Weakly Compact Cardinal

In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; 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'': � ² → there is a set of cardinality κ that is homogeneous for ''f''. In this context, � 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''sup>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 κ:

# κ is weakly compact.
# for every λ<κ, natural number n ≥ 2, and function f: �sup>n → λ, there is a set of cardinality κ that is homogeneous for f.
# 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.
# Every linear order of cardinality κ has an ascending or a descending sequence of order type κ.
# κ is $\Pi^1_1$- indescribable.
# 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.
# 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.
# κ is inaccessible and the infinitary language ''L''_κ,κ satisfies the weak compactness theorem.
# κ is inaccessible and the infinitary language ''L''_κ,ω satisfies the weak compactness theorem.
# κ is inaccessible and for every transitive set $M$ of cardinality κ with κ $\in M$, $^M\subset M$, and satisfying a sufficiently large fragment of ZFC, there is an elementary embedding $j$ from $M$ to a transitive set $N$ of cardinality κ such that $^N\subset N$, with critical point $crit(j)=$κ.

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.

See also

* List of large cardinal properties

References

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* {{citation, last=Kanamori, first=Akihiro, authorlink=Akihiro Kanamori, year=2003, publisher=Springer, title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings, title-link= The Higher Infinite , edition=2nd , isbn= 3-540-00384-3

Large cardinals