Determinacy and weakly Ramsey sets in Banach spaces

Bagaria, Joan; López-Abad, Jordi

doi:10.1090/S0002-9947-01-02926-9

Determinacy and weakly Ramsey sets in Banach spaces
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by Joan Bagaria and Jordi López-Abad PDF

Trans. Amer. Math. Soc. 354 (2002), 1327-1349 Request permission

Abstract:

We give a sufficient condition for a set of block subspaces in an infinite-dimensional Banach space to be weakly Ramsey. Using this condition we prove that in the Levy-collapse of a Mahlo cardinal, every projective set is weakly Ramsey. This, together with a construction of W. H. Woodin, is used to show that the Axiom of Projective Determinacy implies that every projective set is weakly Ramsey. In the case of $c_0$ we prove similar results for a stronger Ramsey property. And for hereditarily indecomposable spaces we show that the Axiom of Determinacy plus the Axiom of Dependent Choices imply that every set is weakly Ramsey. These results are the generalizations to the class of projective sets of some theorems from W. T. Gowers, and our paper “Weakly Ramsey sets in Banach spaces.”

References

Joan Bagaria, Bounded forcing axioms as principles of generic absoluteness, Arch. Math. Logic 39 (2000), no. 6, 393–401. MR 1773776, DOI 10.1007/s001530050154
J. Bagaria and J. López-Abad, Weakly Ramsey sets in Banach spaces. Advances in Mathematics, 160 (2001), no. 2, pp. 133–174.
Joan Bagaria, A characterization of Martin’s axiom in terms of absoluteness, J. Symbolic Logic 62 (1997), no. 2, 366–372. MR 1464104, DOI 10.2307/2275537
James E. Baumgartner, Iterated forcing, Surveys in set theory, London Math. Soc. Lecture Note Ser., vol. 87, Cambridge Univ. Press, Cambridge, 1983, pp. 1–59. MR 823775, DOI 10.1017/CBO9780511758867.002
W. T. Gowers, Lipschitz functions on classical spaces, European J. Combin. 13 (1992), no. 3, 141–151. MR 1164759, DOI 10.1016/0195-6698(92)90020-Z
W. T. Gowers, Analytic sets and games in Banach spaces. Preprint IHES M/94/42.
W. T. Gowers, A new dichotomy for Banach spaces, Geom. Funct. Anal. 6 (1996), no. 6, 1083–1093. MR 1421876, DOI 10.1007/BF02246998
W. T. Gowers, An infinite Ramsey theorem and some Banach-space dichotomies. Preprint.
Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
Akihiro Kanamori, The higher infinite, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1994. Large cardinals in set theory from their beginnings. MR 1321144
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1983. An introduction to independence proofs; Reprint of the 1980 original. MR 756630
A. Levy and R. M. Solovay, On the decomposition of sets of reals to Borel sets, Ann. Math. Logic 5 (1972/73), 1–19. MR 311476, DOI 10.1016/0003-4843(72)90016-2
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
J. López-Abad, Weakly Ramsey sets in Banach Spaces, Ph.D. Thesis. Universitat de Barcelona. 2000.
A. R. D. Mathias, Happy families, Ann. Math. Logic 12 (1977), no. 1, 59–111. MR 491197, DOI 10.1016/0003-4843(77)90006-7
Yiannis N. Moschovakis, Descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 561709
Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206, DOI 10.1007/978-3-662-12831-2
Jack Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970), 60–64. MR 332480, DOI 10.2307/2271156
Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 1–56. MR 265151, DOI 10.2307/1970696
B.S. Tsirelson, Not every Banach space contains $\ell _p$ or $c_0$, Functional Analysis and Applications 8 (1974), pp. 138–141 (translated from Russian).
Boban Veli ković, Forcing axioms and stationary sets, Adv. Math. 94 (1992), no. 2, 256–284. MR 1174395, DOI 10.1016/0001-8708(92)90038-M
W. Hugh Woodin, On the consistency strength of projective uniformization, Proceedings of the Herbrand symposium (Marseilles, 1981) Stud. Logic Found. Math., vol. 107, North-Holland, Amsterdam, 1982, pp. 365–384. MR 757040, DOI 10.1016/S0049-237X(08)71895-0