Mahlo cardinal
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In mathematics, a Mahlo cardinal is a certain kind of large cardinal number.
Formally, a cardinal number κ is called Mahlo iff the set U = {λ < κ: λ is inaccessible} is stationary in κ. Assuming that ZFC is consistent, the existence of Mahlo cardinals provably cannot be proved in ZFC.
Mahlo cardinals were first described in 1911 by mathematician Paul Mahlo.
A cardinal κ is Mahlo of order α≤κ iff κ is inaccessible and for every ordinal β<α, the set of β-Mahlo cardinals below κ is stationary. A cardinal κ is greatly Mahlo, or κ+-Mahlo, iff 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: set of ordinals S → {α∈S: α has uncountable cofinality and S∩α is stationary in α}
The properties of being inaccessible, Mahlo, α-Mahlo, and greatly Mahlo (but not weak compactness) are preserved if we replace the universe by an inner model.