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. (Tarski originally called them "not strongly incompact" cardinals.)

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.

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.
 * 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. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)
 * 5) κ is $$\Pi^1_1$$-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 (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.
 * 11) κ is inaccessible and for every transitive set $$M$$ of cardinality κ with κ $$\in M$$, $${}^{<\kappa}M\subset M$$, and satisfying a sufficiently large fragment of ZFC, there is an elementary embedding $$j$$ from $$M$$ to a transitive set $$N$$ of cardinality κ such that $$^{<\kappa}N\subset N$$, with critical point $$crit(j)=$$κ.
 * 12) κ is a strongly inaccessible ramifiable cardinal. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)
 * 13) $$\kappa=\kappa^{<\kappa}$$ ($$\kappa^{<\kappa}$$ defined as $$\sum{\lambda<\kappa}\kappa^\lambda$$) and every $$\kappa$$-complete filter of a $$\kappa$$-complete field of sets of cardinality $$\leq\kappa$$ is contained in a $$\kappa$$-complete ultrafilter. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)
 * 14) $$\kappa$$ has Alexander's property, i.e. for any space $$X$$ with a $$\kappa$$-subbase $$\mathcal A$$ with cardinality $$\leq\kappa$$, and every cover of $$X$$ by elements of $$\mathcal A$$ has a subcover of cardinality $$<\kappa$$, then $$X$$ is $$\kappa$$-compact. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.182--185)
 * 15) $$(2^{\kappa})_\kappa$$ is $$\kappa$$-compact. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)

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 $$\kappa$$ is weakly compact, then there are chains of well-founded elementary end-extensions of $$(V_\kappa,\in)$$ of arbitrary length $$<\kappa^+$$. p.6

Weakly compact cardinals remain weakly compact in $$L$$. Assuming V = L, a cardinal is weakly compact iff it is 2-stationary.