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The core model for sequences of measures. I

  • William J. Mitchell (a1)

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.

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[1] E. L. Bull Jr. Successive large cardinals. Ann. Math. Logic 15 (1978), 161191.

[3] A. J. Dodd and R. Jensen . The core model. Ann. Math. Logic 20 (1981), 4375.

[5] A. J. Dodd and R. Jensen . The covering lemma for L(U). Ann. Math. Logic 22 (1982), 127135.

[7] R. Jensen . The fine structure of the constructible hierarchy. Ann. Math. Logic 4 (1972), 229309.

[9] A. Kanamori and M. Magidor . The evolution of large cardinals in set theory. In Higher Set Theory, Lecture Notes in Math. vol. 699 (Springer-Verlag, 1978), 99275.

[10] K. Kunen . Some applications of iterated ultrapowers in set theory. Ann. Math. Logic 1 (1970), 179227.

[12] M. Magidor . On the singular cardinals problem, II. Ann. of Math. 106 (1977), 517547.

[18] R. M. Solovay , W. N. Reinhart and A. Kanamori . Strong axioms of infinity and elementary embeddings. Ann. Math. Logic 13 (1978), 73116.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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