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Weakly compact cardinal

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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.

Contents

• Equivalent formulations

• Properties

• See also

• References

• Citations

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, chapter 7 theorem 3.5)

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 Π Π 1 1 {\displaystyle \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 {\displaystyle M} of cardinality κ with κ ∈ ∈ M {\displaystyle \in M} , < κ κ M ⊂ ⊂ M {\displaystyle {}^{<\kappa }M\subset M} , and satisfying a sufficiently large fragment of ZFC, there is an elementary embedding j {\displaystyle j} from M {\displaystyle M} to a transitive set N {\displaystyle N} of cardinality κ such that < κ κ N ⊂ ⊂ N {\displaystyle ^{<\kappa }N\subset N} , with critical point c r i t ( j ) = {\displaystyle crit(j)=} κ. (Hauser 1991, Theorem 1.3)

12. κ κ = κ κ < κ κ {\displaystyle \kappa =\kappa ^{<\kappa }} ( κ κ < κ κ {\displaystyle \kappa ^{<\kappa }} defined as ∑ ∑ λ λ < κ κ κ κ λ λ {\displaystyle \sum _{\lambda <\kappa }\kappa ^{\lambda }} ) and every κ κ {\displaystyle \kappa } -complete filter of a κ κ {\displaystyle \kappa } -complete field of sets of cardinality ≤ ≤ κ κ {\displaystyle \leq \kappa } is contained in a κ κ {\displaystyle \kappa } -complete ultrafilter. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)

13. κ κ {\displaystyle \kappa } has Alexander's property, i.e. for any space X {\displaystyle X} with a κ κ {\displaystyle \kappa } -subbase A {\displaystyle {\mathcal {A}}} with cardinality ≤ ≤ κ κ {\displaystyle \leq \kappa } , and every cover of X {\displaystyle X} by elements of A {\displaystyle {\mathcal {A}}} has a subcover of cardinality < κ κ {\displaystyle <\kappa } , then X {\displaystyle X} is κ κ {\displaystyle \kappa } -compact. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.182--185)

14. ( 2 κ κ ) κ κ {\displaystyle (2^{\kappa })_{\kappa }} is κ κ {\displaystyle \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 κ κ {\displaystyle \kappa } is weakly compact, then there are chains of well-founded elementary end-extensions of ( V κ κ , ∈ ∈ ) {\displaystyle (V_{\kappa },\in )} of arbitrary length < κ κ + {\displaystyle <\kappa ^{+}} .cite-ref-villaveces96-1-0[1]p.6

Weakly compact cardinals remain weakly compact in L {\displaystyle L} .cite-ref-2[2] Assuming V = L, a cardinal is weakly compact iff it is 2-stationary.cite-ref-3[3]