for formulas are allowed, but no explicit quantifiers over them. This allows eliminating all non-integer quantifiers in set-theoretic sentences. The restrictions seem to require almost no changes in mathematical papers, only reinterpreting some formalities.