In mathematics and logic, an axiomatic system or axiom system is a standard type of deductive logical structure, used also in theoretical computer science. It consists of a set of formal statements known as axioms that are used for the logical deduction of other statements. In mathematics these logical consequences of the axioms may be known as lemmas or theorems. A mathematical theory is an expression used to refer to an axiomatic system and all its derived theorems.
A proof within an axiomatic system is a sequence of deductive steps that establishes a new statement as a consequence of the axioms. By itself, the system of axioms is, intentionally, a syntactic construct: when axioms are expressed in natural language, which is normal in books and technical papers, the nouns are intended as placeholder words. The use of an axiomatic approach is a move away from informal reasoning, in which nouns may carry real-world semantic values, and towards formal proof. In a fully formal setting, a logical system such as predicate calculus must be used in the proofs. The contemporary application of formal axiomatic reasoning differs from traditional methods both in the exclusion of semantic considerations, and in the specification of the system of logic in use.
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