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 isdescribed by a single Set Theory formula with parameters unrelated toother formulas. Exotic expressions involving sets related to formulas ofunlimited quantifiers height appear mostly in esoteric or foundationalstudies. Recognizing internal to math (formula-specified) and external (based onparameters in those formulas) aspects of math objects greatly simplifiesfoundations. I postulate external sets (not internally specified,constituting the domain of variables) to be hereditarily countable andindependent of formula-defined classes, i.e. with finite algorithmicinformation about them. This allows to eliminate all non-integer quantifiers in Set Theorysentences. All with seemingly no need to change almost anything inmathematical papers, only to reinterpret some formalities.