$\text{Ord}$ is Mahlo
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 $\text{Ord}$ 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
- $\text{Ord}$ is not Mahlo (the class $C$ itself witnesses it), thus the Vopěnka principle fails,
- but the extension adds no new sets, thus the Vopěnka scheme still holds and $\text{Ord}$ remains definably Mahlo.
- The forcing preserves $\text{GBC}$.
- It is relatively consistent that $\text{GBC}$ and the generic Vopěnka principle holds, yet $\text{Ord}$ is not Mahlo:
- If $0^♯$ (zero sharp) exists in $V$, then there is a class-forcing notion $\mathbb{P}$ definable in the constructible universe $L$, such that in any $L$-generic extension $L[C]$ by this forcing, $\text{GBC}$ and the generic Vopěnka principle hold, yet $\text{Ord}$ is not Mahlo.
- Proof includes a lemma stating: For any ordinal $δ$ and any natural number (of the meta-theory — this lemma is a scheme) $n$, if $D_{δ,n} ⊂ \mathbb{P}$ is the collection of conditions $c$ for which there is an ordinal $θ$ such that
- $L_θ ≺_{Σ_n} L$,
- $c ∩ θ$ is $L_θ$-generic for $\mathbb{P}^{L_θ}$ and
- in some forcing extension of $L$, there is an elementary embedding
- $j : ⟨ L_θ , ∈, c ∩ θ ⟩ → ⟨ L_θ , ∈, c ∩ θ ⟩$
- with critical point above $δ$,
- then $D_{δ,n}$ is a definable dense subclass of $\mathbb{P}$ in $L$.
- Proof includes a lemma stating: For any ordinal $δ$ and any natural number (of the meta-theory — this lemma is a scheme) $n$, if $D_{δ,n} ⊂ \mathbb{P}$ is the collection of conditions $c$ for which there is an ordinal $θ$ such that
- If $0^♯$ (zero sharp) exists in $V$, then there is a class-forcing notion $\mathbb{P}$ definable in the constructible universe $L$, such that in any $L$-generic extension $L[C]$ by this forcing, $\text{GBC}$ and the generic Vopěnka principle hold, yet $\text{Ord}$ is not Mahlo.
- It is relatively consistent that $\text{ZFC}$ and the generic Vopěnka scheme holds, yet $\text{Ord}$ is not definably Mahlo and not even $∆_2$-Mahlo:
- If $0^♯$ exists in $V$, then there is a definable class-forcing notion in $L$, such that in the corresponding $L$-generic extension, $\text{GBC}$ holds, the generic Vopěnka scheme holds, but $\text{Ord}$ is not definably Mahlo, because there is a $∆_2$-definable club class avoiding the regular cardinals.
- 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