- Weakly compact cardinal
In
mathematics , a weakly compact cardinal is a certain kind ofcardinal number introduced by harvtxt|Erdös|Tarski|1961; weakly compact cardinals arelarge cardinal s, 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 cardinal s, 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 κ:# κ is weakly compact.
# 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.
# κ is inaccessible and has thetree 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.
# κ is -indescribable.
# 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.
# Theinfinitary language "L"κ,κ satisfies the weak compactness theorem.
# Theinfinitary 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 cardinal s are defined in a similar way without the restriction on the cardinality of the set of sentences.ee also
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strongly compact cardinal
*list of large cardinal properties References
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*citation|id=MR|0167422|authorlink1=Paul Erdös|authorlink2=Alfred Tarski|last= Erdös|first= Paul|last2=Tarski|first2= Alfred|chapter= On some problems involving inaccessible cardinals|year= 1961 |title= Essays on the foundations of mathematics |pages= 50--82 |publisher=Magnes Press, Hebrew Univ.|publication-place= Jerusalem|url=http://www.renyi.hu/~p_erdos/
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