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 strongly Mahlo if κ is strongly inaccessible and the set U = {λ < κ: λ is strongly 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.
The term "Mahlo cardinal" now usually means "strongly Mahlo cardinal", though the cardinals originally considered by Mahlo were weakly Mahlo cardinals.
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:
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.
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