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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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In what respect are univalent foundations "better" than set theory?
And I call all of this foundations of mathematics for that reason. … At this point I can safely dismiss set theory as foundations of mathematics entirely because it is itself mathematics and plays no part in the above. …