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.
  
'''Morse Kelley''' (commonly abbreviated as ''MK'' or ''KM'') is a first-order countably infinite axiomatic theory which is stronger than [[ZFC|$ZFC$]] in consistency strength. It was named after John L. Kelley and Anthony Morse. It is very similar to the second-order form of [[ZFC]] (known as [[NBG]]), although slightly strengthened with a modification of class comprehension.
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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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In 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 $n$:
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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).  
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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 ==
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== Models of $\text{MK}$ ==
  
In consistency strength, MK is stronger than [[ZFC]] and weaker than [[Positive set theory]]. It directly implies the consistency of ZFC. However, a cardinal $\kappa$ is worldly iff $V_{\kappa+1}\models\mathrm{MK}$.  
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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).
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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).