Metamathematics
Metamathematics is a program in mathematical logic that seeks to determine which axioms are necessary to prove a given theorem. Metalogic is generally the metatheory of logical systems, while metamathematics is the metatheory of mathematical theories; in current usage they overlap heavily, and in many contexts metalogic is treated as the part of metamathematics that focuses specifically on logic. In ordinary usage the choice of term often tracks disciplinary tradition (philosophy/CS vs foundations of mathematics) more than a sharp technical boundary.
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms.
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