Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Each such set is described by a single set theory formula with parameters unrelated to 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 that external sets (not internally specified, constituting the domain of quantifiable variables) are hereditarily countable and independent of purely formula-defined classes, i.e. with finite algorithmic information about them. Variables for classes are not explicitly quantified. This opens a way to eliminate all non-integer quantifiers in set theory sentences. The restrictions seem to require almost no changes in math papers, only reinterpreting some formalities.

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