Weakly compact cardinal


In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; 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 → there is a set of cardinality κ that is homogeneous for f. In this context, 2 means the set of 2-element subsets of κ, and a subset S of κ is homogeneous for f if and only if either all of 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.

Equivalent formulations

The following are equivalent for any uncountable cardinal κ:
  1. κ is weakly compact.
  2. for every λ<κ, natural number n ≥ 2, and function f: n → λ, there is a set of cardinality κ that is homogeneous for f.
  3. κ is inaccessible and has the tree property, that is, every tree of height κ has either a level of size κ or a branch of size κ.
  4. Every linear order of cardinality κ has an ascending or a descending sequence of order type κ.
  5. κ is -indescribable.
  6. κ has the extension property. In other words, for all U ⊂ Vκ there exists a transitive set X with κ ∈ X, and a subset S ⊂ X, such that is an elementary substructure of. Here, U and S are regarded as unary predicates.
  7. For every set S of cardinality κ of subsets of κ, there is a non-trivial κ-complete filter that decides S.
  8. κ is κ-unfoldable.
  9. κ is inaccessible and the infinitary language Lκ,κ satisfies the weak compactness theorem.
  10. κ is inaccessible and the infinitary language Lκ,ω satisfies the weak compactness theorem.
  11. κ is inaccessible and for every transitive set of cardinality κ with κ,, and satisfying a sufficiently large fragment of ZFC, there is an elementary embedding from to a transitive set of cardinality κ such that, with critical point κ.
  12. and every -complete filter of a -complete field of sets of cardinality is contained in a -complete ultrafilter.
  13. has Alexander's property, i.e. for any space with a -subbase with cardinality, and every cover of by elements of has a subcover of cardinality, then is -compact.
  14. is -compact.
A language Lκ,κ is said to satisfy the weak compactness theorem if whenever Σ is a set of sentences of cardinality at most κ and every subset with less than κ elements has a model, then Σ has a model. Strongly compact cardinals are defined in a similar way without the restriction on the cardinality of the set of sentences.

Properties

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.
If is weakly compact, then there are chains of well-founded elementary end-extensions of of arbitrary length.p.6
Weakly compact cardinals remain weakly compact in. Assuming V = L, a cardinal is weakly compact iff it is 2-stationary.