Extender-based Radin forcing
HTML articles powered by AMS MathViewer
- by Carmi Merimovich PDF
- Trans. Amer. Math. Soc. 355 (2003), 1729-1772 Request permission
Abstract:
We define extender sequences, generalizing measure sequences of Radin forcing.
Using the extender sequences, we show how to combine the Gitik-Magidor forcing for adding many Prikry sequences with Radin forcing.
We show that this forcing satisfies a Prikry-like condition, destroys no cardinals, and has a kind of properness.
Depending on the large cardinals we start with, this forcing can blow the power of a cardinal together with changing its cofinality to a prescribed value. It can even blow the power of a cardinal while keeping it regular or measurable.
References
- James Cummings, A model in which GCH holds at successors but fails at limits, Trans. Amer. Math. Soc. 329 (1992), no. 1, 1–39. MR 1041044, DOI 10.1090/S0002-9947-1992-1041044-9
- Menachem Magidor, Changing cofinality of cardinals, Fund. Math. 99 (1978), no. 1, 61–71. MR 465868, DOI 10.4064/fm-99-1-61-71
- Moti Gitik and Menachem Magidor, The singular cardinal hypothesis revisited, Set theory of the continuum (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 26, Springer, New York, 1992, pp. 243–279. MR 1233822, DOI 10.1007/978-1-4613-9754-0_{1}6
- Moti Gitik, Changing cofinalities and the nonstationary ideal, Israel J. Math. 56 (1986), no. 3, 280–314. MR 882254, DOI 10.1007/BF02782938
- William Mitchell, How weak is a closed unbounded ultrafilter?, Logic Colloquium ’80 (Prague, 1980) Studies in Logic and the Foundations of Mathematics, vol. 108, North-Holland, Amsterdam-New York, 1982, pp. 209–230. MR 673794
- K. L. Prikry, Changing measurable into accessible cardinals, Dissertationes Math. (Rozprawy Mat.) 68 (1970), 55. MR 262075
- Lon Berk Radin, Adding closed cofinal sequences to large cardinals, Ann. Math. Logic 22 (1982), no. 3, 243–261. MR 670992, DOI 10.1016/0003-4843(82)90023-7
- M. Segal, On Powers of Singular Cardinals with Cofinality ${}> \omega$, Master’s Thesis, The Hebrew University of Jerusalem (1995)
- Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955, DOI 10.1007/978-3-662-21543-2
- H. Woodin and J. Cummings, Chapters from an unpublished book on Radin forcing
Additional Information
- Carmi Merimovich
- Affiliation: Computer Science Department, The Academic College of Tel-Aviv, 4 Antokolsky St., Tel-Aviv 64044, Israel
- Email: carmi@mta.ac.il
- Received by editor(s): October 19, 1998
- Published electronically: January 8, 2003
- Additional Notes: This work is a part of research which, hopefully, will become the author’s Ph.D. thesis. It was done at Tel-Aviv University under the supervision of M. Gitik. The author thanks M. Gitik for his help with this work, with other works and just in general.
We thank Asaf Sharon for reading and pointing out some problems in a preliminary version of this work. - © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 1729-1772
- MSC (2000): Primary 03E35, 03E55
- DOI: https://doi.org/10.1090/S0002-9947-03-03202-1
- MathSciNet review: 1953523