Zermelo-Fraenkel Axioms, Internal Classes, External Sets

Abstract: Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. They are described by a single Set Theory formula with parameters unrelated to other formulas. Exotic expressions involving sets related to formulas of unlimited complexity or to Powerset axiom appear mostly in esoteric or foundational studies. Recognizing internal to math (formula-specified) and external (based on parameters in those formulas) aspects of math objects greatly simplifies foundations. I postulate external sets (not internally specified, treated as the domain of variables) to be hereditarily countable and independent of formula-defined classes, i.e. with finite Kolmogorov Information about them. This allows elimination of all non-integer quantifiers in Set Theory statements.