One of the standard ways of postulating large cardinal axioms is to consider elementary embeddings, j , from the universe, V , into some transitive submodel, M . See Reinhardt–Solovay [7] for more details. If j is not the identity, and κ is the first ordinal moved by j , then κ is a measurable cardinal. Conversely, Scott [8] showed that whenever κ is measurable, there is such j and M . If we had assumed, in addition, that , then κ would be the κ th measurable cardinal; in general, the wider we assume M to be, the larger κ must be.
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Kenneth Kunen (1971) studied this question.