# ⓘ 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 lar ..

## ⓘ 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