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A minimal Prikry-type forcing for singularizing a measurable cardinal

  • Peter Koepke (a1), Karen Räsch (a1) and Philipp Schlicht (a1)

Abstract

Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by /finite. By modifying the standard Prikry tree forcing we define a Prikry-type forcing which also singularizes a measurable cardinal but which is minimal, i.e., there are no intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting from the normality of the associated measures.

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[1]Dodd, Anthony J. and Jensen, Ronald B., The covering lemma for L[U], Annals of Mathematical Logic, vol. 22 (1982), pp. 127135.
[2]Gitik, Moti, Prikry-type forcings, Handbook of set theory (Foreman, M. and Kanamori, A., editors), vol. 2, Springer, 2010, pp. 13511448.
[3]Gitik, Moti, Kanovei, Vladimir, and Koepke, Peter, Intermediate models of Prikry generic extensions, preprint, 2010.
[4]Gray, C. W., Iterated forcing from the strategic point of view, Ph.D. thesis, University of California, Berkeley, California, 1980.
[5]Jech, Thomas, Set theory, revised and expanded, third millennium ed., Springer, 2006.
[6]Judah, Haim and Shelah, Saharon, -Assets of reals, Annals of Pure and Applied Logic, vol. 42 (1989), pp. 207223.
[7]Kunen, Kenneth and Paris, Jeff B., Boolean extensions and measurable cardinals, Annals of Mathematical Logic, vol. 2 (1970/1971), no. 4, pp. 359377.
[8]Mathias, Adrian R. D., On sequences generic in the sense of Prikry, Journal of the Australian Mathematical Society, vol. 15 (1973), pp. 409416.
[9]Prikry, Karel L., Changing measurable into accessible cardinals, Dissertationes Mathematicae, vol. 68 (1970), pp. 152.
[10]Räsch, Karen, Intermediate models of generic extensions by Prikry forcings, Master's thesis, Universität Bonn, 2010.
[11]Sacks, Gerald E., Forcing with perfect closed sets, Axiomatic set theory (Scott, D., editor), Proceedings of Symposia in Pure Mathematics, vol. XIII, part I, American Mathematical Society, Providence, R.I., 1971, pp. 5868.

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