Weakly compact cardinal

In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by (Erdős Tarski); 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. (Drake 1974)
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 [math]\displaystyle{ \Pi^1_1 }[/math]-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 [math]\displaystyle{ M }[/math] of cardinality κ with κ [math]\displaystyle{ \in M }[/math], [math]\displaystyle{ {}^{\lt \kappa}M\subset M }[/math], and satisfying a sufficiently large fragment of ZFC, there is an elementary embedding [math]\displaystyle{ j }[/math] from [math]\displaystyle{ M }[/math] to a transitive set [math]\displaystyle{ N }[/math] of cardinality κ such that [math]\displaystyle{ ^{\lt \kappa}N\subset N }[/math], with critical point [math]\displaystyle{ crit(j)= }[/math]κ. (Hauser 1991)
12. κ is a strongly inaccessible ramifiable cardinal. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)
13. [math]\displaystyle{ \kappa=\kappa^{\lt \kappa} }[/math] ([math]\displaystyle{ \kappa^{\lt \kappa} }[/math] defined as [math]\displaystyle{ \sum{\lambda\lt \kappa}\kappa^\lambda }[/math]) and every [math]\displaystyle{ \kappa }[/math]-complete filter of a [math]\displaystyle{ \kappa }[/math]-complete field of sets of cardinality [math]\displaystyle{ \leq\kappa }[/math] is contained in a [math]\displaystyle{ \kappa }[/math]-complete ultrafilter. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)
14. [math]\displaystyle{ \kappa }[/math] has Alexander's property, i.e. for any space [math]\displaystyle{ X }[/math] with a [math]\displaystyle{ \kappa }[/math]-subbase [math]\displaystyle{ \mathcal A }[/math] with cardinality [math]\displaystyle{ \leq\kappa }[/math], and every cover of [math]\displaystyle{ X }[/math] by elements of [math]\displaystyle{ \mathcal A }[/math] has a subcover of cardinality [math]\displaystyle{ \lt \kappa }[/math], then [math]\displaystyle{ X }[/math] is [math]\displaystyle{ \kappa }[/math]-compact. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.182--185)
15. [math]\displaystyle{ (2^{\kappa})_\kappa }[/math] is [math]\displaystyle{ \kappa }[/math]-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 [math]\displaystyle{ \kappa }[/math] is weakly compact, then there are chains of well-founded elementary end-extensions of [math]\displaystyle{ (V_\kappa,\in) }[/math] of arbitrary length [math]\displaystyle{ \lt \kappa^+ }[/math].^p.6

Weakly compact cardinals remain weakly compact in [math]\displaystyle{ L }[/math]. Assuming V = L, a cardinal is weakly compact iff it is 2-stationary.

See also

• List of large cardinal properties

References

• Drake, F. R. (1974), Set Theory: An Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics, 76, Elsevier Science Ltd, ISBN 0-444-10535-2 
• Erdős, Paul; Tarski, Alfred (1961), "On some problems involving inaccessible cardinals", Essays on the foundations of mathematics, Jerusalem: Magnes Press, Hebrew Univ., pp. 50–82, http://www.renyi.hu/~p_erdos/ 
• Hauser, Kai (1991), "Indescribable Cardinals and Elementary Embeddings", Journal of Symbolic Logic (Association for Symbolic Logic) 56 (2): 439–457, doi:10.2307/2274692 
• Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Springer, ISBN 3-540-00384-3