Von Neumann–Bernays–Gödel set theory

Von Neumann–Bernays–Gödel set theory, or NBG, is an important area of study in the foundations of mathematics. It is a type of axiomatic set theory that builds on Zermelo–Fraenkel–choice set theory, often called ZFC. One special feature of NBG is that it introduces something called a "class." A class is a big collection of sets that follows a specific rule.

NBG lets mathematicians talk about very large collections, like the class of all sets or the class of all ordinals, which are bigger than any single set. This helps in solving problems that come up when dealing with infinite collections. NBG has a key theorem called the class existence theorem. It says that for any rule that only talks about sets, there is a class made up of all the sets that follow that rule.

The idea of using classes in set theory began with John von Neumann in 1925. Later, Paul Bernays and Kurt Gödel worked on simplifying and improving the theory. Their work helped make NBG easier to use and understand, and it remains important today for studying the basic building blocks of mathematics.

Classes in set theory

Classes help organize sets in a special way. They let us talk about big collections, like all sets together, that are too large to be sets themselves. This helps avoid problems in math where assuming something is a set leads to contradictions.

One important use of classes is to handle tricky situations in set theory, like the class of all ordinals. If we tried to treat this as a set, it would create a problem because it would have to contain itself in a way that breaks the rules of ordering. By calling it a class instead, we avoid this issue. Classes also help mathematicians build important structures, like the constructible universe, used in proving certain ideas in set theory.

Axiomatization of NBG

NBG set theory includes two types of objects: classes and sets. Every set is also considered a class. There are different ways to organize these ideas, but they all lead to the same conclusions. One approach uses a special kind of logic to separate classes and sets, while another uses rules to define what is a class and what is a set. These differences mainly affect how we write statements rather than what we can prove.

The theory uses specific rules to define how classes and sets interact. For example, it includes principles like the axiom of extensionality, which says that if two classes have the same members, they are the same class. There are also rules for creating new classes from existing ones, such as combining classes or finding what’s common between them. These tools help mathematicians work with large collections of sets in a structured way.

Example 1: If the classes F {\displaystyle F} and G {\displaystyle G} are functions, then the composite function G ∘ F {\displaystyle G\circ F} is defined by the formula: ∃ t [ ( x , t ) ∈ F ∧ ( t , y ) ∈ G ] . {\displaystyle \exists t[(x,t)\in F\,\land \,(t,y)\in G].} Since this formula has two free set variables, x {\displaystyle x} and y , {\displaystyle y,} the class existence theorem constructs the class of ordered pairs: G ∘ F = { ( x , y ) : ∃ t [ ( x , t ) ∈ F ∧ ( t , y ) ∈ G ] } . {\displaystyle G\circ F\,=\,\{(x,y):\exists t[(x,t)\in F\,\land \,(t,y)\in G]\}.} Because this formula is built from simpler formulas using conjunction ∧ {\displaystyle \land } and existential quantification ∃ {\displaystyle \exists } , class operations are needed that take classes representing the simpler formulas and produce classes representing the formulas with ∧ {\displaystyle \land } and ∃ {\displaystyle \exists } . To produce a class representing a formula with ∧ {\displaystyle \land } , intersection is used since x ∈ A ∩ B ⟺ x ∈ A ∧ x ∈ B . {\displaystyle x\in A\cap B\iff x\in A\land x\in B.} To produce a class representing a formula with ∃ {\displaystyle \exists } , the domain is used since x ∈ D o m ( A ) ⟺ ∃ t [ ( x , t ) ∈ A ] . {\displaystyle x\in Dom(A)\iff \exists t[(x,t)\in A].} Before taking the intersection, the tuples in F {\displaystyle F} and G {\displaystyle G} need an extra component so they have the same variables. The component y {\displaystyle y} is added to the tuples of F {\displaystyle F} and x {\displaystyle x} is added to the tuples of G {\displaystyle G} : F ′ = { ( x , t , y ) : ( x , t ) ∈ F } {\displaystyle F'=\{(x,t,y):(x,t)\in F\}\,} and G ′ = { ( t , y , x ) : ( t , y ) ∈ G } . {\displaystyle \,G'=\{(t,y,x):(t,y)\in G\}.} In the definition of F ′ , {\displaystyle F',} the variable y {\displaystyle y} is not restricted by the statement ( x , t ) ∈ F , {\displaystyle (x,t)\in F,} so y {\displaystyle y} ranges over the class V {\displaystyle V} of all sets. Similarly, in the definition of G ′ , {\displaystyle G',} the variable x {\displaystyle x} ranges over V . {\displaystyle V.} So an axiom is needed that adds an extra component (whose values range over V {\displaystyle V} ) to the tuples of a given class. Next, the variables are put in the same order to prepare for the intersection: F ″ = { ( x , y , t ) : ( x , t ) ∈ F } {\displaystyle F''=\{(x,y,t):(x,t)\in F\}\,} and G ″ = { ( x , y , t ) : ( t , y ) ∈ G } . {\displaystyle \,G''=\{(x,y,t):(t,y)\in G\}.} To go from F ′ {\displaystyle F'} to F ″ {\displaystyle F''} and from G ′ {\displaystyle G'} to G ″ {\displaystyle G''} requires two different permutations, so axioms that support permutations of tuple components are needed. The intersection of F ″ {\displaystyle F''} and G ″ {\displaystyle G''} handles ∧ {\displaystyle \land } : F ″ ∩ G ″ = { ( x , y , t ) : ( x , t ) ∈ F ∧ ( t , y ) ∈ G } . {\displaystyle F''\cap G''=\{(x,y,t):(x,t)\in F\,\land \,(t,y)\in G\}.} Since ( x , y , t ) {\displaystyle (x,y,t)} is defined as ( ( x , y ) , t ) {\displaystyle ((x,y),t)} , taking the domain of F ″ ∩ G ″ {\displaystyle F''\cap G''} handles ∃ t {\displaystyle \exists t} and produces the composite function: G ∘ F = D o m ( F ″ ∩ G ″ ) = { ( x , y ) : ∃ t ( ( x , t ) ∈ F ∧ ( t , y ) ∈ G ) } {\displaystyle G\circ F=Dom(F''\cap G'')=\{(x,y):\exists t((x,t)\in F\,\land \,(t,y)\in G)\}} So axioms of intersection and domain are needed.

Example 2: Rule 4 is applied to the formula ϕ ( x 1 ) {\displaystyle \phi (x_{1})} that defines the class consisting of all sets of the form { ∅ , { ∅ , … } , … } . {\displaystyle \{\emptyset ,\{\emptyset ,\dots \},\dots \}.} That is, sets that contain at least ∅ {\displaystyle \emptyset } and a set containing ∅ {\displaystyle \emptyset } — for example, { ∅ , { ∅ , a , b , c } , d , e } {\displaystyle \{\emptyset ,\{\emptyset ,a,b,c\},d,e\}} where a , b , c , d , {\displaystyle a,b,c,d,} and e {\displaystyle e} are sets. ϕ ( x 1 ) = ∃ u [ u ∈ x 1 ∧ ¬ ∃ v ( v ∈ u ) ] ∧ ∃ w ( w ∈ x 1 ∧ ∃ y [ ( y ∈ w ∧ ¬ ∃ z ( z ∈ y ) ] ) ϕ r ( x 1 ) = ∃ x 2 [ x 2 ∈ x 1 ∧ ¬ ∃ x 3 ( x 3 ∈ x 2 ) ] ∧ ∃ x 2 ( x 2 ∈ x 1 ∧ ∃ x 3 [ ( x 3 ∈ x 2 ∧ ¬ ∃ x 4 ( x 4 ∈ x 3 ) ] ) {\displaystyle {\begin{aligned}\phi (x_{1})\,&=\,\exists u\;\,[\,u\in x_{1}\,\land \,\neg \exists v\;\,(\;v\,\in \,u\,)]\,\land \,\,\exists w\;{\bigl (}w\in x_{1}\,\land \,\exists y\;\,[(\;y\,\in w\;\land \;\neg \exists z\;\,(\;z\,\in \,y\,)]{\bigr )}\\\phi _{r}(x_{1})\,&=\,\exists x_{2}[x_{2}\!\in \!x_{1}\,\land \,\neg \exists x_{3}(x_{3}\!\in \!x_{2})]\,\land \,\,\exists x_{2}{\bigl (}x_{2}\!\in \!x_{1}\,\land \,\exists x_{3}[(x_{3}\!\in \!x_{2}\,\land \,\neg \exists x_{4}(x_{4}\!\in \!x_{3})]{\bigr )}\end{aligned}}} Since x 1 {\displaystyle x_{1}} is the only free variable, n = 1. {\displaystyle n=1.} The quantified variable x 3 {\displaystyle x_{3}} appears twice in x 3 ∈ x 2 {\displaystyle x_{3}\in x_{2}} at nesting depth 2. Its subscript is 3 because n + q = 1 + 2 = 3. {\displaystyle n+q=1+2=3.} If two quantifier scopes are at the same nesting depth, they are either identical or disjoint. The two occurrences of x 3 {\displaystyle x_{3}} are in disjoint quantifier scopes, so they do not interact with each other.

Example 3: Transforming Y 1 ⊆ Y 2 . {\displaystyle Y_{1}\subseteq Y_{2}.} Y 1 ⊆ Y 2 ⟺ ∀ z 1 ( z 1 ∈ Y 1 ⟹ z 1 ∈ Y 2 ) (rule 1) {\displaystyle Y_{1}\subseteq Y_{2}\iff \forall z_{1}(z_{1}\in Y_{1}\implies z_{1}\in Y_{2})\quad {\text{(rule 1)}}}

Example 4: Transforming x 1 ∩ Y 1 ∈ x 2 . {\displaystyle x_{1}\cap Y_{1}\in x_{2}.} x 1 ∩ Y 1 ∈ x 2 ⟺ ∃ z 1 [ z 1 = x 1 ∩ Y 1 ∧ z 1 ∈ x 2 ] (rule 3b) ⟺ ∃ z 1 [ ∀ z 2 ( z 2 ∈ z 1 ⟺ z 2 ∈ x 1 ∩ Y 1 ) ∧ z 1 ∈ x 2 ] (rule 4) ⟺ ∃ z 1 [ ∀ z 2 ( z 2 ∈ z 1 ⟺ z 2 ∈ x 1 ∧ z 2 ∈ Y 1 ) ∧ z 1 ∈ x 2 ] (rule 3a) {\displaystyle {\begin{alignedat}{2}x_{1}\cap Y_{1}\in x_{2}&\iff \exists z_{1}[z_{1}=x_{1}\cap Y_{1}\,\land \,z_{1}\in x_{2}]&&{\text{(rule 3b)}}\\&\iff \exists z_{1}[\forall z_{2}(z_{2}\in z_{1}\iff z_{2}\in x_{1}\cap Y_{1})\,\land \,z_{1}\in x_{2}]&&{\text{(rule 4)}}\\&\iff \exists z_{1}[\forall z_{2}(z_{2}\in z_{1}\iff z_{2}\in x_{1}\land z_{2}\in Y_{1})\,\land \,z_{1}\in x_{2}]\quad &&{\text{(rule 3a)}}\\\end{alignedat}}} This example illustrates how the transformation rules work together to eliminate an operation.

History

Von Neumann published his ideas about set theory in 1925 and later in 1928. He focused on two main types of objects: functions and arguments. Some objects could be both, called argument-functions. Functions relate to collections called classes, while argument-functions relate to sets.

Von Neumann was influenced by earlier mathematicians like Georg Cantor, Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem. He solved important problems in Zermelo's set theory, such as developing a theory of ordinal numbers and creating ways to identify very large collections. Later, in 1929, von Neumann updated his system of axioms, which eventually helped form what is now called von Neumann–Bernays–Gödel set theory (NBG). Paul Bernays and Kurt Gödel further refined these ideas, with Gödel creating the final version of NBG in 1940.

NBG, ZFC, and MK

NBG, or von Neumann–Bernays–Gödel set theory, is a way to talk about bigger collections of things called "classes" than the sets we usually work with in ZFC (Zermelo–Fraenkel–choice set theory). Even though NBG can discuss classes, it still agrees with ZFC on everything that involves just sets. This means NBG doesn’t prove any new facts about sets that ZFC can’t prove itself.

Because NBG and ZFC agree on sets, they are equally reliable — if one can prove something impossible, like a contradiction, the other can too. This makes them "equiconsistent." NBG can sometimes prove things in a simpler way than ZFC, but both systems are equally strong in what they can achieve about sets.

Morse–Kelley set theory is another set theory that is even stronger than NBG. It can prove things that NBG cannot, including the consistency of NBG itself.

Category theory

The von Neumann–Bernays–Gödel set theory (NBG) helps us talk about very big collections of things without causing problems in math. In category theory, we sometimes talk about "large categories," which are groups of objects and connections between them that are too big to be normal sets. NBG allows us to talk about these large categories safely.

NBG does not let us make a category that includes all categories, because some of these categories are too big to be part of anything else. To handle this, mathematicians use something called a "conglomerate," which is a collection of these big categories. This way, they can still talk about the idea of a category of all categories in a controlled way.