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, treated as the domain of variables) to be hereditarily countable and independent of formula-defined classes, i.e. with finite Algorithmic Information about them.
This allows elimination of 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.
talk video (5 G, 32 minutes),
and its transcript.
Its slides with beamer class (Landscape)
or with slides class (Portrait).
More details in a 6 page article
or (may be older) in arxiv
Outline:
Set Theory: Some History, Self-Referentials
Dealing with the Concerns; Cardinalities
Going at the Self-Referential Root
Radical Computer Theorist Hits Back
Some Complexity Background
Independence Postulate
The Formalities
Reducing All Quantifiers to those on Integers
Consistency: Models in ZFC, Countable, Internal.
A Problem: One-Way Functions
Takeaway: the Issues
Takeaway: a Way to Handle
Some More IP Applications
Appendix: ZFC Axioms
To Modify ZFC