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Elementary embeddings and infinitary combinatorics

  • Kenneth Kunen (a1)
  • DOI: http://dx.doi.org/10.2307/2269948
  • Published online: 01 March 2014
Abstract

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 κth measurable cardinal; in general, the wider we assume M to be, the larger κ must be.

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[3]K. Gödel , The consistency of the continuum hypothesis, Princeton Univ. Press, Princeton, 1940.

[5]K. Kunen , Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179227.

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