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  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  showed that whenever κ is measurable, there is such j and M. If we had assumed, in addition, that , then κ would be the κth measurable cardinal; in general, the wider we assume M to be, the larger κ must be.
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