Weakly compact cardinal

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.

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 ${\displaystyle \Pi _{1}^{1}}${\displaystyle \Pi _{1}^{1}}-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 ${\displaystyle M}${\displaystyle M} of cardinality κ with κ ${\displaystyle \in M}${\displaystyle \in M}, ${\displaystyle {}^{<\kappa }M\subset M}${\displaystyle {}^{<\kappa }M\subset M}, and satisfying a sufficiently large fragment of ZFC, there is an elementary embedding ${\displaystyle j}${\displaystyle j} from ${\displaystyle M}${\displaystyle M} to a transitive set ${\displaystyle N}${\displaystyle N} of cardinality κ such that ${\displaystyle ^{<\kappa }N\subset N}${\displaystyle ^{<\kappa }N\subset N}, with critical point ${\displaystyle crit(j)=}${\displaystyle crit(j)=}κ. (Hauser 1991, Theorem 1.3)
12. ${\displaystyle \kappa =\kappa ^{<\kappa }}${\displaystyle \kappa =\kappa ^{<\kappa }} (${\displaystyle \kappa ^{<\kappa }}${\displaystyle \kappa ^{<\kappa }} defined as ${\displaystyle \sum _{\lambda <\kappa }\kappa ^{\lambda }}${\displaystyle \sum _{\lambda <\kappa }\kappa ^{\lambda }}) and every ${\displaystyle \kappa }${\displaystyle \kappa }-complete filter of a ${\displaystyle \kappa }${\displaystyle \kappa }-complete field of sets of cardinality ${\displaystyle \leq \kappa }${\displaystyle \leq \kappa } is contained in a ${\displaystyle \kappa }${\displaystyle \kappa }-complete ultrafilter. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.185)
13. ${\displaystyle \kappa }${\displaystyle \kappa } has Alexander's property, i.e. for any space ${\displaystyle X}${\displaystyle X} with a ${\displaystyle \kappa }${\displaystyle \kappa }-subbase ${\displaystyle {\mathcal {A}}}${\displaystyle {\mathcal {A}}} with cardinality ${\displaystyle \leq \kappa }${\displaystyle \leq \kappa }, and every cover of ${\displaystyle X}${\displaystyle X} by elements of ${\displaystyle {\mathcal {A}}}${\displaystyle {\mathcal {A}}} has a subcover of cardinality ${\displaystyle <\kappa }${\displaystyle <\kappa }, then ${\displaystyle X}${\displaystyle X} is ${\displaystyle \kappa }${\displaystyle \kappa }-compact. (W. W. Comfort, S. Negrepontis, The Theory of Ultrafilters, p.182--185)
14. ${\displaystyle (2^{\kappa })_{\kappa }}${\displaystyle (2^{\kappa })_{\kappa }} is ${\displaystyle \kappa }${\displaystyle \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.

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 ${\displaystyle \kappa }${\displaystyle \kappa } is weakly compact, then there are chains of well-founded elementary end-extensions of ${\displaystyle (V_{\kappa },\in )}${\displaystyle (V_{\kappa },\in )} of arbitrary length ${\displaystyle <\kappa ^{+}}${\displaystyle <\kappa ^{+}}.^{[1] p.6}

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

• List of large cardinal properties

• Drake, F. R. (1974), Set Theory: An Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics, vol. 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, MR 0167422
• Hauser, Kai (1991), "Indescribable Cardinals and Elementary Embeddings", Journal of Symbolic Logic, 56 (2), Association for Symbolic Logic: 439–457, doi:10.2307/2274692, JSTOR 2274692, S2CID 288779
• Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Springer, ISBN 3-540-00384-3

• Large cardinals

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