Difference between revisions of "ORD is Mahlo"
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Relation to the [[Vopenka|Vopěnka principle]]:<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite> | Relation to the [[Vopenka|Vopěnka principle]]:<cite>GitmanHamkins2018:GenericVopenkaPrincipleNotMahlo</cite> | ||
* The [[Vopenka|Vopěnka principle]] implies that $Ord$ is Mahlo: every club class contains a regular cardinal and indeed, an [[extendible]] cardinal and more. | * The [[Vopenka|Vopěnka principle]] implies that $Ord$ is Mahlo: every club class contains a regular cardinal and indeed, an [[extendible]] cardinal and more. | ||
+ | * If the Vopěnka scheme holds, then there is a class-forcing extension $V[C]$ where it continues to hold, yet in which the Vopěnka principle fails and Ord is not Mahlo, although it remains definably Mahlo. | ||
* It is relatively consistent that GBC and the generic Vopěnka principle holds, yet $Ord$ is not Mahlo. | * It is relatively consistent that GBC and the generic Vopěnka principle holds, yet $Ord$ is not Mahlo. | ||
* It is relatively consistent that ZFC and the generic Vopěnka scheme holds, yet $Ord$ is not definably Mahlo and not even $∆_2$-Mahlo. | * It is relatively consistent that ZFC and the generic Vopěnka scheme holds, yet $Ord$ is not definably Mahlo and not even $∆_2$-Mahlo. |
Revision as of 10:43, 10 October 2019
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.
If there is a pseudo uplifting (proof in that article) cardinal, or indeed, merely a pseudo $0$-uplifting cardinal, then there is a transitive set model of ZFC with a reflecting cardinal and consequently also a transitive model of ZFC plus $\text{Ord}$ is Mahlo.[1]
Relation to the Vopěnka principle:[2]
- The Vopěnka principle implies that $Ord$ is Mahlo: every club class contains a regular cardinal and indeed, an extendible cardinal and more.
- If the Vopěnka scheme holds, then there is a class-forcing extension $V[C]$ where it continues to hold, yet in which the Vopěnka principle fails and Ord is not Mahlo, although it remains definably Mahlo.
- It is relatively consistent that GBC and the generic Vopěnka principle holds, yet $Ord$ is not Mahlo.
- It is relatively consistent that ZFC and the generic Vopěnka scheme holds, yet $Ord$ is not definably Mahlo and not even $∆_2$-Mahlo.
- In such a model, there can be no $Σ_2$-reflecting cardinals and therefore also no remarkable cardinals.
References
- Hamkins, Joel David and Johnstone, Thomas A. Resurrection axioms and uplifting cardinals. , 2014. www arχiv bibtex
- Gitman, Victoria and Hamkins, Joel David. A model of the generic Vopěnka principle in which the ordinals are not Mahlo. , 2018. arχiv bibtex