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143:. 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.
150:. 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.
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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
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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.
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An uncountable cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ-complete ultrafilter.
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Set Theory: An
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if and only if it is κ-compact; this was the original definition of that concept.
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which holds for an inaccessible cardinal if and only if it is strongly compact.
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The consistency strength of strong compactness is strictly above that of a
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Strongly compact cardinals were originally defined in terms of
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are allowed to take infinitely many operands. The logic on a
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192:Hachtman, Sherwood; Sinapova, Dima (2020).
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