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The core model for sequences of measures. I

Published online by Cambridge University Press:  24 October 2008

William J. Mitchell
William J. Mitchell
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, U.S.A.
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Extract

The model K() presented in this paper is a new inner model of ZFC which can contain measurable cardinals of high order. Like the model L() of [14], from which it is derived, K() is constructed from a sequence of filters such that K() satisfies for each (α, β) ε domain () that (α,β) is a measure of order β on α and the only measures in K() are the measures (α,β). Furthermore K(), like L(), has many of the basic properties of L: the GCH and ⃟ hold and there is a definable well ordering which is on the reals. The model K() is derived from L() by using techniques of Dodd and Jensen [2–5] to build in absoluteness for measurability and related properties.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 95 , Issue 2 , March 1984 , pp. 229 - 260
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1]Bull, E. L. Jr. Successive large cardinals. Ann. Math. Logic 15 (1978), 161–191.CrossRefGoogle Scholar
[2]Dodd, A. J.. The Core Model. L.M.S. Lecture Notes series no. 61 (Cambridge University Press, 1982).CrossRefGoogle Scholar
[3]Dodd, A. J. and Jensen, R.. The core model. Ann. Math. Logic 20 (1981), 43–75.CrossRefGoogle Scholar
[4]Dodd, A. J. and Jensen, R.. The covering lemma for K. Ann. Math. Logic 22 (1982) 1–30.CrossRefGoogle Scholar
[5]Dodd, A. J. and Jensen, R.. The covering lemma for L(U). Ann. Math. Logic 22 (1982), 127–135.CrossRefGoogle Scholar
[6]Jech, T.. Set Theory. (Academic Press 1978.)Google Scholar
[7]Jensen, R.. The fine structure of the constructible hierarchy. Ann. Math. Logic 4 (1972), 229–309.CrossRefGoogle Scholar
[8]Kanamori, A.. Ultrafilters over a measurable cardinal. Ann. Math. Logic 11 (1976), 315–356.CrossRefGoogle Scholar
[9]Kanamori, A. and Magidor, M.. The evolution of large cardinals in set theory. In Higher Set Theory, Lecture Notes in Math. vol. 699 (Springer-Verlag, 1978), 99–275.CrossRefGoogle Scholar
[10]Kunen, K.. Some applications of iterated ultrapowers in set theory. Ann. Math. Logic 1 (1970), 179–227.CrossRefGoogle Scholar
[11]Kunen, K.. Saturated ideals. J. Symbolic Logic 43 (1978), 65–76.CrossRefGoogle Scholar
[12]Magidor, M.. On the singular cardinals problem, II. Ann. of Math. 106 (1977), 517–547.CrossRefGoogle Scholar
[13]Magidor, M.. Changing cofinality of cardinals. Fund. Math. 99 (1978), 61–71.CrossRefGoogle Scholar
[14]Mitchell, W. J.. Sets constructible from sequences of ultrafilters. J. Symbolic Logic 39 (1974), 57–66.CrossRefGoogle Scholar
[15]Mitchell, W. J.. Indiscernibles, skies and ideals. In Proceedings of the 1983 Boulder Summer Conference in Set Theory. Contemporary Mathematics (American Mathematical Society, in the Press).Google Scholar
[16]Mitchell, W. J.. How weak is a closed unbounded ultrafilter? In Logic Colloquium 1980, Studies in Logic vol. 108 (North-Holland, 1982), 209–230.Google Scholar
[17]Silver, J.. On the singular cardinals problem. In Proc. Internat. Congr. Math., Vancouver 1974, vol. 1, pp. 265–268.Google Scholar
[18]Solovay, R. M., Reinhart, W. N. and Kanamori, A.. Strong axioms of infinity and elementary embeddings. Ann. Math. Logic 13 (1978), 73–116.CrossRefGoogle Scholar
[19]Woodin, H.. Hypermeasures and the closed, unbounded filter. (In the Press.)Google Scholar
[20]Woodin, H.. Private communication.Google Scholar