Weakly compact cardinal

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In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erdös & Tarski (1961); weakly compact cardinals are large cardinals, meaning that their existence can neither be proven nor disproven from the standard axioms of set theory.

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.

Weakly compact cardinals are Mahlo cardinals, and the set of Mahlo cardinals less than a given weakly compact cardinal is stationary.

[edit] 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: κⁿ → 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.
5. κ is -indescribable.
6. For every set S of cardinality κ of subsets of κ, there is a non-trivial κ-complete filter that decides S.
7. κ is κ-unfoldable.
8. The infinitary language L_κ,κ satisfies the weak compactness theorem.
9. The infinitary language L_κ,ω satisfies the weak compactness theorem.

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.

[edit] See also

• strongly compact cardinal
• list of large cardinal properties

[edit] References

• Drake (1974), Set Theory: An Introduction to Large Cardinals, vol. 76, Studies in Logic and the Foundations of Mathematics, Elsevier Science Ltd, ISBN 0-444-10535-2 
• Erdös, Paul & Tarski, Alfred (1961), “On some problems involving inaccessible cardinals”, Essays on the foundations of mathematics, Jerusalem: Magnes Press, Hebrew Univ., pp. 50--82, MR0167422, <http://www.renyi.hu/~p_erdos/> 
• Kanamori, Akihiro (2003), The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Springer, ISBN 3-540-00384-3