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The Pythagorean theorem has at least 370 known proofs. [1]In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. [a] [2] [3] The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
Löb's theorem (mathematical logic) Lochs's theorem (number theory) Looman–Menchoff theorem (complex analysis) Łoś' theorem (model theory) Lovelock's theorem ; Löwenheim–Skolem theorem (mathematical logic) Lucas's theorem (number theory) Lukacs's proportion-sum independence theorem (probability) Lumer–Phillips theorem (semigroup theory)
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first understood from context, after which an element ϕ ∈ T {\displaystyle \phi \in T} of a deductively closed theory T {\displaystyle T} is then called a theorem of the theory.
The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. For a well-formed formula to qualify as part of a proof, it must be the result of applying a rule of the deductive apparatus (of some formal system) to the previous well-formed formulas in the proof sequence.
The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. [3] In modern logic, an axiom is a premise or starting point for reasoning. [4] In mathematics, an axiom may be a "logical axiom" or a "non-logical axiom".
compactness theorem A theorem in logic stating that if every finite subset of a set of sentences has a model, then the entire set has a model. complete infinity A concept in philosophy and mathematics referring to an actual infinity that is considered as a completed whole, contrasting with potential infinities that are indefinitely extendable.
Bet hedging (biology) Bishop–Cannings theorem; F. Fisher's fundamental theorem of natural selection; L. Lewis' law; M. Marginal value theorem; Monodomain model
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 ...