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 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.
In mathematics, a Mahlocardinal is a certain kind of large cardinal number. Mahlocardinals were first described by Paul Mahlo (1911, 1912, 1913). As...
{\displaystyle \pi } ) is Π 1 {\displaystyle \Pi _{1}} . Worldly cardinal, a weaker notion Mahlocardinal, a stronger notion Club set Inner model Von Neumann universe...
worldly cardinals weakly and strongly inaccessible, α-inaccessible, and hyper inaccessible cardinals weakly and strongly Mahlo, α-Mahlo, and hyper Mahlo cardinals...
(See also strong cardinal.) A Woodin cardinal is preceded by a stationary set of measurable cardinals, and thus it is a Mahlocardinal. However, the first...
also that weakly compact cardinals are Mahlocardinals, and the set of Mahlocardinals less than a given weakly compact cardinal is stationary. If κ{\displaystyle...
"Hyper-inaccessible cardinal" occasionally means a Mahlocardinal hyper-Mahlo A hyper-Mahlocardinal is a cardinal κ that is a κ-Mahlocardinal hyperset A set...
κ is called greatly Mahlo if it is κ+-Mahlo (Mekler & Shelah 1989). An inaccessible reflecting cardinal is not in general Mahlo however, see https://mathoverflow...
Mahlo introduced Mahlo cardinals in 1911. He also showed that the continuum hypothesis implies the existence of a Luzin set. Mahlo, Paul (1908), Topologische...
is the first Mahlocardinal. This is the proof-theoretic ordinal of KPM, an extension of Kripke-Platek set theory based on a Mahlocardinal. Its value is...
each k {\displaystyle k} , "there is a strongly k {\displaystyle k} -Mahlocardinal", and S M A H + {\displaystyle {\mathsf {SMAH^{+}}}} represents the...
the first Mahlocardinal. Uses Rathjen's ψ rather than Buchholz's ψ. 12.^ K {\displaystyle K} represents the first weakly compact cardinal. Uses Rathjen's...
inaccessible cardinals Existence of Mahlocardinals Existence of measurable cardinals (first conjectured by Ulam) Existence of supercompact cardinals The following...
consistency is implied by an I3 rank-into-rank cardinal. Add an axiom saying that Ord is a Mahlocardinal — for every closed unbounded class of ordinals...
\omega _{2}} -Aronszajn trees is equiconsistent with the existence of a Mahlocardinal, the non-existence of ω 2 {\displaystyle \omega _{2}} -Aronszajn trees...
Arithmetic (2009, p.387) M. Rathjen, Ordinal notations based on a weakly Mahlocardinal, (1990, p.251). Accessed 16 August 2022. M. Rathjen, "The Art of Ordinal...
"hyper-inaccessible cardinal", different authors conflict on this terminology. An ordinal α {\displaystyle \alpha } is called recursively Mahlo if it is admissible...
that real valued measurable cardinals are weakly inaccessible (they are in fact weakly Mahlo). All measurable cardinals are real-valued measurable, and...
Weakly Mahlocardinals become strongly Mahlo. And more generally, any large cardinal property weaker than 0# (see the list of large cardinal properties)...
collapse of a Mahlocardinal to describe the ordinal-theoretic strength of Kripke–Platek set theory augmented by the recursive Mahloness of the class of...
membership becomes a (proper class) model of ZFC (in which there are n-Mahlocardinals for each n; this extension of NFU is strictly stronger than ZFC). This...
material includes inaccessible cardinals, Mahlocardinals, measurable cardinals, compact cardinals and indescribable cardinals. The chapter covers the constructible...
manner that is highly parallel to that of (small) large cardinals (one can define recursively Mahlo ordinals, for example). But all these ordinals are still...
freedoms. Above all, I want to show that an agenda, based only on seven cardinal pillars of the EFF, can take South Africa to economic freedom and sustainable...
Polokwane, Limpopo Province. South Africa. The parasite was named after Mahlo Mokgalong for his contribution to the field of bird parasitology. Mediorhynchus...
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