On Power Set Axiom

Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Generic sets dependent on Power Set axiom appear mostly in esoteric areas, logic of Set Theory (ST), etc. Recognizing internal to math (formula-specified) and external (based on parameters in those formulas) aspects of sets greatly simplifies the 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. have only finite Kolmogorov Information about them. This allows elimination of all non-integer quantifiers in ST formulas.