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Prikry on extenders, revisited

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Abstract

The extender based forcing of Gitik and Magidor is generalized to yield, given any extender j: V å M with critical point κ, a cardinal preserving generic extension with no new bounded subset of κ in which cf(κ) = ω and \(\kappa ^\omega = |j(\kappa )|\).

Assuming a superstrong cardinal exists, the forcing notion is used to construct a model in which the added Prikry sequences are a scale in the normal Prikry sequence.

In addition, several ways to produce generic filter over an iterated ultrapower are presented.

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References

  1. L. Bukovsky, Changing cofinality of a measurable cardinal, Commentationes Mathematicae Universitatis Carolinae 14 (1973), 689–697.

    MATH  MathSciNet  Google Scholar 

  2. J. Cummings and H. Woodin, A book on Radin forcing, In preparation.

  3. P. Dehornoy, Iterated ultrapowers and Prikry forcing, Annals of Mathematical Logic 15 (1978), 109–160.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Gitik, Prikry type forcings, in Handbook of Set Theory (M. Foreman, A. Kanamori and M. Magidor, eds.), Springer, New-York, 2006.

    Google Scholar 

  5. M. Gitik and M. Magidor, The singular cardinal hypothesis revisited, in Set Theory of the Continuum (H. Judah, W. Just and H. Woodin, eds.), Springer, Berlin, 1992, pp. 243–278.

    Google Scholar 

  6. T. Jech, On the cofinality of countable products of cardinal numbers, in A Tribute to Paul Erdős, (A. Baker, B. Bollobás and A. Hajnal, eds.), Cambridge University Press, Cambridge, 1990, pp. 289–305.

    Google Scholar 

  7. P. Koepke, An Iteration Model Violating the Singular Cardinals Hypothesis, in Sets and Proofs (S. Barry Cooper and J. K. Truss, eds.), volume 258 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1999, pp. 95–102.

    Google Scholar 

  8. M. Magidor, On the singular cardinal problem I, Israel Journal of Mathematics 28 (1977), 1–31.

    MATH  MathSciNet  Google Scholar 

  9. K. Prikry, Changing measurable into accessible cardinal, Dissertations Mathematics 68 (1970), 5–52.

    MathSciNet  Google Scholar 

  10. A. Sharon, Generators of pcf, Master’s thesis, Tel-Aviv Univeristy, 2000.

  11. S. Shelah, Proper Forcing, Lecture Notes in Mathematics, volume 940, Springer, Berlin, 1982.

    Google Scholar 

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The writing of this paper was partially supported by the Israel Science Foundation

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Merimovich, C. Prikry on extenders, revisited. Isr. J. Math. 160, 253–280 (2007). https://doi.org/10.1007/s11856-007-0063-1

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  • DOI: https://doi.org/10.1007/s11856-007-0063-1

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