The unstable pillar

When reading about the foundations of mathematics, one is bound to stumble upon the notion of “the four pillars of the foundations of mathematics”; namely, proof theory, model theory, recursion theory and axiomatic set theory. But what exactly is it about these areas that makes them get this prestigious title? One can argue that these “pillars” have a metamathematical nature, in that they try to describe methods of existing mathematics instead of describing elements of the mathematical realm. Proof theory about the way of proofs, model theory about how we construct new structures and recursion theory deals with questions about which parts of mathematics can be computed. I would argue that the last “pillar”, axiomatic set theory, does not fall into this category of metamathematics.

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