Set Theory in the Foundation of Math; Internal Classes and External Sets

Abstract

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

8 pages article + 17 pages slides; talk video: https://doi.org/10.5281/zenodo.18626525