ⓘ Weakly compact cardinal
In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erdos & Tarski ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory.
Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: 2 maps to 0 or all of it maps to 1.
The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below.
Every weakly 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 the set of Mahlo cardinals less than a given weakly compact cardinal is stationary.
 Weakly compact can refer to: Weakly compact cardinal an infinite cardinal number on which every binary relation has an equally large homogeneous subset
 Compact cardinal may refer to: Weakly compact cardinal Subcompact cardinal Supercompact cardinal Strongly compact cardinal
 stationary in α. Reflecting cardinals were introduced by Mekler Shelah 1989 Every weakly compact cardinal is a reflecting cardinal and is also a limit
 is a Mahlo cardinal. However, the first Woodin cardinal is not even weakly compact Woodin cardinals are important in descriptive set theory. By a result
 statements has cardinality below a certain cardinal λ we may then refer to λ  compactness A cardinal is weakly compact if and only if it is κ  compact this was
 shrewd cardinals that makes sense when α κ: there is λ κ and β such that φ U Vλ holds in Vλ β. Π1 1  indescribable cardinals are the same as weakly compact
 strength of Morse Kelley set theory with the proper class ordinal a weakly compact cardinal The universal set is a proper set in this theory. The sets of
 Subtle cardinal Supercompact cardinal Superstrong cardinal Totally indescribable cardinal Weakly compact cardinal Weakly hyper  Woodin cardinal Weakly inaccessible
 theory L L R Large cardinal property Inaccessible cardinal Mahlo cardinal Measurable cardinal Supercompact cardinal Weakly compact cardinal Linear partial

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