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...

{\displaystyle \pi } ) is Π 1 {\displaystyle \Pi _{1}} . Worldly cardinal, a weaker notion Mahlo cardinal, a stronger notion Club set Inner model Von Neumann universe...

(See also strong cardinal.) A Woodin cardinal is preceded by a stationary set of measurable cardinals, and thus it is a Mahlo cardinal. However, the first...

worldly cardinals weakly and strongly inaccessible, α-inaccessible, and hyper inaccessible cardinals weakly and strongly Mahlo, α-Mahlo, and hyper Mahlo cardinals...

also that weakly compact cardinals are Mahlo cardinals, and the set of Mahlo cardinals less than a given weakly compact cardinal is stationary. If κ {\displaystyle...

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...

is the first Mahlo cardinal. This is the proof-theoretic ordinal of KPM, an extension of Kripke-Platek set theory based on a Mahlo cardinal. Its value is...

inaccessible cardinals Existence of Mahlo cardinals Existence of measurable cardinals (first conjectured by Ulam) Existence of supercompact cardinals The following...

Ramsey cardinal Erdős cardinal Extendible cardinal Huge cardinal Hyper-Woodin cardinal Inaccessible cardinal Ineffable cardinal Mahlo cardinal Measurable...

that real valued measurable cardinals are weakly inaccessible (they are in fact weakly Mahlo). All measurable cardinals are real-valued measurable, and...

  "Hyper-inaccessible cardinal" occasionally means a Mahlo cardinal hyper-Mahlo A hyper-Mahlo cardinal is a cardinal κ that is a κ-Mahlo cardinal hyperset A set...

Mahlo introduced Mahlo cardinals in 1911. He also showed that the continuum hypothesis implies the existence of a Luzin set. Mahlo, Paul (1908), Topologische...

consistency is implied by an I3 rank-into-rank cardinal. Add an axiom saying that Ord is a Mahlo cardinal — for every closed unbounded class of ordinals...

Weakly Mahlo cardinals become strongly Mahlo. And more generally, any large cardinal property weaker than 0# (see the list of large cardinal properties)...

theory) L L(R) Large cardinal property Inaccessible cardinal Mahlo cardinal Measurable cardinal Supercompact cardinal Weakly compact cardinal Linear partial...

Arithmetic (2009, p.387) M. Rathjen, Ordinal notations based on a weakly Mahlo cardinal, (1990, p.251). Accessed 16 August 2022. M. Rathjen, "The Art of Ordinal...

the first Mahlo cardinal. Uses Rathjen's ψ rather than Buchholz's ψ. 12.^ K {\displaystyle K} represents the first weakly compact cardinal. Uses Rathjen's...

infinite family of axioms "there exists a strongly k {\displaystyle k} -Mahlo cardinal for all positive integers k {\displaystyle k} . and S M A H + {\displaystyle...

"hyper-inaccessible cardinal", different authors conflict on this terminology. An ordinal α {\displaystyle \alpha } is called recursively Mahlo if it is admissible...

collapse of a Mahlo cardinal to describe the ordinal-theoretic strength of Kripke–Platek set theory augmented by the recursive Mahloness of the class of...

\omega _{2}} -Aronszajn trees is equiconsistent with the existence of a Mahlo cardinal, the non-existence of ω 2 {\displaystyle \omega _{2}} -Aronszajn trees...

It collapses weakly Mahlo cardinals M {\displaystyle M} to generate large countable ordinals. A weakly Mahlo cardinal is a cardinal such that the set of...

membership becomes a (proper class) model of ZFC (in which there are n-Mahlo cardinals for each n; this extension of NFU is strictly stronger than ZFC). This...

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...

material includes inaccessible cardinals, Mahlo cardinals, measurable cardinals, compact cardinals and indescribable cardinals. The chapter covers the constructible...

Polokwane, Limpopo Province. South Africa. The parasite was named after Mahlo Mokgalong for his contribution to the field of bird parasitology. Mediorhynchus...