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 formulationsEdit
The following are equivalent for any uncountable cardinal κ:
- κ is weakly compact.
- 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. Template:Harv
- 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.
- 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)
- κ is <math>\Pi^1_1</math>-indescribable.
- 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.
- For every set S of cardinality κ of subsets of κ, there is a non-trivial κ-complete filter that decides S.
- κ is κ-unfoldable.
- κ is inaccessible and the infinitary language Lκ,κ satisfies the weak compactness theorem.
- κ is inaccessible and the infinitary language Lκ,ω satisfies the weak compactness theorem.
- κ is inaccessible and for every transitive set <math>M</math> of cardinality κ with κ <math>\in M</math>, <math>{}^{<\kappa}M\subset M</math>, and satisfying a sufficiently large fragment of ZFC, there is an elementary embedding <math>j</math> from <math>M</math> to a transitive set <math>N</math> of cardinality κ such that <math>^{<\kappa}N\subset N</math>, with critical point <math>crit(j)=</math>κ. Template:Harv
- <math>\kappa=\kappa^{<\kappa}</math> (<math>\kappa^{<\kappa}</math> defined as <math>\sum_{\lambda<\kappa}\kappa^\lambda</math>) and every <math>\kappa</math>-complete filter of a <math>\kappa</math>-complete field of sets of cardinality <math>\leq\kappa</math> is contained in a <math>\kappa</math>-complete ultrafilter. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)
- <math>\kappa</math> has Alexander's property, i.e. for any space <math>X</math> with a <math>\kappa</math>-subbase <math>\mathcal A</math> with cardinality <math>\leq\kappa</math>, and every cover of <math>X</math> by elements of <math>\mathcal A</math> has a subcover of cardinality <math><\kappa</math>, then <math>X</math> is <math>\kappa</math>-compact. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.182--185)
- <math>(2^{\kappa})_\kappa</math> is <math>\kappa</math>-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.
PropertiesEdit
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>\kappa</math> is weakly compact, then there are chains of well-founded elementary end-extensions of <math>(V_\kappa,\in)</math> of arbitrary length <math><\kappa^+</math>.<ref name="Villaveces96">Template:Cite arXiv</ref>p.6
Weakly compact cardinals remain weakly compact in <math>L</math>.<ref>T. Jech, 'Set Theory: The third millennium edition' (2003)</ref> Assuming V = L, a cardinal is weakly compact iff it is 2-stationary.<ref>Bagaria, Magidor, Mancilla. On the Consistency Strength of Hyperstationarity, p.3. (2019)</ref>