ПАРАДОКСЫ

Paradoxes (in logic and the set theory)

Formal logical contradictions which arise in the set theory and in formal logic, while preserving the correct line of reasoning; they are akin to Zeno's aporias and semantic antinomies known since antiquity. In modern science Paradoxes were discovered in the 19th century in some branches of the set theory—for example, by George Cantor in 1895 and Cesare Burali-Forti in 1897. One of the best known Paradoxes was discovered by Bertrand Russell in 1902 when two mutually exclusive (contradictory) propositions are equally demonstrable. They can appear both in a scientific theory and in ordinary arguments—for example, Russell's rewording of his paradox about a set in all normal sets: "Barber in a certain village who shaves all and only those persons in the village who do not shave themselves. Does he shave himself?"

Since a formal logical contradiction destroys inference as a means of finding and demonstrating truth—in a theory in which Paradoxes appear, any proposition both true and false is equally demonstrable—the task arises of revealing the sources of Paradoxes and finding ways of eliminating them. A dialectical materialist analysis shows that Paradoxes are an expression of profoundly dialectical and epistemological difficulties associated with concepts of an object and the objective sphere in formal logic, of a set or class in logic and in the set theory, with the employment of the principle of abstraction which makes it possible to introduce new abstract objects, and with methods of defining abstract objects in science, etc. That is why there can be no universal method of removing all Paradoxes.

Various ways are possible for solving the problem of removing Paradoxes from scientific theories: construction of the theory of types, or hierarchy of types, restriction of the principle of abstraction, etc. Thus, to remove Paradoxes from the set theory, axiomatic set theories were created in which restrictions were introduced sufficient for excluding the known Paradoxes—the first system was proposed by E. Zermelo in 1908. The problem of philosophical understanding and finding concrete solution of Paradoxes is an important methodological problem of formal logic and the logical principle of mathematics.