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A proof system includes the components: [1] [2] Formal language: The set L of formulas admitted by the system, for example, propositional logic or first-order logic. Rules of inference: List of rules that can be employed to prove theorems from axioms and theorems. Axioms: Formulas in L assumed to be valid. All theorems are derived from axioms.
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 proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases ...
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.
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (known as well-formed formulas when relating to formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence, according to the rule of inference.
An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system (usually formulated within model theory) that describes a set of sentences that is closed under logical implication. [1] A formal proof is a complete rendition of a mathematical proof within a formal system.
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