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 The term α-Mahlo is ambiguous and different authors give inequivalent definitions. One definition is that a cardinal κ is called α-Mahlo for some ordinal α if κ is strongly inaccessible and for every ordinal β<α, the set of β-Mahlo cardinals below κ is stationary in κ. However the condition "κ is strongly inaccessible" is sometimes replaced by other conditions, such as "κ is regular" or "κ is weakly inaccessible" or "κ is Mahlo". We can define "hyper-Mahlo", "α-hyper-Mahlo", "hyper-hyper-Mahlo", "weakly α-Mahlo", "weakly hyper-Mahlo", "weakly α-hyper-Mahlo", and so on, by analogy with the definitions for inaccessibles, so for example a cardinal κ is called hyper-Mahlo if it is κ-Mahlo. 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 {αS: 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. Every reflecting cardinal has strictly more consistency strength than a greatly Mahlo, but inaccessible reflecting cardinals aren't in general Mahlo -- see https://mathoverflow.net/q/212597 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