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Mathematical logic, Set theory, Peano arithmetic, Model theory, Proof theory, Recursion theory, Computability theory, Univalent foundations, Reverse mathematics, Frege foundation of arithmetic, Goedel's incompleteness and Mathematics, Structural set theory, Category theory, Type theory.

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This is not yet a very important consideration, but is something univalent foundations and other type-theory foundations are explicitly trying to do well. … Update: John Baez gave a good alternative word for "foundation" in Foundations of Mathematics: Personally I don’t think the metaphor of “foundations” is even appropriate for this approach. …
answered May 20 '20 by MaudPieTheRocktorate