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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.

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 $\text{MK}$:

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:

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 $B$,

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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