Mahlo cardinal

A Mahlo cardinal is a certain type of large cardinal used in the study of reflection principles and consistency strength. These are much stronger than inaccessible cardinals, 1-inaccessible cardinals, hyper-inaccessible cardinals, and more. However, like with inaccessible cardinals, there are two primary types.

Weakly Mahlo

Analogously to how weakly inaccessible cardinals are more popular than strongly inaccessible cardinals in apeirological circles, but less popular in the literature, weakly Mahlo cardinals are more popular than strongly Mahlo cardinals in apeirological circles, but less popular in the literature. This is essentially due to the fact that weakly Mahlo cardinals are defined in terms of weakly inaccessibles, and strongly Mahlo cardinals are defined in terms of strongly inaccessibles. Essentially, we say a cardinal \( \kappa \) is weakly Mahlo if every club \( C \subseteq \kappa \) contains a regular cardinal.

First, you can see that any weakly Mahlo cardinal is regular. Assume \( \kappa \) is weakly Mahlo but not regular. Let \( \lambda_i \) be a sequence of cardinals with limit \( \kappa \) and length \(\eta < \kappa \). Let \( C^* = \{\lambda_i+1: i < \eta\} \) and let \( C \) be the closure of \( C^* \). You can verify that \( C^* \) is unbounded, and thus \( C \) is club, however \( C \) doesn't contain any regular cardinals. Contradiction! Similarly, you can show that any weakly Mahlo cardinal is a limit cardinal. Assume \( \kappa = \lambda^+ \) for some \( \lambda \). Let \( C = \{\lambda + \eta: 0 < \eta < \kappa \} \). Then \( C \) is club, however \( C \) doesn't contain any regular cardinals. Contradiction! You can continue on to show that the least inaccessible cardinal isn't weakly Mahlo, and any weakly Mahlo cardinal has to be a limit of weakly inaccessibles: if \( \kappa \) is the least weakly inaccessible, then the set of limit cardinals below \( \kappa \) is club but doesn't contain any regulars, and similarly if \( \kappa \) is the next weakly inaccessible after \( \lambda \), then the set of limit cardinals in-between \( \lambda \) and \( \kappa \) is club but doesn't contain any regulars.

Continuing on this way, a weakly Mahlo cardinal can be shown to be weakly hyper-inaccessible. There's a convenient characterisation of weakly Mahlo which explains why they're so large. Recall that \( \omega_1^{\mathrm{CK}} \) is the least ordinal \( \alpha > \omega \) so that, for any \( \Delta_1(L_\alpha) \)-definable function \( f: \alpha \to \alpha \), there is a closure point of \( f \) below \( \alpha \). By weakening the definability condition, you get gap ordinals, and removing it altogether grants you a condition equivalent to regularity! Then, being Mahlo is obtained by adding the condition that such a closure point must be regular: in other words, \( \kappa \) is Mahlo iff. for any function \( f: \kappa \to \kappa \), there is a regular closure point of \( f \) below \( \kappa \). You can see that this is equivalent to Mahloness by noting that the set of closure points of any function is club, and any club is equal to the set of closure points of some function.

Weakly Mahlos see some proof-theoretical usage in ordinal-analysis of extensions of Kripke-Platek set theory, such as KPM, since, like how \( \Omega \) acts as a "diagonalizer" over the Veblen hierarchy, warranting its use in OCFs, a Mahlo cardinal can be thought to act as a "diagonalizer" over the inaccessible hierarchy.

Strongly Mahlo

Strong Mahloness is obtained by replacing "contains a regular cardinal" with "contains a strongly inaccessible cardinal". Clearly any strongly Mahlo cardinal is weakly Mahlo, since every strongly inaccessible cardinal is regular, and the results above can be generalized to show any strongly Mahlo cardinal is strongly hyper-inaccessible. Like the situation between weakly and strongly inaccessible cardinals, \( \mathrm{GCH} \) implies weakly and strongly Mahlo cardinals are the same, while other axioms imply that \( 2^{\aleph_0} \) can be weakly Mahlo.

Ord is Mahlo

"Ord is Mahlo" is an assertion that, as one can likely guess, asserts that every function \( f: \mathrm{Ord} \to \mathrm{Ord} \) has a strongly inaccessible closure point. Clearly, "Ord is Mahlo" implies that there is a proper class of inaccessible cardinals, 1-inaccessible cardinals, and more. However, if \( \kappa \) is Mahlo, then \( V_\kappa \) satisfies "Ord is Mahlo", and thus "Ord is Mahlo" has consistency strength squashed between the inaccessible hierarchy and strongly Mahlo cardinals.

Ord is Mahlo has interesting consistency strength, as we've mentioned. Recall from here that a cardinal \( \kappa \) is sound if \( V_\kappa \) is a full elementary substructure of \( V \). Such cardinals are massive, but their existence is provable in \( \mathrm{ZFC} \), due to the reflection principle. In particular, for all \(n\), we have a club of cardinals which are \(\Sigma_n\)-sound, and thus their intersection is also club. Meanwhile, say a cardinal \( \kappa \) is totally reflecting if it is sound and strongly inaccessible. Such cardinals are hyper-inaccessible and larger than virtually any other large cardinal axiom size-wise other than possibly stationary superhuges or Reinhardt cardinals. However, their consistency strength is not particularly high: it turns out that Ord is Mahlo has the same consistency strength as the existence of a totally reflecting cardinal, which shows that slight modifications of reflection principles can give large consistency strength.

Furthermore, let \( \mathrm{MP}(\mathbb{R}) \), the maximality principle for the real numbers be the following statement: "assume \( r \) is a real number and \( \varphi \) is a formula. Then if there is a forcing extension \( V[G] \) so that \( \varphi(r) \) and \( \varphi(r) \) persists, i.e. remains true in all subsequent extensions \( V[G][H] \), then \( \varphi(r) \) is already true in the universe". Essentially, the theory of the real numbers is already maximal, and it's not possible to for a statement with real numbers to be false but, in extensions of the universe, possibly necessary. The statement \( \mathrm{MP}(\mathbb{R}) \) has less consistency strength than \( \mathrm{MP}(V) \), where \( r \) is an arbitrary set, and is actually equiconsistent with Ord is Mahlo.