Weakly compact cardinal

In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; 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 -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 of cardinality κ with κ , , and satisfying a sufficiently large fragment of ZFC, there is an elementary embedding from to a transitive set of cardinality κ such that , with critical point κ.
12. ( defined as ) and every -complete filter of a -complete field of sets of cardinality is contained in a -complete ultrafilter. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)
13. has Alexander's property, i.e. for any space with a -subbase with cardinality , and every cover of by elements of has a subcover of cardinality , then is -compact. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.182--185)
14. is -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 is weakly compact, then there are chains of well-founded elementary end-extensions of of arbitrary length .^p.6

Weakly compact cardinals remain weakly compact in . Assuming V = L, a cardinal is weakly compact iff it is 2-stationary.

See also

• List of large cardinal properties

References

Citations