Second-order

From Cantor's Attic
Revision as of 14:27, 11 November 2017 by Wabb2t (Talk | contribs)

Jump to: navigation, search

Morse Kelley (commonly abbreviated as $\text{MK}$ or $\text{KM}$) is a first-order countably infinite axiomatic theory which is stronger than $\text{ZFC}$ in consistency strength. It was named after John L. Kelley and Anthony Morse. It is very similar to the second-order form of [[ZFC]|$\text{ZFC}$] (known as [[$\text{NBG}$]]), although slightly strengthened with a modification of class comprehension.

In $\text{MK}$, the sets are precisely the classes which are in another class. $X$ is a set iff $\exists W(X\in W)$. This is often abbreviated $MX$.

Axioms

The axioms of $\text{MK}$ are exactly those of $\text{NBG}$ except with a modified Axiom Schema of Class Comprehension. This modified version allows classes to be defined using other classes, as long as none of them are that class. Specifically, for any $\phi$ and any $n$:

$$\forall W_1\forall W_2...\forall W_n(\neg\forall x(\phi(x,W_1...W_n)\Leftrightarrow x\in W_1)\land \neg\forall x(\phi(x,W_1...W_n)\Leftrightarrow x\in W_2)...\neg\forall x(\phi(x,W_1...W_n)\Leftrightarrow x\in W_n)$$ $$\rightarrow\exists Y(\forall x(\phi(x,W_1...W_n)\Leftrightarrow x\in Y))$$

In other words, for any formula $\phi$ with variables $x$ and $W_1,...,W_n$, there is a class $Y=\{x:\phi(x,W_1...W_n)\}$ as long as no $W_m=Y$ and $Y$ does not occur in $\phi$. This is because, if we allowed $Y$ to occur in $\phi$ or if we allowed $Y$ to be passed as a parameter for $\phi$, then we could create a sentence $\phi$ such that $\phi(x,W_1,...,W_n)\iff x\not\in Y$, and then we would have $\forall x(x\in Y\iff x\not\in Y)$, a contradiction. This is generally considered to be a useful definition for a class, allowing almost all known "useful" classes to be created.

The other most important axiom (which is also in $\text{NBG}$) is the Axiom of Limitation of Size, which asserts that a class is not a set iff it has a bijection onto $V$. This is a particularly strong axiom, implying the Axiom of Choice. Furthermore, it implies that every class can be well-ordered (known as the axiom of global choice).

Models of $\text{MK}$

In consistency strength, $\text{MK}$ is stronger than $\text{ZFC}$ and weaker than the positive set theory $\text{GPK}^{+}_\infty$. It directly implies the consistency of $\text{ZFC}$. However, if a cardinal $\kappa$ is worldly then $V_{\kappa+1}\models\text{MK}$.

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 (which is so much stronger that it actually implies the consistency of a supercompact cardinal).