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Predicate logic. First-order logic. Infinitary logic; Many-sorted logic; Higher-order logic. Lindström quantifier; Second-order logic; Soundness theorem; Gödel's completeness theorem. Original proof of Gödel's completeness theorem; Compactness theorem; Löwenheim–Skolem theorem. Skolem's paradox; Gödel's incompleteness theorems; Structure ...
Axiomatic proofs have been used in mathematics since the famous Ancient Greek textbook, Euclid's Elements of Geometry, c. 300 BC. But the first known fully formalized proof system that thereby qualifies as a Hilbert system dates back to Gottlob Frege's 1879 Begriffsschrift.
The notion of analytic proof was introduced by Gentzen for the sequent calculus; there the analytic proofs are those that are cut-free. Much of the interest in cut-free proofs comes from the subformula property: every formula in the end sequent of a cut-free proof is a subformula of one of the premises. This allows one to show consistency of ...
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.
Jankov logic (KC) is an extension of intuitionistic logic, which can be axiomatized by the intuitionistic axiom system plus the axiom [13] ¬ A ∨ ¬ ¬ A . {\displaystyle \neg A\lor \neg \neg A.} Gödel–Dummett logic (LC) can be axiomatized over intuitionistic logic by adding the axiom [ 13 ]
This is a list of notable theorems. Lists of theorems and similar statements include: List of algebras; List of algorithms; List of axioms; List of conjectures; List of data structures; List of derivatives and integrals in alternative calculi; List of equations; List of fundamental theorems; List of hypotheses; List of inequalities; Lists of ...
Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational; Proof that the sum of the reciprocals of the primes diverges
In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally ...