Difference between revisions of "ORD is Mahlo"

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The assertion ''Ord is Mahlo'' is the scheme expressing that the proper class [[REG]] consisting of all regular cardinals is a [[stationary]] proper class, meaning that it has elements from every definable (with parameters) [[closed unbounded]] proper class of ordinals. In other words, the scheme asserts for every formula $\varphi$, that if for some parameter $z$ the class $\{\alpha\mid \varphi(\alpha,z)\}$ is a closed unbounded class of ordinals, then it contains a regular cardinal.
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{{DISPLAYTITLE: $\text{Ord}$ is Mahlo}}
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The assertion ''$\text{Ord}$ is Mahlo'' is the scheme expressing that the proper class [[REG]] consisting of all regular cardinals is a [[stationary]] proper class, meaning that it has elements from every definable (with parameters) [[closed unbounded]] proper class of ordinals. In other words, the scheme asserts for every formula $\varphi$, that if for some parameter $z$ the class $\{\alpha\mid \varphi(\alpha,z)\}$ is a closed unbounded class of ordinals, then it contains a regular cardinal.
  
* If $\kappa$ is [[Mahlo]], then $V_\kappa\models\text{ORD is Mahlo}$.  
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* If $\kappa$ is [[Mahlo]], then $V_\kappa\models\text{Ord is Mahlo}$.  
* Consequently, the existence of a Mahlo cardinal implies the consistency of Ord is Mahlo, and the two notions are not equivalent.
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* Consequently, the existence of a Mahlo cardinal implies the consistency of $\text{Ord}$ is Mahlo, and the two notions are not equivalent.
 
* Moreoever, since the ORD is Mahlo scheme is expressible as a first-order theory, it follows that whenever $V_\gamma\prec V_\kappa$, then also $V_\gamma$ satisfies the Levy scheme.  
 
* Moreoever, since the ORD is Mahlo scheme is expressible as a first-order theory, it follows that whenever $V_\gamma\prec V_\kappa$, then also $V_\gamma$ satisfies the Levy scheme.  
 
* Consequently, if there is a Mahlo cardinal, then there is a club of cardinals $\gamma\lt\kappa$ for which $V_\gamma\models\text{Ord is Mahlo}$.  
 
* Consequently, if there is a Mahlo cardinal, then there is a club of cardinals $\gamma\lt\kappa$ for which $V_\gamma\models\text{Ord is Mahlo}$.  
  
A simple compactness argument establishes that Ord is Mahlo is equiconsistent over ZFC with the existence of an [[inaccessible reflecting cardinal]]. On the one hand, if $\kappa$ is an inaccessible reflecting cardinal, then since $V_\kappa\prec V$ it follows that any class club definable in $V$ with parameters below $\kappa$ will be unbounded in $\kappa$ and hence contain $\kappa$ as an element and consequently contain an inaccessible cardinal. On the other hand, if Ord is Mahlo is consistent, then every finite fragment of the theory asserting that $\kappa$ is an inaccessible reflecting cardinal (which is after all asserted as a scheme) is consistent, and hence by compactness the whole theory is consistent.
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A simple compactness argument establishes that $\text{Ord}$ is Mahlo is equiconsistent over $\text{ZFC}$ with the existence of an [[inaccessible reflecting cardinal]]. On the one hand, if $\kappa$ is an inaccessible reflecting cardinal, then since $V_\kappa\prec V$ it follows that any class club definable in $V$ with parameters below $\kappa$ will be unbounded in $\kappa$ and hence contain $\kappa$ as an element and consequently contain an inaccessible cardinal. On the other hand, if $\text{Ord}$ is Mahlo is consistent, then every finite fragment of the theory asserting that $\kappa$ is an inaccessible reflecting cardinal (which is after all asserted as a scheme) is consistent, and hence by compactness the whole theory is consistent.

Revision as of 14:28, 11 November 2017

The assertion $\text{Ord}$ is Mahlo is the scheme expressing that the proper class REG consisting of all regular cardinals is a stationary proper class, meaning that it has elements from every definable (with parameters) closed unbounded proper class of ordinals. In other words, the scheme asserts for every formula $\varphi$, that if for some parameter $z$ the class $\{\alpha\mid \varphi(\alpha,z)\}$ is a closed unbounded class of ordinals, then it contains a regular cardinal.

  • If $\kappa$ is Mahlo, then $V_\kappa\models\text{Ord is Mahlo}$.
  • Consequently, the existence of a Mahlo cardinal implies the consistency of $\text{Ord}$ is Mahlo, and the two notions are not equivalent.
  • Moreoever, since the ORD is Mahlo scheme is expressible as a first-order theory, it follows that whenever $V_\gamma\prec V_\kappa$, then also $V_\gamma$ satisfies the Levy scheme.
  • Consequently, if there is a Mahlo cardinal, then there is a club of cardinals $\gamma\lt\kappa$ for which $V_\gamma\models\text{Ord is Mahlo}$.

A simple compactness argument establishes that $\text{Ord}$ is Mahlo is equiconsistent over $\text{ZFC}$ with the existence of an inaccessible reflecting cardinal. On the one hand, if $\kappa$ is an inaccessible reflecting cardinal, then since $V_\kappa\prec V$ it follows that any class club definable in $V$ with parameters below $\kappa$ will be unbounded in $\kappa$ and hence contain $\kappa$ as an element and consequently contain an inaccessible cardinal. On the other hand, if $\text{Ord}$ is Mahlo is consistent, then every finite fragment of the theory asserting that $\kappa$ is an inaccessible reflecting cardinal (which is after all asserted as a scheme) is consistent, and hence by compactness the whole theory is consistent.