Weakly Compact Cardinal - Equivalent Formulations

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. (Drake 1974, chapter 7 theorem 3.5)
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 ⊂ V_κ there exists a transitive set X with κ ∈ X, and a subset S ⊂ X, such that (V_κ, ∈, U) is an elementary substructure of (X, ∈, S). 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 L_κ,κ satisfies the weak compactness theorem.
10. κ is inaccessible and the infinitary language L_κ,ω satisfies the weak compactness theorem.

A language L_κ,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.