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The relationship between the large cardinal notions of strong compactness andsupercompactness cannot be determined under the standard ZFC axioms of settheory. Under a hypothesis called the Ultrapower Axiom, we prove that thenotions are equivalent except for a class of counterexamples identified byMenas. This is evidence that strongly compact and supercompact cardinals areequiconsistent.

The Ultrapower Axiom and the equivalence between strong compactness and supercompactness