The generalized Borel conjecture and strongly proper orders
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- by Paul Corazza PDF
- Trans. Amer. Math. Soc. 316 (1989), 115-140 Request permission
Abstract:
The Borel Conjecture is the statement that $C = {[\mathbb {R}]^{ < {\omega _1}}}$, where $C$ is the class of strong measure zero sets; it is known to be independent of ZFC. The Generalized Borel Conjecture is the statement that $C = {[\mathbb {R}]^{ < {\mathbf {c}}}}$. We show that this statement is also independent. The construction involves forcing with an ${\omega _2}$-stage iteration of strongly proper orders; this latter class of orders is shown to include several well-known orders, such as Sacks and Silver forcing, and to be properly contained in the class of $\omega$-proper, ${\omega ^\omega }$-bounding orders. The central lemma is the observation that A. W. Miller’s proof that the statement $({\ast })$ "Every set of reals of power c can be mapped (uniformly) continuously onto $[0,1]$" holds in the iterated Sacks model actually holds in several other models as well. As a result, we show for example that $({\ast })$ is not restricted by the presence of large universal measure zero $({{\text {U}}_0})$ sets (as it is by the presence of large $C$ sets). We also investigate the $\sigma$-ideal $\mathcal {J} = \{ X \subset \mathbb {R}:X\;{\text {cannot be mapped uniformly continuously onto }}[0,1]\}$ and prove various consistency results concerning the relationships between $\mathcal {J},\;{{\text {U}}_0}$, and AFC (where $\operatorname {AFC} = \{ X \subset \mathbb {R}:X\;{\text {is always first category\} }}$). These latter results partially answer two questions of J. Brown.References
- 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
- James E. Baumgartner and Richard Laver, Iterated perfect-set forcing, Ann. Math. Logic 17 (1979), no. 3, 271–288. MR 556894, DOI 10.1016/0003-4843(79)90010-X J. B. Brown, Countable Baire order and singular sets, unpublished manuscript.
- Jack B. Brown and Karel Prikry, Variations on Lusin’s theorem, Trans. Amer. Math. Soc. 302 (1987), no. 1, 77–86. MR 887497, DOI 10.1090/S0002-9947-1987-0887497-6 J. B. Brown and C. Cox, Classical theory of totally imperfect sets, Real Anal. Exchange 7 (1982).
- Kenneth Kunen and Jerry E. Vaughan (eds.), Handbook of set-theoretic topology, North-Holland Publishing Co., Amsterdam, 1984. MR 776619 D. H. Fremlin, Cichon’s diagram, presented at the Séminaire Initiation a l’Analyse, G. Choquet, M. Rogalski, J. Saint Raymond, at the Universite Pierre et Marie Curie, Paris, 23e annee, 1983/1984, #5, 13 pp.
- Arnold W. Miller and David H. Fremlin, On some properties of Hurewicz, Menger, and Rothberger, Fund. Math. 129 (1988), no. 1, 17–33. MR 954892, DOI 10.4064/fm-129-1-17-33
- Edward Grzegorek, Solution of a problem of Banach on $\sigma$-fields without continuous measures, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), no. 1-2, 7–10 (1981) (English, with Russian summary). MR 616191
- E. Grzegorek, Always of the first category sets, Proceedings of the 12th winter school on abstract analysis (Srní, 1984), 1984, pp. 139–147. MR 782712 —, Always of the first category sets. II, unpublished manuscript, 1985.
- Fred Galvin and Arnold W. Miller, $\gamma$-sets and other singular sets of real numbers, Topology Appl. 17 (1984), no. 2, 145–155. MR 738943, DOI 10.1016/0166-8641(84)90038-5
- J. R. Isbell, Spaces without large projective subspaces, Math. Scand. 17 (1965), 89–105. MR 196695, DOI 10.7146/math.scand.a-10766
- Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
- T. Jech, Multiple forcing, Cambridge Tracts in Mathematics, vol. 88, Cambridge University Press, Cambridge, 1986. MR 895139
- Kenneth Kunen and Jerry E. Vaughan (eds.), Handbook of set-theoretic topology, North-Holland Publishing Co., Amsterdam, 1984. MR 776619
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- Richard Laver, On the consistency of Borel’s conjecture, Acta Math. 137 (1976), no. 3-4, 151–169. MR 422027, DOI 10.1007/BF02392416
- Arnold W. Miller, Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), no. 1, 93–114. MR 613787, DOI 10.1090/S0002-9947-1981-0613787-2
- Arnold W. Miller, Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), no. 3, 575–584. MR 716618, DOI 10.2307/2273449
- Arnold W. Miller, Special subsets of the real line, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 201–233. MR 776624 E. Szpilrajn (Marczewski), On absolutely measurable sets and functions, C. R. Soc. Sci. Varsovie (3) 30 (1937), 39-68. (Polish)
- Janusz Pawlikowski, Why Solovay real produces Cohen real, J. Symbolic Logic 51 (1986), no. 4, 957–968. MR 865922, DOI 10.2307/2273908 F. Rothberger, Eine Verscharfung dei Eigenschaft $C$, Fund. Math. 30 (1938), 50-55.
- Gerald E. Sacks, Forcing with perfect closed sets, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 331–355. MR 0276079
- Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955
- W. Sierpiński, Sur la non-invariance topologique de la propriété $\lambda ’$, Fund. Math. 33 (1945), 264–268 (French). MR 17332, DOI 10.4064/fm-33-1-264-268 J. Walsh, Marczewski sets, measure and the Baire property, Dissertation, Auburn University, 1984.
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 316 (1989), 115-140
- MSC: Primary 03E35; Secondary 04A15, 26A21, 28A05
- DOI: https://doi.org/10.1090/S0002-9947-1989-0982239-X
- MathSciNet review: 982239