Second-order set theories
In construction.
$\text{ZFC}$, the usual first-order axiomatic set theory, can only manipulate sets, and cannot formalize the notion of a proper class (e.g. the class of all sets, $V$), so when one needs to manipulate proper class objects, it is tempting to switch to a second-order logic form of $\text{ZFC}$. However, because many useful theorems such as Gödel's Completeness Theorem does not apply to second-order logic theories, it is more convenient to use first-order two-sorted axiomatic theories with two types of variables, one for sets and one for classes. Two such "false" second-order theories, $\text{NBG}$ and $\text{MK}$, are the most widely used extensions of $\text{ZFC}$.
Throughout this article, "first-order (set theory) formula" means a formula in the language of $\text{ZFC}$, eventually with class parameters, but only set quantifiers. "second-order (set theory) formula" means a formula in the language of $\text{NBC}$ and $\text{MK}$, i.e. it can contain class quantifiers.
Von Neumann-Bernays-Gödel set theory (commonly abbreviated as $\text{NBG}$ or $\text{GCB}$) is a conservative extension of $\text{ZFC}$ - that is, it proves the same first-order sentences as $\text{ZFC}$. It is equiconsistent with $\text{ZFC}$. However, unlike $\text{ZFC}$ and $\text{MK}$, it is possible to finitely axiomatize $\text{NBG}$. It was named after John von Neumann, Paul Bernays and Kurt Gödel.
Morse-Kelley set theory (commonly abbreviated as $\text{MK}$ or $\text{KM}$) is an extension of $\text{NBG}$ which is stronger than $\text{ZFC}$ and $\text{NBG}$ in consistency strength. It was named after John L. Kelley and Anthony Morse. It only differs from $\text{NBG}$ by a single axiom-schema. It is not finitely axiomatizable.
It is possible to turn both theories into first-order axiomatic set theories with only class variables, a class $X$ is called a set (abbreviated $\text{M}X$) iff $\exists Y(X\in Y)$. One can also define $\text{M}X\equiv X\in V$ with $V$ a symbol of the language representing the universal class containing all sets. That is, a set is a class member of another class.
Contents
Axioms of $\text{NBG}$ and $\text{MK}$
We will be using capital letters to denote classes and lowercase letters to denote sets. The following axioms are common to both $\text{NBG}$ and $\text{MK}$:
- Extensionality: two classes are equal iff they have the same elements: $\forall X\forall Y(X=Y\iff\forall z(z\in X\iff z\in Y))$.
- Regularity: every class is disjoint from one of its elements: $\forall X(\exists a(a\in X)\implies\exists x(x\in X\land\forall y(y\in x\implies y\not\in X)))$.
- Infinity: there exists an infinite set. For example, $\exists x(\exists a(a\in x)\land\forall y(y\in x\implies\exists z(y\in z\land z\in x))$.
- Union: the union of the elements of a set is a set : $\forall x\exists y\forall z(z\in y\iff\exists w(z\in w\land w\in x))$.
- Pairing: the pair of two sets is itself a set: $\forall x\forall y\exists z\forall w(w\in z\iff(w=x\lor w=y))$.
- Powerset: the powerset of a set is a set: $\forall x\exists y\forall z(z\in y\iff\forall w(w\in z\implies w\in x))$.
- Limitation of Size: a class $X$ is proper if and only if there is bijection between $X$ and the universal class $V$.
The axiom of limitation of size implies the axiom of global choice ("there is a well-ordering of the universe") and $\text{ZFC}$'s axiom of replacement. Using the one-sorted version of those theories, pairing becomes $\forall X\forall Y(\text{M}X\land\text{M}Y\Rightarrow\exists Z(\text{M}Z\land X\in Z\land Y\in Z))$. The other axioms are modified similarly.
$\text{NBG}$ the the theory obtained by adding the following axiom-scheme:
- Class comprehension: for every first-order formula $\varphi(x)$ with a free variables $x$ and class parameters, there exists a class containing all sets $x$ such that $\varphi(x)$.
Not that the created class only contains sets, in particular there is no class of all classes.
$\text{MK}$ is obtained by removing the "first-order" restriction in class comprehension, i.e. second-order formulas are now allowed.
Finitely axiomatizing $\text{NBG}$
$\text{NBG}$ can be finitely axiomatized as its class comprehension axiom can be replaced by the following set of axioms: For every classes $A$ and $B$,
- The complement $V\setminus A=\{x:x\not\in A\}$ exists.
- The intersection $A\cap B=\{x:x\in A\land x\in B\}$ exists.
- The range $\{y:\exists x((x,y)\in A\}$ exists, using $(x,y)=\{\{x\},\{x,y\}\}$.
- The product $A\times B=\{(a,b):a\in A\land b\in B\}$ exists.
- The classes $\{(x,y):x\in y\}$ and $\{(x,y):x=y\}$ exist.
- The following classes exist: $\{(b,a):(a,b)\in A\}$, $\{(b,(a,c)):(a,(b,c))\in A\}$.
- The following classes exist: $\{((a,b),c):(a,(b,c))\in A\}$, $\{(d,(a,(b,c))):(d,((a,b),c))\in A\}$.
Models of $\text{MK}$
In consistency strength, $\text{MK}$ is stronger than $\text{ZFC}$ and weaker than the existence of an inaccessible cardinal. It directly implies the consistency of $\text{ZFC}$ plus "there is a proper class of worldly cardinals stationary in $\text{Ord}$". However, if a cardinal $\kappa$ is inaccessible then $V_{\kappa+1}\models\text{MK}$, also $\text{def}(V_\kappa)$ satisfies $\text{NBG}$ minus global choice (i.e. replacing limitation of size by $\text{ZFC}$'s axiom of replacement).
Because it uses classes, set models of $\text{MK}$ are generally taken to be the powerset of some model of $\text{ZFC}$. For this reason, a large cardinal axiom with $V_\kappa$ having some elementary property can be strengthened by instead using $V_{\kappa+1}$. When doing this with $\Pi_m^0$-indescribability, one achieves $\Pi_m^1$-indescribability (which is considerably stronger). When doing this with 0-extendibility (which is equiconsistent with $\text{ZFC}$), one achieves 1-extendibility.
Between $\text{NBG}$ and $\text{MK}$
Class forcing theorem and equivalents
Class determinacy of open games
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