# Difference between revisions of "Second-order"

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+ | '''Morse Kelley''' (commonly abbreviated as ''$\text{MK}$'' or ''$\text{KM}$'') is a first-order countably infinite axiomatic theory which is stronger than [[ZFC|$\text{ZFC}$]] 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 $MX$. | |

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== Axioms == | == Axioms == | ||

− | The axioms of MK are exactly those of [[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 $n$: | + | The axioms of $\text{MK}$ are exactly those of [[NBG|$\text{NBG}$]] 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 $n$: |

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

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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. | 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 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). | + | 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 MK == | + | == Models of $\text{MK}$ == |

− | In consistency strength, MK is stronger than [[ZFC]] and weaker than [[ | + | In consistency strength, $\text{MK}$ is stronger than [[ZFC|$\text{ZFC}$]] 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 MK are generally taken to be the powerset of some model of 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 [[indescribable|$\Pi_m^0$-indescribability]], one achieves [[indescribable|$\Pi_m^1$-indescribability]] (which is considerably stronger). When doing this with [[extendible|$0$-extendibility]] (which is equiconsistent with ZFC), one achieves [[extendible|$1$-extendibility]] (which is so much stronger that it actually implies the consistency of a [[supercompact]] cardinal). | + | 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 [[indescribable|$\Pi_m^0$-indescribability]], one achieves [[indescribable|$\Pi_m^1$-indescribability]] (which is considerably stronger). When doing this with [[extendible|$0$-extendibility]] (which is equiconsistent with $\text{ZFC}$), one achieves [[extendible|$1$-extendibility]] (which is so much stronger that it actually implies the consistency of a [[supercompact]] cardinal). |

## Revision as of 14:27, 11 November 2017

**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).