Weakly compact cardinal explained

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 → 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

\Pi ^{1} _{1} -indescribable.

1. 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.
2. For every set S of cardinality κ of subsets of κ, there is a non-trivial κ-complete filter that decides S.
3. κ is κ-unfoldable.
4. κ is inaccessible and the infinitary language L_κ,κ satisfies the weak compactness theorem.
5. κ is inaccessible and the infinitary language L_κ,ω satisfies the weak compactness theorem.

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.

1. κ is a strongly inaccessible ramifiable cardinal. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)

\kappa = \kappa ^{<\kappa} (\kappa ^{<\kappa} defined as \sum{ λ <\kappa}\kappa ^{λ} ) and every \kappa -complete filter of a \kappa -complete field of sets of cardinality \leq\kappa is contained in a \kappa -complete ultrafilter. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)\kappa has Alexander's property, i.e. for any space X with a \kappa -subbase l A with cardinality \leq\kappa , and every cover of X by elements of l A has a subcover of cardinality <\kappa , then X is \kappa -compact. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.182--185)( 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 ^{+} .^{[1] p.6}

Weakly compact cardinals remain weakly compact in

L .^[2] Assuming V = L, a cardinal is weakly compact iff it is 2-stationary.^[3]

See also

• List of large cardinal properties

Notes and References

1. math/9611209 . Villaveces . Andres . Chains of End Elementary Extensions of Models of Set Theory . 1996 .
2. T. Jech, 'Set Theory: The third millennium edition' (2003)
3. Bagaria, Magidor, Mancilla. On the Consistency Strength of Hyperstationarity, p.3. (2019)