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Elementary embeddings and infinitary combinatorics

  • Kenneth Kunen (a1)

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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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[1]Erdös, P. and Hajnal, A., On a problem of B. Jónsson, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 14 (1966), pp. 1923.
[2]Gaifman, H., Pushing up the measurable cardinal, Lecture notes 1967 Institute on Axiomatic Set Theory (University of California, Los Angeles, Calif., 1967), American Mathematical Society, Providence, R.I., 1967, pp. IV R 1–16. (Mimeographed).
[3]Gödel, K., The consistency of the continuum hypothesis, Princeton Univ. Press, Princeton, 1940.
[4]Kelley, J. L., General topology, D. Van Nostrand Co., Inc., Princeton, 1965.
[5]Kunen, K., Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179227.
[6]Kunen, K., On the GCH at measurable cardinals, Proceedings of 1969 Logic Summer School at Manchester, pp. 107110.
[7]Reinhardt, W. and Solovay, R., Strong axioms of infinity and elementary embeddings, to appear.
[8]Scott, D., Measurable cardinals and constructible sets, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 9 (1961) pp. 521524.

Elementary embeddings and infinitary combinatorics

  • Kenneth Kunen (a1)

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