Difference between revisions of "ORD is Mahlo"

From Cantor's Attic
Jump to: navigation, search
Line 1: Line 1:
 +
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.
  
The ''Levy scheme'', also known as 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.
+
* 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.
 +
* 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 $V_\gamma\models\text{ORD is Mahlo}$.  
  
* If $\kappa$ is [[Mahlo]], then $V_\kappa$ is a model of the Levy scheme.
+
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.
* Consequently, the existence of a Mahlo cardinal implies the consistency of the Levy scheme, and the two notions are not equivalent.
+
* Moreoever, since the Levy 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 $V_\gamma$ satisfies the Levy scheme.
+
 
+
==Maximality Principle==
+
 
+
The Levy scheme is equiconsistent with the boldface maximality principle $\text{MP}(\mathbb{R})$, which asserts of any statement $\varphi(r)$ with parameter $r\in\mathbb{R}$ that if $\varphi(r)$ is forceable in such a way that it remains true in all subsequent forcing extensions, then it is already true; in short, $\text{MP}(\mathbb{R})$ asserts that every possibly necessary statement with real parameters is already true. Hamkins showed that if the Levy scheme holds, then there is a forcing extension with $\text{MP}(\mathbb{R})$, and conversely, whenever $\text{MP}(\mathbb{R})$ holds, then there is an inner model of the Levy scheme.
+
 
+
Note: this needs fixing. I should add an entry for inaccessible reflecting cardinal. that is what is used in the MP argument, and it is equiconsistent with ORD is Mahlo.
+

Revision as of 18:43, 29 December 2011

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.

  • 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.
  • 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 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.