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 Gödel because it builds on the work of these mathematicians. In terms of quantities , it is equivalent to the more widely used Zermelo-Fraenkel set theory (ZFC). In contrast to ZFC, NBG's objects are not just sets, but rather classes . Sets are specially defined classes: A class is called a set if it is a member of a class. NBG's classes can only contain sets as elements. There are also classes that are not sets; they are referred to as real classes (also, somewhat jokingly, tons).

About history

John von Neumann laid the first foundation stone for the Neumann-Bernays-Gödel set theory in 1925/1927 in his axiomatization of set theory. Here he took up Abraham Fraenkel's criticism of the Zermelo set theory and designed the first calculus with a deducible substitution axiom and an axiom of limitation that excludes circular set formations and anticipates Zermelo's foundation axiom in the ZF system of 1930. In contrast to ZF, whose substitution scheme produces an infinite number of axioms, he formulated a substitution axiom with the concept of function and thus achieved a finite system of 23 axioms. He based this entirely on functions (II.-things) and arguments (I.-things). He equated functions that are also arguments (I-II things) with sets. His difficult-to-read, very technical-looking function calculation did not catch on. Paul Bernays transferred von Neumann's ideas into a system of axioms with classes and sets in his set theory , which he worked out from 1937. Here he strictly separated classes and sets and used two types of variables and two types of element predicates ε and η for sets and classes. He later saw this separation as a dead end; In 1958 he formulated a simplified set theory with classes that are no longer quantifiable individuals. The modern NBG set theory did not take up this late modification, but followed the simplification that Kurt Gödel published in 1940 as part of his famous work on the continuum hypothesis . It only removed the second element predicate for classes, but kept different types of variables for classes and sets.

The NBG axioms

Modern versions of the Neumann-Bernays-Gödel set theory are based on a first-level predicate logic with equality and element predicate. Their variables generally stand for classes and are written in uppercase. The above quantity definition captures a formula through which specific quantity variables can be introduced, which are written as lower case letters:

${\displaystyle \operatorname {Mg} (X)\colon \Leftrightarrow \exists Y\colon X\in Y}$

${\displaystyle \forall x\colon A(x)}$ stands for ${\displaystyle \forall X\colon (\operatorname {Mg} (X)\rightarrow A(X))}$

${\displaystyle \exists x\colon A(x)}$ stands for ${\displaystyle \exists X\colon (\operatorname {Mg} (X)\land A(X))}$

The NBG axioms get a clear form with analogous abbreviations for other quantity variables:

• Axiom of extensionality: Two classes are equal if and only if they contain the same elements.

${\displaystyle \forall X,Y\colon (X=Y\leftrightarrow \forall z\colon (z\in X\leftrightarrow z\in Y))}$

According to the axiom of extensionality, classes are uniquely determined if their elements are described by an equivalent property. An abbreviated notation can be specified for such classes. This happens with some of the following axioms for the classes or sets postulated as existent.

• Axiom of the Empty Set: There is a class that contains no elements.

${\displaystyle \exists X\colon \forall y\colon \lnot (y\in X)}$

Notation: ${\displaystyle \varnothing }$

• Pair set axiom: For every two sets there is a set whose elements are exactly the two sets.

${\displaystyle \forall x,y\colon \exists z\colon \forall w\colon (w\in z\leftrightarrow (w=x\lor w=y))}$

Notation: ${\displaystyle \,\{x,y\}}$

Special case (single element set): ${\displaystyle \,\{x\}:=\{x,x\}}$

Special case ( ordered pair ): ${\displaystyle \,(a,b):=\{\{a,b\},\{a\}\}}$

• Axiom of union: For every set there is a set whose elements are exactly the elements of the elements from the first set.

${\displaystyle \forall x\colon \exists y\colon \forall z\colon (z\in y\leftrightarrow \exists w\colon (w\in x\land z\in w))}$

Notation: ${\displaystyle \bigcup x}$

Special case: ${\displaystyle x\cup y:=\bigcup \{x,y\}}$

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

${\displaystyle \forall x\colon \exists y\colon \forall z\colon (z\in y\leftrightarrow \forall w\colon (w\in z\rightarrow w\in x))}$

Notation: ${\displaystyle {\mathcal {P}}(x)}$

• Infinity axiom: There is a set that contains the empty set and with every element also the set (see inductive set ). ${\displaystyle \,y}$ ${\displaystyle y\cup \{y\}}$

${\displaystyle \exists x\colon (\varnothing \in x\land \forall y\colon (y\in x\rightarrow y\cup \{y\}\in x))}$

• Regularity axiom (foundation axiom): Every nonempty class contains an element that is disjoint to this class .

${\displaystyle \forall X\colon (X\neq \varnothing \rightarrow \exists y\colon (y\in X\land \lnot \exists z\colon (z\in X\land z\in y)))}$

• Comprehension scheme: For every property there is a class of all sets that fulfill this property; any formula is permitted as a property in which quantifiers occur only in front of set variables: ${\displaystyle \varphi }$

${\displaystyle \forall X_{1},\ldots ,X_{n}\colon \exists Y\colon \forall z\colon (z\in Y\leftrightarrow \varphi (z,X_{1},\ldots ,X_{n}))}$

Notation: ${\displaystyle \{z\mid \varphi (z,X_{1},\ldots ,X_{n})\}}$

• Replacement axiom: The image of a set under a function is again a set.

${\displaystyle \forall F,x\colon (F\operatorname {Funktion} \rightarrow \exists y\colon \forall z\colon (z\in y\leftrightarrow \exists w\colon (w\in x\land (w,z)\in F)))}$

Notation: ${\displaystyle \{F(w)\mid w\in x\}}$

The function operator in the replacement axiom is defined as follows:

${\displaystyle F\operatorname {Funktion} :\leftrightarrow \forall p\colon (p\in F\rightarrow \exists x,y\colon p=(x,y))\land \;\forall x,y,z\colon {\bigl (}(x,y)\in F\land (x,z)\in F\rightarrow y=z{\bigr )}}$

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

${\displaystyle \exists F\colon (F\operatorname {Funktion} \land \forall x\colon (x\not =\varnothing \rightarrow \exists y\colon (y\in x\land (x,y)\in F)))}$

Finite axiomatizability

NBG can be represented by a finite number of axioms. In his 1937 essay, Bernays proved in a sentence he called the Class Theorem that the only axiom scheme of NBG, the comprehension scheme, can be generated by a finite number of individual axioms. In 1954 he showed that one of these axioms can be derived from the rest. If one wants to get by with as few axioms as possible when proving the class theorem, one must orient oneself to the proof of Gödel in his article from 1940.

Resolving the contradictions of naive set theory

Classes that were classified as sets in naive set theory and then led to contradictions, turn out to be real classes in NBG. The Russell paradox dissolves, for example, like so: If one forms the class of all sets that do not contain after Komprehensionsschema itself

${\displaystyle R=\{x\mid x\not \in x\}}$,

there is no set, because otherwise the contradiction would arise . So is a real class (it actually contains all sets), and it holds because the members of a class are sets by definition. ${\displaystyle R}$ ${\displaystyle R\in R\Leftrightarrow R\notin R}$ ${\displaystyle R}$ ${\displaystyle R\notin R}$

Russell's antinomy can also be avoided, analogously to type theory or Quine's New Foundations, by explicitly requiring , when building the language of NBG for atomic expressions , with the additional agreement that differently designated variables also stand for different classes. ${\displaystyle X\in Y}$

The class of all classes cannot be formed according to the definition of the term class, since classes only contain sets. As soon as one writes, it must be proven or assumed that there is a lot. ${\displaystyle x\in X}$ ${\displaystyle x}$

literature

• Elliot Mendelson: Introduction to Mathematical Logic , Fourth Edition, 1997, ISBN 0-412-80830-7 . Chapter 4, Axiomatic Set Theory, pp. 225-304

Individual evidence

1. ↑ John von Neumann: An axiomatization of set theory , in: Journal for pure and applied mathematics 154 (1925), 219-240. Precise calculation in: John von Neumann: Die Axiomatisierung der Setlehre , 1927, in: Mathematische Zeitschrift 27 (1928) 669–752
2. ^ Paul Bernays: A System of Axiomatic Set Theory I , in: Journal of Symbolic Logic 2 (1937), pp. 65-77. Parts II-VII in the same journal in the years 1941–1943, 1948, 1954.
3. ^ Abraham Adolf Fraenkel, Paul Bernays: Axiomatic Set Theory , Amsterdam 1958
4. ↑ Kurt Gödel: The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory , Annals of Mathematical Studies, Volume 3, Princeton NJ, 1940
5. ^ G. Keene: Abstract sets and finite ordinals , Dover Books Oxford, London, New York, Paris 1961. There the proof of Bernays is also presented.
6. ^ Paul Bernays: A System of Axiomatic Set Theory VII , in: Journal of Symbolic Logic 19 (1954), pp. 81-96