Von Neumann–Bernays–Gödel set theory

Von Neumann–Bernays–Gödel set theory, or NBG, is a special kind of axiomatic set theory. It builds on another theory called ZFC. One special part of NBG is the idea of a "class." A class is a very big group of sets that follows a certain rule.

NBG helps mathematicians talk about huge collections, like all sets or all ordinals. These collections are bigger than any single set. This helps solve problems with infinite groups. NBG has an important rule called the class existence theorem. It says that if you have a rule about sets, there is a class of all sets that follow that rule.

The idea of classes in set theory started with John von Neumann in 1925. Later, Paul Bernays and Kurt Gödel helped make the theory simpler and easier to understand. Their work still matters today for learning about the basic parts of mathematics.

Classes in set theory

Classes help organize sets in a special way. They let us talk about very big groups, like all sets together, that are too large to be sets themselves. This helps avoid problems in math.

One important use of classes is to handle tricky situations in set theory, like the class of all ordinals. By calling it a class instead of a set, we avoid problems. Classes also help mathematicians build important structures, like the constructible universe.

Axiomatization of NBG

NBG set theory talks about two kinds of objects: classes and sets. Every set is also a class. There are different ways to organize these ideas, but they all lead to the same results. One way uses special logic to tell classes and sets apart. Another way uses rules to decide what is a class and what is a set. These differences mostly change how we write things, not what we can prove.

The theory has special rules for how classes and sets work together. For example, it has the axiom of extensionality. This rule says that if two classes have the same members, they are the same class. There are also rules for making new classes from old ones, like combining classes or finding what they share. These tools help mathematicians handle big groups of sets in an organized 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 shared his ideas about set theory in 1925 and again in 1928. He talked about two main types of objects: functions and arguments. Some objects could be both types, called argument-functions. Functions are related to groups called classes, while argument-functions are related to sets.

Von Neumann was inspired by earlier mathematicians like Georg Cantor, Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem. He solved big problems in Zermelo's set theory. This included making a theory about ordinal numbers and finding ways to describe very large groups. In 1929, von Neumann changed his rules, or axioms. These changes helped create what we now call von Neumann–Bernays–Gödel set theory (NBG). Paul Bernays and Kurt Gödel improved these ideas more. Gödel made 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 groups of things called "classes." These are larger 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 when it comes to sets. This means NBG doesn’t prove any new facts about sets that ZFC can’t prove.

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.

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." These 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. Some categories are too big to be part of anything else. To handle this, mathematicians use something called a "conglomerate." This 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.