Mahlo cardinal

In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by Paul Mahlo (1911, 1912, 1913). As with all large cardinals, none of these varieties of Mahlo cardinals can be proved to exist by ZFC (assuming ZFC is consistent).

A cardinal number κ is called Mahlo if κ is inaccessible and the set U = {λ < κ: λ is inaccessible} is stationary in κ.

A cardinal κ is called weakly Mahlo if κ is weakly inaccessible and the set of weakly inaccessible cardinals less than κ is stationary in κ.

Minimal condition sufficient for a Mahlo cardinal

If κ is a limit ordinal and the set of regular ordinals less than κ is stationary in κ, then κ is weakly Mahlo.

The main difficulty in proving this is to show that κ is regular. We will suppose that it is not regular and construct a club set which gives us a μ such that:

μ = cf(μ) < cf(κ) < μ < κ which is a contradiction.

If κ were not regular, then cf(κ) < κ. We could choose a strictly increasing and continuous cf(κ)-sequence which begins with cf(κ)+1 and has κ as its limit. The limits of that sequence would be club in κ. So there must be a regular μ among those limits. So μ is a limit of an initial subsequence of the cf(κ)-sequence. Thus its cofinality is less than the cofinality of κ and greater than it at the same time; which is a contradiction. Thus the assumption that κ is not regular must be false, i.e. κ is regular.

No stationary set can exist below $\aleph_0$ with the required property because {2,3,4,...} is club in ω but contains no regular ordinals; so κ is uncountable. And it is a regular limit of regular cardinals; so it is weakly inaccessible. Then one uses the set of uncountable limit cardinals below κ as a club set to show that the stationary set may be assumed to consist of weak inaccessibles.

If κ is weakly Mahlo and also a strong limit, then κ is Mahlo.

κ is weakly inaccessible and a strong limit, so it is strongly inaccessible.

We show that the set of uncountable strong limit cardinals below κ is club in κ. Let μ₀ be the larger of the threshold and ω₁. For each finite n, let μ_n+1 = 2^μ_n which is less than κ because it is a strong limit cardinal. Then their limit is a strong limit cardinal and is less than κ by its regularity. The limits of uncountable strong limit cardinals are also uncountable strong limit cardinals. So the set of them is club in κ. Intersect that club set with the stationary set of weakly inaccessible cardinals less than κ to get a stationary set of strongly inaccessible cardinals less than κ.

Example: showing that Mahlo cardinals are hyper-inaccessible

Suppose κ is Mahlo. We proceed by transfinite induction on α to show that κ is α-inaccessible for any α ≤ κ. Since κ is Mahlo, κ is inaccessible; and thus 0-inaccessible, which is the same thing.

If κ is α-inaccessible, then there are β-inaccessibles (for β < α) arbitrarily close to κ. Consider the set of simultaneous limits of such β-inaccessibles larger than some threshold but less than κ. It is unbounded in κ (imagine rotating through β-inaccessibles for β < α ω-times choosing a larger cardinal each time, then take the limit which is less than κ by regularity (this is what fails if α ≥ κ)). It is closed, so it is club in κ. So, by κ's Mahlo-ness, it contains an inaccessible. That inaccessible is actually an α-inaccessible. So κ is α+1-inaccessible.

If λ ≤ κ is a limit ordinal and κ is α-inaccessible for all α < λ, then every β < λ is also less than α for some α < λ. So this case is trivial. In particular, κ is κ-inaccessible and thus hyper-inaccessible.

To show that κ is a limit of hyper-inaccessibles and thus 1-hyper-inaccessible, we need to show that the diagonal set of cardinals μ < κ which are α-inaccessible for every α < μ is club in κ. Choose a 0-inaccessible above the threshold, call it α₀. Then pick an α₀-inaccessible, call it α₁. Keep repeating this and taking limits at limits until you reach a fixed point, call it μ. Then μ has the required property (being a simultaneous limit of α-inaccessibles for all α < μ) and is less than κ by regularity. Limits of such cardinals also have the property, so the set of them is club in κ. By Mahlo-ness of κ, there is an inaccessible in this set and it is hyper-inaccessible. So κ is 1-hyper-inaccessible. We can intersect this same club set with the stationary set less than κ to get a stationary set of hyper-inaccessibles less than κ.

The rest of the proof that κ is α-hyper-inaccessible mimics the proof that it is α-inaccessible. So κ is hyper-hyper-inaccessible, etc..

α-Mahlo, hyper-Mahlo and greatly Mahlo cardinals

A cardinal κ is α-Mahlo for some ordinal α if and only if κ is Mahlo and for every ordinal β<α, the set of β-Mahlo cardinals below κ is stationary in κ. We can define "hyper-Mahlo", "α-hyper-Mahlo", "weakly α-Mahlo", "weakly hyper-Mahlo", "weakly α-hyper-Mahlo", etc. by analogy with the definitions for inaccessibles.

A cardinal κ is greatly Mahlo or κ⁺-Mahlo if and only if it is inaccessible and there is a normal (i.e. nontrivial and closed under diagonal intersections) κ-complete filter on the power set of κ that is closed under the Mahlo operation, which maps the set of ordinals S to {α $\in$ S: α has uncountable cofinality and S∩α is stationary in α}

The properties of being inaccessible, Mahlo, weakly Mahlo, α-Mahlo, greatly Mahlo, etc. are preserved if we replace the universe by an inner model.

The Mahlo operation

If X is a class of ordinals, them we can form a new class of ordinals M(X) consisting of the ordinals α of uncountable cofinality such that α∩X is stationary in α. This operation M is called the Mahlo operation. It can be used to define Mahlo cardinals: for example, if X is the class of regular cardinals, then M(X) is the class of weakly Mahlo cardinals. The condition that α has uncountable cofinality ensures that the closed unbounded subsets of α are closed under intersection and so form a filter; in practice the elements of X often already have uncountable cofinality in which case this condition is redundant. Some authors add the condition that α is in X, which in practice usually makes little difference as it is often automatically satisfied.

For a fixed regular uncountable cardinal κ, the Mahlo operation induces an operation on the Boolean algebra of all subsets of κ modulo the non-stationary ideal.

The Mahlo operation can be iterated transfinitely as follows:

M⁰(X) = X
M^α+1(X) = M(M^α(X))
If α is a limit ordinal then M^α(X) is the intersection of M^β(X) for β<α

These iterated Mahlo operations produce the classes of α-Mahlo cardinals starting with the class of strongly inaccessible cardinals.

It is also possible to diagonalize this process by defining

M^Δ(X) is the set of ordinals α that are in M^β(X) for β<α.

And of course this diagonalization process can be iterated too. The diagonalized Mahlo operation produces the hyper-Mahlo cardinals, and so on.

References