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 math objects 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. with only finite Kolmogorov Information about them. This allows
elimination of all non-integer quantifiers in ST formulas.Comment: 5 page