Extender-based Radin forcing
Trans. Amer. Math. Soc. 355 (2003), 1729-1772
Primary 03E35, 03E55
January 8, 2003
Full-text PDF Free Access
Similar Articles |
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.
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
Magidor, Changing cofinality of cardinals, Fund. Math.
99 (1978), no. 1, 61–71. MR 0465868
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
Gitik, Changing cofinalities and the nonstationary ideal,
Israel J. Math. 56 (1986), no. 3, 280–314. MR 882254
Mitchell, How weak is a closed unbounded ultrafilter?, Logic
Colloquium ’80 (Prague, 1980) Stud. Logic Foundations Math.,
vol. 108, North-Holland, Amsterdam, 1982, pp. 209–230. MR 673794
L. Prikry, Changing measurable into accessible cardinals,
Dissertationes Math. Rozprawy Mat. 68 (1970), 55. MR 0262075
Berk Radin, Adding closed cofinal sequences to large
cardinals, Ann. Math. Logic 22 (1982), no. 3,
670992 (83m:03062), http://dx.doi.org/10.1016/0003-4843(82)90023-7
M. Segal, On Powers of Singular Cardinals with Cofinality , Master's Thesis, The Hebrew University of Jerusalem (1995)
Shelah, Proper forcing, Lecture Notes in Mathematics,
vol. 940, Springer-Verlag, Berlin, 1982. MR 675955
H. Woodin and J. Cummings, Chapters from an unpublished book on Radin forcing
- J. Cummings, A Model in which GCH Holds at Successors but Fails at Limits, Transactions of the American Mathematical Society 329(1992), Number 1, 1-39 MR 92h:03076
- M. Magidor, Changing Cofinality Of Cardinals, Fundamenta Mathematicae 99 (1978), 61-71 MR 57:5754
- M. Gitik and M. Magidor, The Singular Cardinal Hypothesis Revisited, in Set Theory of the Continuum, H. Judah, W. Just, H. Woodin, (Eds.), MSRI Publ., vol. 26, Springer-Verlarg (1992), 243-278 MR 95c:03131
- M. Gitik, Changing Cofinalities and the Non-Stationary Ideal, Israel Journal of Mathematics 56 (1986), 280-314 MR 89b:03086
- W. Mitchell, How Weak is a Closed Unbounded Ultrafilter?, in Logic Colloquium '80, D. van Dalen, D. Lascar, J. Smiley, (Eds.), North-Holland Publishing Company (1982), 209-230 MR 84f:03047
- K. Prikry, Changing measurable into accessible cardinals, Diss. Math. 68 (1970), 5-52 MR 41:6685
- L. B. Radin, Adding Closed Cofinal Sequences to Large Cardinals, Annals of Mathematical Logic 22(1982), 243-261 MR 83m:03062
- M. Segal, On Powers of Singular Cardinals with Cofinality , Master's Thesis, The Hebrew University of Jerusalem (1995)
- S. Shelah, Proper Forcing, Lecture Notes in Mathematics, Vol. 940, Springer, Berlin, 1982 MR 84h:03002
- H. Woodin and J. Cummings, Chapters from an unpublished book on Radin forcing
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
Retrieve articles in all journals
with MSC (2000):
Computer Science Department, The Academic College of Tel-Aviv, 4 Antokolsky St., Tel-Aviv 64044, Israel
generalized continuum hypothesis,
singular cardinal hypothesis
Received by editor(s):
October 19, 1998
January 8, 2003
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