# 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 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:
* κ is weakly compact.
* 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)
* 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.
* 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)
* κ is $\Pi _{1}^{1}$-indescribable.
* 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.
* 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.
* κ is inaccessible and the infinitary language L$_{κ,κ}$ satisfies the weak compactness theorem.
* κ is inaccessible and the infinitary language L$_{κ,ω}$ satisfies the weak compactness theorem.
* 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. (Hauser 1991, Theorem 1.3)
* $\kappa =\kappa ^{<\kappa }$ ($\kappa ^{<\kappa }$ 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)
* $\kappa$ 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)
* $(2^{\kappa })_{\kappa }$ is $\kappa$-compact. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)
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.

## 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.