Metamathematics
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Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Metamathematical metatheorems about mathematics itself were originally differentiated from ordinary mathematical theorems in the 19th century, to focus on what was then called the foundational crisis of mathematics. Richard's paradox (Richard 1905) concerning certain 'definitions' of real numbers in the English language is an example of the sort of contradictions which can easily occur if one fails to distinguish between mathematics and metamathematics. Something similar can be said around the well-known Russell's paradox (Does the set of all those sets that do not contain themselves contain itself?).
The term "metamathematics" is sometimes used as a synonym for certain elementary parts of formal logic, including propositional logic and predicate logic.
Milestones
- Principia Mathematica (Whitehead and Russell 1925)
- Gödel's completeness theorem, 1930
- Gödel's incompleteness theorem, 1931
- Tarski's definition of model-theoretic satisfaction, now called the T-schema
- The proof of the impossibility of the Entscheidungsproblem, obtained independently in 1936–1937 by Church and Turing.
See also