Weakly Compact Cardinal

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## Equivalent formulations

## See also

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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.

The following are equivalent for any uncountable cardinal ?:

- ? is weakly compact.
- 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) - ? is inaccessible and has the tree property, that is, every tree of height ? has either a level of size ? or a branch of size ?.
- Every linear order of cardinality ? has an ascending or a descending sequence of order type ?.
- ? is -indescribable.
- ? 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. - For every set S of cardinality ? of subsets of ?, there is a non-trivial ?-complete filter that decides S.
- ? is ?-unfoldable.
- ? is inaccessible and the infinitary language
*L*_{?,?}satisfies the weak compactness theorem. - ? is inaccessible and the infinitary language
*L*_{?,?}satisfies the weak compactness theorem. - ? is inaccessible and for every transitive set of cardinality ? with ? , , and satisfying a sufficiently large fragment of ZFC, there is an elementary embedding from to a transitive set of cardinality ? such that , with critical point ?. (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.

- 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

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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