The assertion *$\text{Ord}$ is Mahlo* is the scheme expressing that the
proper class
<a href="index.php?title=REG&action=edit&redlink=1" class="new" title="REG (page does not exist)">REG</a>
consisting of all regular cardinals is a
[[Club sets and stationary sets|stationary]]
proper class, meaning that it has elements from every definable (with
parameters)
<a href="index.php?title=Closed_unbounded&action=edit&redlink=1" class="new" title="Closed unbounded (page does not exist)">closed unbounded</a>
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](Mahlo.md "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 %}