In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erdős & 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 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.
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a weaklycompactcardinal is a certain kind of cardinal number introduced by Erdős & Tarski (1961); weaklycompactcardinals are large cardinals, meaning...
Weaklycompact can refer to: Weaklycompactcardinal, an infinite cardinal number on which every binary relation has an equally large homogeneous subset...
large cardinal number can be both and thus weakly inaccessible. An ordinal is a weakly inaccessible cardinal if and only if it is a regular ordinal and...
statements has cardinality below a certain cardinal λ; we may then refer to λ-compactness. A cardinal is weaklycompact if and only if it is κ-compact; this was...
strength of Morse–Kelley set theory with the proper class ordinal a weaklycompactcardinal. The universal set is a proper set in this theory. The sets of...
stationary in α.) Reflecting cardinals were introduced by (Mekler & Shelah 1989). Every weaklycompactcardinal is a reflecting cardinal, and is also a limit...
stationary). The least n {\displaystyle n} -subtle cardinal is not even weaklycompact (and unlike ineffable cardinals, the least n {\displaystyle n} -almost ineffable...
and thus it is a Mahlo cardinal. However, the first Woodin cardinal is not even weaklycompact. The hierarchy V α {\displaystyle V_{\alpha }} (known as...
satisfies the weakcompactness theorem 3. A weakly Mahlo cardinal is a cardinal κ that is weakly inaccessible and such that the set of weakly inaccessible...
stationarily many weakcompactnesscardinals. Vopenka's principle itself may be stated as the existence of a strong compactnesscardinal for each logic....
Silver showed that it is consistent (relative to the existence of a weaklycompactcardinal) that no ℵ 2 {\displaystyle \aleph _{2}} -Aronszajn trees exist...
only if it is weaklycompact. A κ+ω-unfoldable cardinal is indescribable and preceded by a stationary set of totally indescribable cardinals.[citation needed]...
and |x| = |y|. K computes the successors of singular and weaklycompactcardinals correctly (Weak Covering Property). Moreover, if |κ| > ω1, then cofinality((κ+)K) ≥ |κ|...
Mahlo cardinal, the non-existence of ω 2 {\displaystyle \omega _{2}} -Aronszajn trees is equiconsistent with the existence of a weaklycompactcardinal. Large...
and Shelah) Several large cardinal properties can be defined using this notation. In particular: Weaklycompactcardinals κ {\displaystyle \kappa } are...
function is based on the least weaklycompactcardinal to create large countable ordinals. For a weaklycompactcardinal K, the functions M α {\displaystyle...
Π 1 1 {\displaystyle \Pi _{1}^{1}} -indescribable cardinals are the same as weaklycompactcardinals. The indescribability condition is equivalent to V...
{\displaystyle X} has a weakly convergent subsequence. A weaklycompact subset A {\displaystyle A} in ℓ 1 {\displaystyle \ell ^{1}} is norm-compact. Indeed, every...
the first Mahlo cardinal. Uses Rathjen's ψ rather than Buchholz's ψ. 12.^ K {\displaystyle K} represents the first weaklycompactcardinal. Uses Rathjen's...
model of the extended theory has cardinality at least κ {\displaystyle \kappa } . A third application of the compactness theorem is the construction of...
"recursive analogues" of some uncountable cardinals such as weaklycompactcardinals and indescribable cardinals. For example, an ordinal which Π 3 {\displaystyle...
\left(S,{\mathcal {T}}|_{S}\right)} has property P . {\displaystyle P.} Weakly hereditary, if for every topological space ( X , T ) {\displaystyle (X,{\mathcal...
include: Every small category has a skeleton. If two small categories are weakly equivalent, then they are equivalent. Every continuous functor on a small-complete...
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