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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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Could groups be used instead of sets as a foundation of mathematics?

Of course there are alternative foundations that don't use sets, but as far as I know all these foundations are still based on things that are quite similar to sets (for example $\sf HoTT$ uses $\infty …