НЕПРОТИВОРЕЧИВОСТЬ АКСИОМАТИЧЕСКОЙ ТЕОРИИ

Axiomatic Theory, Non-Contradiction of

A condition which must be fulfilled by any axiomatic theory and according to which a proposition P and its negation not-P cannot be simultaneously deduced within the framework of the given theory. In view of the difference between the syntactic and semantic aspects of axiomatic theories (see Axiomatic Method), non-contradictoriness is formulated in two ways: a theory is syntactically non-contradictory if a proposition and its negation are not simultaneously deduced in it; a theory is semantically non-contradictory if it has at least one model, i.e., a certain sphere of objects satisfying the given theory. Of all the conditions for axiomatic constructions (see Axiomatic Theory, Completeness of; Axiomatic System, Independence of) non-contradiction is the leading one: its violation makes the theory invalid, because it becomes possible to prove any proposition in it.