Russell's paradox

The Russell's antinomy is discovered by Bertrand Russell and Ernst Zermelo paradox of naive set theory, the Russell 1903 published and therefore bears his name.

Term and problem

Russell made ​​his antinomy with the help of the " class of all classes not contain itself as an element of the " who is called Russell's class; he defined it formally as follows:

Often the Russell class is defined as a " set of all sets that do not contain themselves as an element "; corresponding to the former set theory that have not yet distinguished between Classes and Sets. The Russell's antinomy, however, unlike the older antinomies of naive set theory ( Burali - Forti paradox and Cantor's paradoxes ) purely logical in nature and independent of quantity axioms. Therefore, it has very strong effect and abruptly brought about the end of naive set theory.

Russell led his antinomy from logically to suppose contains itself, then applies due to the class property is defined with the that does not contain, which contradicts the assumption. Suppose it is the opposite and does not contain yourself, then met the class property so that but even contains contrary to the assumption. Mathematically expressed this following contradictory equivalence of:

To derive this contradiction, no axioms and theorems of set theory are used, but other than the definition only Frege's abstraction principle, which took over Russell in his theory of types:

History and Solutions

It is not known exactly when Russell discovered his paradox, probably in the spring of 1901 while working on his book, The Principles of Mathematics, where in 1903 he published the antinomy. There he wrote that he found the antinomy in employment with an indirect proof of Cantor, which is the first Cantor's antinomy of 1897 based. Early as 1902, he shared the antinomy Gottlob Frege with a letter. He was referring to Frege's first volume of the basic laws of arithmetic of 1893, tried to build up the arithmetic on a set-theoretical axioms in Frege. The Russell's antinomy showed that this axiom system was contradictory. Frege responded in the epilogue of the second volume of his basic laws of arithmetic from 1903:

Probably Frege gave up his work in the area of axiomatic logic due to the discovery of the paradox.

Russell solved the paradox in 1903 by its type theory; in her class always has a higher type than its elements; Statements such as " a class contains itself ", with which he made ​​his antinomy can then be no longer formulate. So he tried, as he clung to Frege's principle of abstraction to solve the problem through a restricted syntax of permissible class statements. However, the restricted syntax proved to be complicated and inadequate for construction of mathematics and has not prevailed permanently.

Parallel to Russell developed Zermelo who found the antinomy independently by Russell and Russell before publication knew, the first axiomatic set theory with full syntax. The axiom of Zermelo set theory of 1907 allows only a restricted class education within a given set. He showed by an indirect proof of this antinomy that Russell's class is not a lot. His approach has prevailed. In the extended Zermelo -Fraenkel set theory (ZF ), which today serves as the foundation of mathematics, in addition, the foundation axiom ensures that no amount of can contain themselves, so that here the Russell class is identical to the universal class.

Since Russell's antinomy of pure logical in nature and does not depend on quantity of axioms, is already at the level of non-contradictory first-order logic provable that the Russell class is non-existent as a set. This makes the following argument understandable that converts a second indirect evidence Russell in a direct proof:

This phrase means in the language of predicate logic: There is no set of all sets that do not contain themselves as an element. It applies in all modern axiomatic set teachings that are based on the first order predicate logic, for example in ZF. It also applies in the Neumann - Bernays - Gödel set theory, in Russell's class but exists as a real class. In the class of logic Oberschelp, which is a proven consistent extension of first-order predicate logic, also any class terms can be formed to any defining statements; specifically there is also the Russell class a correct term with provable nonexistence. In this class logic axiom systems such as the ZF set theory can be integrated.

Since the set was derived in a direct proof, it is also valid in the intuitionistic logic.

Variants of Russell's antinomy

The Grelling -Nelson's antinomy of 1908 is an inspired by Russell's antinomy semantic paradox.

There are numerous popular varieties of Russell's antinomy. Most famous is the barber paradox, with the Russell 1918 even his train of thought illustrated and generalized.

Curry's paradox from 1942 includes as a special case a generalization of Russell's antinomy.