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Richard's paradox (Richard 1905) concerning certain 'definitions' of real numbers in the English language is an example of the sort of contradictions that can easily occur if one fails to distinguish between mathematics and metamathematics.
Analyses or descriptions of an existing theory would be considered meta-theories. [2] If the subject matter of a theoretical statement consists of one or multiple theories, it would also be called a meta-theory. [3] For mathematics and mathematical logic, a metatheory is a mathematical theory about another mathematical theory. [4]
Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory. [citation needed] A formal system is determined by a formal language and a deductive system (axioms and rules of inference). The formal system can be used to ...
The deduction theorem for predicate logic is similar, but comes with some extra constraints (that would for example be satisfied if is a closed formula). In general a deduction theorem needs to take into account all logical details of the theory under consideration, so each logical system technically needs its own deduction theorem, although ...
Almgren–Pitts min-max theory; Approximation theory; Arakelov theory; Asymptotic theory; Automata theory; Bass–Serre theory; Bifurcation theory; Braid theory; Brill–Noether theory; Catastrophe theory; Category theory; Chaos theory; Character theory; Choquet theory; Class field theory; Cobordism theory; Coding theory; Cohomology theory ...
Introduction to Meta-Mathematics (Tenth impression 1991 ed.). Amsterdam NY: North-Holland Pub. Co. ISBN 0-7204-2103-9. In Chapter III A Critique of Mathematic Reasoning, §11. The paradoxes, Kleene discusses Intuitionism and Formalism in depth. Throughout the rest of the book he treats, and compares, both Formalist (classical) and Intuitionist ...
Meta’s AI research group has developed a generalized AI system that requires only a handful of specific examples in order to respond to new and emerging forms of harmful content.
Cantor–Bernstein–Schröder theorem (set theory, cardinal numbers) Cantor's theorem (set theory, Cantor's diagonal argument) Church–Rosser theorem (lambda calculus) Compactness theorem (mathematical logic) Conservativity theorem (mathematical logic) Craig's theorem (mathematical logic) Craig's interpolation theorem (mathematical logic)