Weakly compact cardinal

In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erdős & Tarski (1961); weakly compact cardinals are large cardinals, meaning that their existence can not 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: [κ] 2 → {0, 1} there is a set of cardinality κ that is homogeneous for f. In this context, [κ] 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 [S]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.

Weakly compact cardinals are Mahlo cardinals, and the set of Mahlo cardinals less than a given weakly compact cardinal is stationary.

Some authors use a weaker definition of weakly compact cardinals, such as one of the conditions below with the condition of inaccessibility dropped.

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: [κ]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 $\Pi^1_1$-indescribable.
6. κ has the extension property. In other words, for all UVκ there exists a transitive set X with κ ∈ X, and a subset SX, such that (Vκ, ∈, 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 Lκ,κ satisfies the weak compactness theorem.
10. κ is inaccessible and the infinitary language Lκ,ω satisfies the weak compactness theorem.

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.