Set Theory in the Foundation of Math; Internal Classes and External Sets
This work addresses foundational issues in mathematics for set theorists and logicians, but it appears incremental as it reinterprets existing formalities rather than introducing a new paradigm.
The paper tackles the complexity of set theory foundations by distinguishing between internal (formula-specified) and external (parameter-based) sets, proposing that external sets be hereditarily countable and algorithmically finite, which eliminates non-integer quantifiers in set theory sentences without altering most mathematical papers.
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 other formulas. Exotic expressions involving sets related to formulas of unlimited quantifiers height 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, constituting the domain of variables) to be hereditarily countable and independent of formula-defined classes, i.e. with finite algorithmic information about them. This allows to eliminate all non-integer quantifiers in Set Theory sentences. All with seemingly no need to change almost anything in mathematical papers, only to reinterpret some formalities.