ⓘ Mahlo cardinal
In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by Paul Mahlo. As with all large cardinals, none of these varieties of Mahlo cardinals can be proved to exist by ZFC.
A cardinal number κ {\displaystyle \kappa } is called strongly Mahlo if κ {\displaystyle \kappa } is strongly inaccessible and the set U = { λ < κ ∣ λ is strongly inaccessible } {\displaystyle U=\{\lambda
 Mahlo introduced Mahlo cardinals in 1911. He also showed that the continuum hypothesis implies the existence of a Luzin set. Mahlo Paul 1908 Topologische
 worldly cardinals weakly and strongly inaccessible, α  inaccessible, and hyper inaccessible cardinals weakly and strongly Mahlo α  Mahlo and hyper Mahlo cardinals
 existence of an inaccessible cardinal is a stronger hypothesis than the existence of a standard model of ZFC. Mahlo cardinal Club set Inner model Von Neumann
 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
 compact cardinal is a reflecting cardinal and is also a limit of reflecting cardinals This means also that weakly compact cardinals are Mahlo cardinals and
 Cantor Cardinal number Inaccessible cardinal Infinity plus one Infinitesimal Large cardinal Large countable ordinal Limit ordinal Mahlo cardinal Measurable
 theory L L R Large cardinal property Inaccessible cardinal Mahlo cardinal Measurable cardinal Supercompact cardinal Weakly compact cardinal Linear partial
 Mahlo cardinal But note that we are still talking about possibly countable ordinals here. While the existence of inaccessible or Mahlo cardinals cannot
 Hyper  inaccessible cardinal occasionally means a Mahlo cardinal hyper  Mahlo A hyper  Mahlo cardinal is a cardinal κ that is a κ  Mahlo cardinal hyperverse The
 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
 Ramsey cardinal Erdos 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

Large cardinal 
Axiom of determinacy 

Berkeley cardinal 
Core model 
Critical point (set theory) 
Extender (set theory) 

Extendible cardinal 
Grothendieck universe 
Huge cardinal 
Indescribable cardinal 
Ineffable cardinal 
Iterable cardinal 
Kunens inconsistency theorem 
Measurable cardinal 

Rankintorank 
Reinhardt cardinal 
Remarkable cardinal 
Shelah cardinal 
Shrewd cardinal 
Strong cardinal 
Strongly compact cardinal 
Quasicompact cardinal 
Unfoldable cardinal 
Weakly compact cardinal 
Wholeness axiom 
Woodin cardinal 
Zero sharp 

Film 

Television show 

Game 

Sport 

Science 

Hobby 

Travel 

Technology 

Brand 

Outer space 

Cinematography 

Photography 

Music 

Literature 

Theatre 

History 

Transport 

Visual arts 

Recreation 

Politics 

Religion 

Nature 

Fashion 

Subculture 

Animation 

Award 

Interest 