Von Neumann–Bernays–Gödel set theory

The Neumann - Bernays - Gödel set theory ( NBG ) is an axiomatization of set theory. It is named after John von Neumann, Paul Bernays and Kurt Godel, as it is based on the work of these mathematicians. The amount range is equivalent to the more widespread Zermelo -Fraenkel set theory ( ZFC ). In contrast to ZFC are the objects of NBG not only quantity, but rather classes. Quantities are defined special classes: A class is called set if it is member of a class. The classes of NBG can thus only contain sets as elements. There are also classes that are not sets; they are called proper classes (somewhat jokingly as tons ).

The History

The first foundation stone for the Neumann - Bernays - Gödel set theory put John von Neumann 1925/1927 in his axiomatization of set theory. He grabbed on her criticism of Abraham Fraenkel at the Zermelo set theory and designed the first calculation with a derivable » replacement axiom " and a " boundedness axiom " that excludes compass illustrative set of and Zermelo foundation axiom anticipates the ZF system of 1930. In contrast to ZF, the replacement scheme produces infinite axioms, he formulated a substitution axiom with the concept of function and thus achieved a finite system of axioms 23. This he founded entirely on functions ( II - things ) and arguments ( I. - things ). Functions, the arguments are ( I-II- things ) at the same time, he continued with the same quantities. Its hard to read, very technical -sounding function calculus but not prevailed. Paul Bernays transferred Neumann's ideas in an axiom system with classes and amounts in its set- theory, which he developed in 1937. He separated classes here and quantities strictly and used two variables places and two predicates element ε and η for quantities and classes. He saw but later this separation as a dead end; 1958 he formulated a simplified set theory with classes that are not quantifiable individuals more. The modern NBG set theory not picked up on this late Modifikition, but followed the simplification that Kurt Gödel published in 1940 as part of his famous work on the continuum hypothesis. It simply eliminates the second element predicate for classes, but retained different variable types for classes and quantities of.

The NBG axioms

Modern versions of the Neumann - Bernays - Gödel set theory apply a first-order logic with equality and underlying predicate element. Your variables are generally for classes and are listed in uppercase. The above definition covers a lot of formula, can be introduced on the specific amount of variables that are written in lowercase:

With analog abbreviated notations for other variables, the amount NBG axioms will receive a clear form:

• Extensionality: Two classes are equal if and only if they contain the same elements.

• Axiom of the empty set: There is a class that contains no elements.

• Pair axiom: For any two sets there is a set whose elements are precisely the two quantities.

• Union axiom: For each class there is a class whose elements are exactly the elements of the elements of the first class.

• Power set axiom: For every set is a set whose elements are exactly the subsets of the first set.

• Axiom of infinity: There exists a set which contains the empty set and with each item and the amount (see Inductive amount ).

• Of regularity ( the foundation axiom ): Every non-empty class contains a disjoint class to this element.

• Komprehensionsschema: For every property there is the class of all sets that satisfy this property; as property is approved each formula, only occur before quantity variables in the quantifiers:

• Substitution axiom: The image of a quantity under a function is a set back.

• Axiom of Choice: There is a function that assigns to each non-empty set of its elements.

Resolution of the contradictions of naive set theory

Classes that were classified as amounts in the naive set theory, and then led to contradictions prove in NBG as real classes. The Russell's antinomy dissolves example, like this: If one forms after Komprehensionsschema the class of all sets that do not themselves contain

So is not a lot, because otherwise the opposition would. So is a real class ( it even contains any quantity ), and it is considered as the elements of a class are by definition amounts.

The class of all classes can not form because classes contain only quantities according to the definition of the class. As soon as you write, must be proved or assumed that a lot is.