Weakly compact cardinal
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In mathematics, a weakly compact cardinal is a certain kind of cardinal number; weakly compact cardinals are large cardinals, meaning that their existence can neither be proven nor disproven from the standard axioms of set theory.
Formally, a cardinal κ is weakly compact iff for every function f: κ 2 → {0, 1} there is a set of cardinality κ that is homogeneous for f.
Theorem
The following are equivalent for any uncountable cardinal κ:
- κ is weakly compact.
- for every λ<κ, integer n, and function f: κn → λ, there is a set of cardinality κ that is homogeneous for f.
- κ is inaccessible and every tree of height κ either has a path or a level of cardinality at least κ.
- Every linear order of cardinality κ has an ascending or a descending sequence of order type κ.
- κ is Π11 indescribable.
- For every set S of cardinality κ of subsets of κ, there is a non-trivial κ-complete filter that decides S.
- κ is κ-unfoldable.