Weakly compact cardinal

Template:Short description In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Template:Harvtxt; 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 ${\mathrm{\Pi }}_1^1$-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.
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. $\kappa ={\kappa }^{<\kappa }$ (${\kappa }^{<\kappa }$ defined as $\sum _{\lambda <\kappa }{\kappa }^{\lambda }$) and every $\kappa$-complete filter of a $\kappa$-complete field of sets of cardinality $\le \kappa$ is contained in a $\kappa$-complete ultrafilter. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)
13. $\kappa$ has Alexander's property, i.e. for any space $X$ with a $\kappa$-subbase ${𝒜}$ with cardinality $\le \kappa$, and every cover of $X$ by elements of ${𝒜}$ has a subcover of cardinality $<\kappa$, then $X$ is $\kappa$-compact. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.182--185)
14. $(2^{\kappa }{)}_{\kappa }$ 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.

See also

• List of large cardinal properties