The assertion *$\text{Ord}$ is Mahlo* is the scheme expressing that the proper class $REG$ consisting of all [[Cardinal|regular cardinals]] is a [[Club sets and stationary sets|stationary]] proper class, meaning that it has elements from every definable (with parameters) [[Club sets and stationary sets|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 <a href="Inaccessible_reflecting_cardinal" class="mw-redirect" title="Inaccessible reflecting cardinal">inaccessible reflecting cardinal</a>. 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](Uplifting.md "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](Reflecting.md "Reflecting") cardinal and consequently also a transitive model of ZFC plus $\text{Ord}$ is Mahlo. {% cite Hamkins2014 %}