High Quality Content by WIKIPEDIA articles! In mathematical set theory, a strongly compact cardinal is a certain kind of large cardinal number; their existence can neither be proven nor disproven from the standard axioms of set theory. A cardinal ? is strongly compact if and only if every ?-complete filter can be extended to a ? 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 ? is defined by requiring the number of operands for each operator to be less than ?; then ? 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 ?.

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