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the mathematical discipline that applies mathematical methods to the study of mathematical theories themselves.

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Is there any current development of a first order formalization of metamathematics?

There are provability logics and extensions in modal logic's that seem to attempt to address this question, and model theory implies answering questions about metamathematics. … But I'm looking for an explicitly first order theory that can be used to prove things about metamathematics. Basically, I'm wondering if there are first order formalizations of metamathematics. …