4. Strongly compact cardinals and
Definition 1 A cardinal
is strongly compact iff it is uncountable, and any
-complete filter (over any set
) can be extended to a
-complete ultrafilter over
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The notion of strong compactness has its origin in infinitary logic, and was formulated by Tarski as a natural generalization of the compactness of first order logic. Many distinct characterizations have been found.