Weakly Compact Cardinal
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Weakly Compact Cardinal

In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erd?s & 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 ? is defined to be weakly compact if it is uncountable and for every function f: [?] 2 -> {0, 1} there is a set of cardinality ? that is homogeneous for f. In this context, [?] 2 means the set of 2-element subsets of ?, and a subset S of ? 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.

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.

## Equivalent formulations

The following are equivalent for any uncountable cardinal ?:

1. ? is weakly compact.
2. for every ?<?, natural number n >= 2, and function f: [?]n -> ?, there is a set of cardinality ? that is homogeneous for f. (Drake 1974, chapter 7 theorem 3.5)
3. ? is inaccessible and has the tree property, that is, every tree of height ? has either a level of size ? or a branch of size ?.
4. Every linear order of cardinality ? has an ascending or a descending sequence of order type ?.
5. ? is ${\displaystyle \Pi _{1}^{1}}$-indescribable.
6. ? has the extension property. In other words, for all U ? V? there exists a transitive set X with ? ? X, and a subset S ? X, such that (V?, ?, U) is an elementary substructure of (X, ?, S). Here, U and S are regarded as unary predicates.
7. For every set S of cardinality ? of subsets of ?, there is a non-trivial ?-complete filter that decides S.
8. ? is ?-unfoldable.
9. ? is inaccessible and the infinitary language L?,? satisfies the weak compactness theorem.
10. ? is inaccessible and the infinitary language L?,? satisfies the weak compactness theorem.
11. ? is inaccessible and for every transitive set ${\displaystyle M}$ of cardinality ? with ? ${\displaystyle \in M}$, ${\displaystyle {}^{<\kappa }M\subset M}$, and satisfying a sufficiently large fragment of ZFC, there is an elementary embedding ${\displaystyle j}$ from ${\displaystyle M}$ to a transitive set ${\displaystyle N}$ of cardinality ? such that ${\displaystyle ^{<\kappa }N\subset N}$, with critical point ${\displaystyle crit(j)=}$?. (Hauser 1991, Theorem 1.3)

A language L?,? is said to satisfy the weak compactness theorem if whenever ? is a set of sentences of cardinality at most ? and every subset with less than ? elements has a model, then ? has a model. Strongly compact cardinals are defined in a similar way without the restriction on the cardinality of the set of sentences.

## References

• Drake, F. R. (1974), Set Theory: An Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics, 76, Elsevier Science Ltd, ISBN 0-444-10535-2
• Erd?s, 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, Association for Symbolic Logic, 56: 439-457, doi:10.2307/2274692
• Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Springer, ISBN 3-540-00384-3