All-class

The universal class denotes the class that contains all elements of a mathematical theory; in set theory this is the class of all sets. The universal class is now precisely defined by any property that meet all the elements, that is, a tautology , or about as the class or , . ${\ displaystyle \ {x \ mid x = x \}}$ ${\ displaystyle \ {x \ mid \ top \}}$ ${\ displaystyle \ {x \ mid \ bot \ rightarrow \ bot \}}$

The all-class was already formed by Georg Cantor as a “system of all conceivable classes”. In 1899 he showed with an indirect proof that the universal class is not a set : If the universal class were a set, then the power set of the universal class would be a subset of the universal class and thus not a more powerful set, as Cantor's theorem requires. This contradiction is Cantor's second antinomy , which shows that there is no universal set or set of all sets, but that this set formation is contradictory to naive set theory . The all class is therefore a very simple example of a real class.

The all-class is to be distinguished from Russell's class, which, when classified as a set, produces Russell's antinomy . Within the framework of usual set theory, the universal class cannot be a set either, since then, due to the axiom of separation, Russell's class would also be a set. Since most of the current axiomatic set theories contain the axiom of exclusion or an equivalent, the universal class is not a set in them, more precisely there is no universal set, that is

{\ displaystyle \ exists A \ forall B \ quad B \ in A}

is refutable in them. Also building on the axiom of separation, Cantor's antinomy leads to the same result. Some set theories, however, allow the handling of real classes and thus also the universal class as objects separate from the sets ( see also class logic ).

If one accepts the axiom of foundation , as it happens in the Zermelo-Fraenkel set theory and the Neumann-Bernays-Gödel set theory , then the all-class is equal to Russell's class.

literature

Arnold Oberschelp : General set theory. BI-Wiss.-Verl., Mannheim / Leipzig / Vienna / Zurich 1994, ISBN 3-411-17271-1 .

Individual evidence

^ Letter from Cantor to Dedekind of August 31, 1899, in: Georg Cantor: Collected treatises of mathematical and philosophical content. With explanatory notes and additions from the Cantor – Dedekind correspondence. Published by Ernst Zermelo . In addition to a résumé of Cantor by Adolf Fraenkel . Springer, Berlin 1932, p. 448.

[1] Letter from Cantor to Dedekind of August 31, 1899, in: Georg Cantor: Collected treatises of mathematical and philosophical content. With explanatory notes and additions from the Cantor – Dedekind correspondence. Published by Ernst Zermelo . In addition to a résumé of Cantor by Adolf Fraenkel . Springer, Berlin 1932, p. 448.