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Together with model theory, axiomatic set theory, and recursion theory, proof theory is one of the so-called "four pillars" of the foundations of mathematics.
en.wikipedia.org The mirrors of the title refer to recursion.
en.wikipedia.org Recursive functions and loops are equivalent in expression; any expression involving loops can be written using recursion, and vice versa.
en.wikipedia.org Recursion theorists in mathematical logic often study the theory of relative computability, reducibility notions and degree structures described in this article.
en.wikipedia.org This shows the usefulness of corecursion rather than recursion for dealing with infinite data structures.
en.wikipedia.org