In
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, a strongly compact cardinal is a certain kind of
large cardinal
In the mathematical 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 name suggests, generally very "large" (for example, bigger than the least ...
.
An uncountable 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
An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. The concept was introduced by Zermelo in the 1930s.
Some infinitary logics may have different properties from those of standard first-order lo ...
, where
logical operator
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the ...
s 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
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...
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 κ 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.
is a second-order analog of strong compactness.
See also
*
List of large cardinal properties
Footnotes
References
*
Large cardinals
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