This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations.(September 2023) (Learn how and when to remove this message)
In set theory, a branch of mathematics, a strongly compact cardinal is a certain kind of large cardinal.
A cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ-complete ultrafilter.
Strongly compact cardinals were originally defined in terms of infinitary logic, where logical operators are allowed to take infinitely many operands. The logic on a regular cardinal κ is defined by requiring the number of operands for each operator to be less than κ; then κ is strongly compact if its logic satisfies an analog of the compactness property of finitary logic.
Specifically, a statement which follows from some other collection of statements should also follow from some subcollection having cardinality less than κ.
The property of strong compactness may be weakened by only requiring this compactness property to hold when the original collection of statements has cardinality below a certain cardinal λ; we may then refer to λ-compactness. A cardinal is weakly compact if and only if it is κ-compact; this was the original definition of that concept.
Strong compactness implies measurability, and is implied by supercompactness. Given that the relevant cardinals exist, it is consistent with ZFC either that the first measurable cardinal is strongly compact, or that the first strongly compact cardinal is supercompact; these cannot both be true, however. A measurable limit of strongly compact cardinals is strongly compact, but the least such limit is not supercompact.
The consistency strength of strong compactness is strictly above that of a Woodin cardinal. Some set theorists conjecture that existence of a strongly compact cardinal is equiconsistent with that of a supercompact cardinal. However, a proof is unlikely until a canonical inner model theory for supercompact cardinals is developed.
Jech obtained a variant of the tree property which holds for an inaccessible cardinal if and only if it is strongly compact.[1]
Extendibility is a second-order analog of strong compactness.
^Hachtman, Sherwood; Sinapova, Dima (2020). "The super tree property at the successor of a singular" (PDF). Israel Journal of Mathematics. 236 (1): 473–500. arXiv:1806.00820. doi:10.1007/s11856-020-2000-5.
and 26 Related for: Strongly compact cardinal information
a branch of mathematics, a stronglycompactcardinal is a certain kind of large cardinal. A cardinal κ is stronglycompact if and only if every κ-complete...
(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...
arranges cardinals in order of consistency strength, with size of the cardinal used as a tiebreaker. In a few cases (such as stronglycompactcardinals) the...
) A cardinal which is weakly inaccessible and also a strong limit cardinal is strongly inaccessible. The assumption of the existence of a strongly inaccessible...
consistency of the existence of a stronglycompactcardinal imply the consistent existence of a supercompact cardinal? Does there exist a Jónsson algebra...
holds for an inaccessible cardinal iff it is supercompact. Indestructibility Stronglycompactcardinal List of large cardinal properties Drake, F. R. (1974)...
This prevents them from witnessing even a κ+ stronglycompactcardinal κ. Subcompact and quasicompact cardinals were defined by Ronald Jensen. "Square in...
stationarily many weak compactnesscardinals. Vopenka's principle itself may be stated as the existence of a strongcompactnesscardinal for each logic. There...
a cardinal is strongly unfoldable if and only if it is strongly λ-unfoldable for all λ. These properties are essentially weaker versions of strong and...
Solovay showed that large cardinals almost imply SCH—in particular, if κ {\displaystyle \kappa } is stronglycompactcardinal, then the SCH holds above...
Vλ 3. A strong limit cardinal is a (usually nonzero) cardinal that is larger than the powerset of any smaller cardinalstrongly 1. A strongly inaccessible...
regular cardinal. He proved that the least stronglycompactcardinal can be equal to the least measurable cardinal or to the least supercompact cardinal (but...
\kappa } is a stronglycompactcardinal then there is an inner model of set theory with κ {\displaystyle \kappa } many measurable cardinals. He proved Kunen's...
and thus it is a Mahlo cardinal. However, the first Woodin cardinal is not even weakly compact. The hierarchy V α {\displaystyle V_{\alpha }} (known as...
uncountable cardinals are singular" (a strong negation of the axiom of choice) from the consistency of "there is a proper class of stronglycompactcardinals"....
dispersion character of X . {\displaystyle X.} Strongly discrete. Set D {\displaystyle D} is strongly discrete subset of the space X {\displaystyle X}...
S2CID 208867731. J. Bagaria & M. Magidor (2014). "Group radicals and stronglycompactcardinals". Transactions of the American Mathematical Society. 366 (4):...
can be complete without being strongly complete. A cardinal κ ≠ ω {\displaystyle \kappa \neq \omega } is weakly compact when for every theory T in L κ...
itself, but requires a mild large cardinal assumption (the existence of an inaccessible cardinal). The much stronger axiom of determinacy, or AD, implies...
field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the...
categoricity conjecture for a successor from large cardinals: If there are class-many stronglycompactcardinals, then Shelah's categoricity conjecture holds...
Π 1 1 {\displaystyle \Pi _{1}^{1}} -indescribable cardinals are the same as weakly compactcardinals. The indescribability condition is equivalent to V...
be tame are known: Tameness is a large cardinal axiom: There are class-many almost stronglycompactcardinals iff any abstract elementary class is tame...
theories is called strong minimality: A theory T is called strongly minimal if every model of T is minimal. A structure is called strongly minimal if the...
Global Information