Mahlo Cardinal - Minimal Condition Sufficient For A Mahlo Cardinal

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 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 = 2μ_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.