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On generic elementary embeddings

  • Moti Gitik (a1)

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Suppose that I is a precipitous ideal over a cardinal κ and j is a generic embedding of I. What is the nature of j? If we assume the existence of a supercompact cardinal then, by Foreman, Magidor and Shelah [FMS], it is quite unclear where some of such j's are coming from. On the other hand, if ¬∃κ0(κ) = κ++, then, by Mitchell [Mi], the restriction of j to the core model is its iterated ultrapower by measures of it. A natural question arising here is if each iterated ultrapower of can be obtained as the restriction of a generic embedding of a precipitous ideal. Notice that there are obvious limitations. Thus the ultrapower of by a measure over λ cannot be obtained as a generic embedding by a precipitous ideal over κλ. But if we fix κ and use iterated ultrapowers of which are based on κ, then the answer is positive. Namely a stronger statement is true:

Theorem. Let τ be an ordinal and κ a measurable cardinal. There exists a generic extension V* of V so that NSℵ1 (the nonstationary ideal on ℵ1) is precipitous and, for every iterated ultrapower i of V of length ≤ τ by measures of V based on κ, there exists a stationary set forcing “the generic ultrapower restricted to V is i”.

Our aim will be to prove this theorem. We assume that the reader is familiar with the paper [JMMiP] by Jech, Magidor, Mitchell and Prikry. We shall use the method of that paper for constructing precipitous ideals. Ideas of Levinski [L] for blowing up 21 preserving precipitousness and of our own earlier paper [Gi] for linking together indiscernibles will be used also.

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[CK]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.
[FMS]Foreman, M., Magidor, M. and Shelah, S., Martin's maximum, saturated ideals, and nonregular ultrafilters, Annals of Mathematics, ser. 2, vol. 127 (1988), pp. 147.
[G]Gaifman, H., Elementary embeddings of models of set-theory and certain subtheories, Axiomatic set theory (Jech, T., editor), Proceedings of Symposia in Pure Mathematics, vol. 13, part 2, American Mathematical Society, Providence, Rhode Island, 1974, pp. 33101.
[Gi]Gitik, M., On nonminimal p-points over a measurable cardinal, Annals of Mathematical Logic, vol. 20 (1981), pp. 269288.
[JMMiP]Jech, T., Magidor, M., Mitchell, W. and Prikry, K., Precipitous ideals, this Journal, vol. 45 (1980), pp. 18.
[K]Kunen, K., Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179227.
[L]Levinski, J.-P., Thèse du Troisième Cycle, Université Paris-VII, Paris, 1980.
[Mi]Mitchell, W., The core model for sequences of measures. I, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 95 (1984), pp. 229260.

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On generic elementary embeddings

  • Moti Gitik (a1)

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