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Concept
Mahlo cardinal
Summary
In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by . As with all large cardinals, none of these varieties of Mahlo cardinals can be proven to exist by ZFC (assuming ZFC is consistent).
A cardinal number is called strongly Mahlo if is strongly inaccessible and the set is stationary in κ.
A cardinal is called weakly Mahlo if is weakly inaccessible and the set of weakly inaccessible cardinals less than is stationary in .
The term "Mahlo cardinal" now usually means "strongly Mahlo cardinal", though the cardinals originally considered by Mahlo were weakly Mahlo cardinals.
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.
Official source
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