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GraziePrego (talk | contribs) Adding local short description: "Concept in mathematics", overriding Wikidata description "inaccessible cardinal number 𝜅 such that the set of inaccessibles less than 𝜅 is stationary in 𝜅" |
→α-Mahlo, hyper-Mahlo and greatly Mahlo cardinals: added section explaining κ which are δ-Mahlo for δ ≤ κ+ |
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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
For α < κ<sup>+</sup>, define the subsets M<sub>α</sub>(κ) ⊆ κ inductively as follows:
*M<sub>0</sub>(κ) is the set of regular cardinals below κ,
*M<sub>α+1</sub>(κ) is the set of regular λ < κ such that M<sub>α</sub>(κ) ∩ λ is stationary in λ,
*for limits α with cf(α) < κ, M<sub>α</sub>(κ) is the intersection of M<sub>β</sub>(κ) over all β < α, and
*for limits α with cf(α) = κ, pick an enumeration f : κ → α of a cofinal subset. Then, M<sub>α</sub>(κ) is the set of all λ < κ such that λ ∈ M<sub>f(γ)</sub>(κ) for all γ < λ.
Although the exact definition depends on a choice of cofinal subset for each α < κ<sup>+</sup> of cofinality κ, any choice will give the same sequence of subsets modulo the nonstationary ideal.
For δ ≤ κ<sup>+</sup>, κ is then called '''δ-Mahlo''' if and only if M<sub>α</sub>(κ) is stationary in κ for all α < δ. A cardinal κ is κ<sup>+</sup>-Mahlo if and only if it is greatly Mahlo.
The properties of being inaccessible, Mahlo, weakly Mahlo, α-Mahlo, greatly Mahlo, etc. are preserved if we replace the universe by an [[inner model]].
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