The generalized Borel conjecture and strongly proper orders

Author:
Paul Corazza

Journal:
Trans. Amer. Math. Soc. **316** (1989), 115-140

MSC:
Primary 03E35; Secondary 04A15, 26A21, 28A05

MathSciNet review:
982239

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Abstract: The Borel Conjecture is the statement that , where is the class of strong measure zero sets; it is known to be independent of ZFC. The Generalized Borel Conjecture is the statement that . We show that this statement is also independent. The construction involves forcing with an -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 -proper, -bounding orders. The central lemma is the observation that A. W. Miller's proof that the statement "Every set of reals of power c can be mapped (uniformly) continuously onto " holds in the iterated Sacks model actually holds in several other models as well. As a result, we show for example that is not restricted by the presence of large universal measure zero sets (as it is by the presence of large sets). We also investigate the -ideal and prove various consistency results concerning the relationships between , and AFC (where ). These latter results partially answer two questions of J. Brown.

**[Ba]**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**, 10.1017/CBO9780511758867.002**[BL]**James E. Baumgartner and Richard Laver,*Iterated perfect-set forcing*, Ann. Math. Logic**17**(1979), no. 3, 271–288. MR**556894**, 10.1016/0003-4843(79)90010-X**[Br1]**J. B. Brown,*Countable Baire order and singular sets*, unpublished manuscript.**[Br2]**Jack B. Brown and Karel Prikry,*Variations on Lusin’s theorem*, Trans. Amer. Math. Soc.**302**(1987), no. 1, 77–86. MR**887497**, 10.1090/S0002-9947-1987-0887497-6**[BrC]**J. B. Brown and C. Cox,*Classical theory of totally imperfect sets*, Real Anal. Exchange**7**(1982).**[vD]**Kenneth Kunen and Jerry E. Vaughan (eds.),*Handbook of set-theoretic topology*, North-Holland Publishing Co., Amsterdam, 1984. MR**776619****[F]**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.**[FM]**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****[G1]**Edward Grzegorek,*Solution of a problem of Banach on 𝜎-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****[G2]**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****[G3]**-,*Always of the first category sets*. II, unpublished manuscript, 1985.**[GM]**Fred Galvin and Arnold W. Miller,*𝛾-sets and other singular sets of real numbers*, Topology Appl.**17**(1984), no. 2, 145–155. MR**738943**, 10.1016/0166-8641(84)90038-5**[Is]**J. R. Isbell,*Spaces without large projective subspaces*, Math. Scand.**17**(1965), 89–105. MR**0196695****[J1]**Thomas Jech,*Set theory*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics. MR**506523****[J2]**T. Jech,*Multiple forcing*, Cambridge Tracts in Mathematics, vol. 88, Cambridge University Press, Cambridge, 1986. MR**895139****[K]**Kenneth Kunen and Jerry E. Vaughan (eds.),*Handbook of set-theoretic topology*, North-Holland Publishing Co., Amsterdam, 1984. MR**776619****[Ku]**K. Kuratowski,*Topology. Vol. I*, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR**0217751****[La]**Richard Laver,*On the consistency of Borel’s conjecture*, Acta Math.**137**(1976), no. 3-4, 151–169. MR**0422027****[M1]**Arnold W. Miller,*Some properties of measure and category*, Trans. Amer. Math. Soc.**266**(1981), no. 1, 93–114. MR**613787**, 10.1090/S0002-9947-1981-0613787-2**[M2]**Arnold W. Miller,*Mapping a set of reals onto the reals*, J. Symbolic Logic**48**(1983), no. 3, 575–584. MR**716618**, 10.2307/2273449**[M3]**Arnold W. Miller,*Special subsets of the real line*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 201–233. MR**776624****[Ma]**E. Szpilrajn (Marczewski),*On absolutely measurable sets and functions*, C. R. Soc. Sci. Varsovie (3)**30**(1937), 39-68. (Polish)**[P]**Janusz Pawlikowski,*Why Solovay real produces Cohen real*, J. Symbolic Logic**51**(1986), no. 4, 957–968. MR**865922**, 10.2307/2273908**[R]**F. Rothberger,*Eine Verscharfung dei Eigenschaft*, Fund. Math.**30**(1938), 50-55.**[S]**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****[Sh]**Saharon Shelah,*Proper forcing*, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR**675955****[Si]**W. Sierpiński,*Sur la non-invariance topologique de la propriété 𝜆’*, Fund. Math.**33**(1945), 264–268 (French). MR**0017332****[W]**J. Walsh,*Marczewski sets, measure and the Baire property*, Dissertation, Auburn University, 1984.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0982239-X

Keywords:
Borel's conjecture,
strong measure zero,
universal measure zero,
iterated forcing Sacks order,
uniformly continuous maps,
maps onto ,
proper orders,
-bounding orders

Article copyright:
© Copyright 1989
American Mathematical Society