Editing Weakly compact cardinal (section)

== 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]<sup>n</sup> → λ, there is a set of cardinality κ that is [[Homogeneous (large cardinal property)|homogeneous]] for f. {{harv|Drake|1974|loc=chapter 7 theorem 3.5}}
# κ is [[Inaccessible cardinal|inaccessible]] and has the [[tree property]], that is, every [[Tree (set theory)|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 <math>\Pi^1_1</math>-[[totally indescribable cardinal|indescribable]].
# κ has the extension property. In other words, for all ''U'' ⊂ ''V''<sub>κ</sub> there exists a transitive set ''X'' with κ ∈ ''X'', and a subset ''S'' ⊂ ''X'', such that (''V''<sub>κ</sub>, ∈, ''U'') is an [[elementary substructure]] of (''X'', ∈, ''S''). Here, ''U'' and ''S'' are regarded as unary [[Predicate (mathematical logic)|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 cardinal|unfoldable]].
# κ is inaccessible and the [[infinitary language]] ''L''<sub>κ,κ</sub> satisfies the weak compactness theorem. 
# κ is inaccessible and the [[infinitary language]] ''L''<sub>κ,ω</sub> satisfies the weak compactness theorem.
# κ is inaccessible and for every [[transitive set]] <math>M</math> of cardinality κ with κ <math>\in M</math>, <math>{}^{<\kappa}M\subset M</math>, and satisfying a sufficiently large fragment of [[ZFC]], there is an [[elementary embedding]] <math>j</math> from <math>M</math> to a transitive set <math>N</math> of cardinality κ such that <math>^{<\kappa}N\subset N</math>, with [[Critical point (set theory)|critical point]] <math>crit(j)=</math>κ. {{harv|Hauser|1991|loc=Theorem 1.3}}
# <math>\kappa=\kappa^{<\kappa}</math> (<math>\kappa^{<\kappa}</math> defined as <math>\sum_{\lambda<\kappa}\kappa^\lambda</math>) and every <math>\kappa</math>-complete filter of a <math>\kappa</math>-complete [[field of sets]] of cardinality <math>\leq\kappa</math> is contained in a <math>\kappa</math>-complete ultrafilter. (W. W. Comfort, S. Negrepontis, ''The Theory of Ultrafilters'', p.185)
# <math>\kappa</math> has Alexander's property, i.e. for any space <math>X</math> with a <math>\kappa</math>-subbase <math>\mathcal A</math> with cardinality <math>\leq\kappa</math>, and every cover of <math>X</math> by elements of <math>\mathcal A</math> has a subcover of cardinality <math><\kappa</math>, then <math>X</math> is <math>\kappa</math>-compact. (W. W. Comfort, S. Negrepontis, ''The Theory of Ultrafilters'', p.182--185)
# <math>(2^{\kappa})_\kappa</math> is <math>\kappa</math>-compact. (W. W. Comfort, S. Negrepontis, ''The Theory of Ultrafilters'', p.185)

A language ''L''<sub>κ,κ</sub> 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 cardinal]]s are defined in a similar way without the restriction on the cardinality of the set of sentences.