The generalized Borel conjecture and strongly proper orders
Author:
Paul Corazza
Journal:
Trans. Amer. Math. Soc. 316 (1989), 115140
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 wellknown 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
(87c:03099), http://dx.doi.org/10.1017/CBO9780511758867.002
 [BL]
James
E. Baumgartner and Richard
Laver, Iterated perfectset forcing, Ann. Math. Logic
17 (1979), no. 3, 271–288. MR 556894
(81a:03050), http://dx.doi.org/10.1016/00034843(79)90010X
 [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J. B. Brown, Countable Baire order and singular sets, unpublished manuscript.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 (88e:26003), http://dx.doi.org/10.1090/S00029947198708874976
 [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 settheoretic topology,
NorthHolland Publishing Co., Amsterdam, 1984. MR 776619
(85k:54001)
 [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 (89g:54061)
 [G1]
Edward
Grzegorek, Solution of a problem of Banach on 𝜎fields
without continuous measures, Bull. Acad. Polon. Sci. Sér. Sci.
Math. 28 (1980), no. 12, 7–10 (1981) (English,
with Russian summary). MR 616191
(82h:04005)
 [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, 𝛾sets and other singular sets of real
numbers, Topology Appl. 17 (1984), no. 2,
145–155. MR
738943 (85f:54011), http://dx.doi.org/10.1016/01668641(84)900385
 [Is]
J.
R. Isbell, Spaces without large projective subspaces, Math.
Scand. 17 (1965), 89–105. MR 0196695
(33 #4882)
 [J1]
Thomas
Jech, Set theory, Academic Press [Harcourt Brace Jovanovich
Publishers], New York, 1978. Pure and Applied Mathematics. MR 506523
(80a:03062)
 [J2]
T.
Jech, Multiple forcing, Cambridge Tracts in Mathematics,
vol. 88, Cambridge University Press, Cambridge, 1986. MR 895139
(89h:03001)
 [K]
Kenneth
Kunen and Jerry
E. Vaughan (eds.), Handbook of settheoretic topology,
NorthHolland Publishing Co., Amsterdam, 1984. MR 776619
(85k:54001)
 [Ku]
K.
Kuratowski, Topology. Vol. I, New edition, revised and
augmented. Translated from the French by J. Jaworowski, Academic Press, New
York, 1966. MR
0217751 (36 #840)
 [La]
Richard
Laver, On the consistency of Borel’s conjecture, Acta
Math. 137 (1976), no. 34, 151–169. MR 0422027
(54 #10019)
 [M1]
Arnold
W. Miller, Some properties of measure and
category, Trans. Amer. Math. Soc.
266 (1981), no. 1,
93–114. MR
613787 (84e:03058a), http://dx.doi.org/10.1090/S00029947198106137872
 [M2]
Arnold
W. Miller, Mapping a set of reals onto the reals, J. Symbolic
Logic 48 (1983), no. 3, 575–584. MR 716618
(84k:03125), http://dx.doi.org/10.2307/2273449
 [M3]
Arnold
W. Miller, Special subsets of the real line, Handbook of
settheoretic topology, NorthHolland, Amsterdam, 1984,
pp. 201–233. MR 776624
(86i:54037)
 [Ma]
E. Szpilrajn (Marczewski), On absolutely measurable sets and functions, C. R. Soc. Sci. Varsovie (3) 30 (1937), 3968. (Polish)
 [P]
Janusz
Pawlikowski, Why Solovay real produces Cohen real, J. Symbolic
Logic 51 (1986), no. 4, 957–968. MR 865922
(88e:03076), http://dx.doi.org/10.2307/2273908
 [R]
F. Rothberger, Eine Verscharfung dei Eigenschaft , Fund. Math. 30 (1938), 5055.
 [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
(43 #1827)
 [Sh]
Saharon
Shelah, Proper forcing, Lecture Notes in Mathematics,
vol. 940, SpringerVerlag, Berlin, 1982. MR 675955
(84h:03002)
 [Si]
W.
Sierpiński, Sur la noninvariance topologique de la
propriété 𝜆’, Fund. Math.
33 (1945), 264–268 (French). MR 0017332
(8,140c)
 [W]
J. Walsh, Marczewski sets, measure and the Baire property, Dissertation, Auburn University, 1984.
 [Ba]
 J. Baumgartner, Iterated forcing, Surveys in Set Theory (A.R.D. Mathias, ed.), Cambridge Univ. Press, 1983, pp. 159. MR 823775 (87c:03099)
 [BL]
 J. Baumgartner and R. Laver, Iterated perfect set forcing, Ann. Math. Logic 17 (1979), 271288. MR 556894 (81a:03050)
 [Br1]
 J. B. Brown, Countable Baire order and singular sets, unpublished manuscript.
 [Br2]
 J. B. Brown and K. Prikry, Variation on Lusin's Theorem, Trans. Amer. Math. Soc. 302 (1987), 7785. MR 887497 (88e:26003)
 [BrC]
 J. B. Brown and C. Cox, Classical theory of totally imperfect sets, Real Anal. Exchange 7 (1982).
 [vD]
 E. van Douwen, The integers and topology, Handbook of Set Theoretic Topology (K. Kunen and J. Vaughan, eds.), NorthHolland, 1984. MR 776619 (85k:54001)
 [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]
 D. H. Fremlin and A. W. Miller, On some properties of Hurewicz, Menger, and Rothberger Fund. Math. 129 (1988), 1733. MR 954892 (89g:54061)
 [G1]
 E. Grzegorek, Solution of a problem of Banach on fields without continuous measures, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), 710. MR 616191 (82h:04005)
 [G2]
 , Always of the first category sets, Proceedings of the 12th Winter School on Abstract Analysis, Srni (Bohemian Weald), January 1528, 1984, Section of Topology, Supplemento ai Rend. Circ. Mat. Palermo (2) 6 (1984), 139147. MR 782712
 [G3]
 , Always of the first category sets. II, unpublished manuscript, 1985.
 [GM]
 F. Galvin and A. W. Miller, sets and other singular sets, Topology Appl. 17 (1984), 145155. MR 738943 (85f:54011)
 [Is]
 J. R. Isbell, Spaces without large projective subspaces, Math. Scand. 17 (1965), 80105. MR 0196695 (33:4882)
 [J1]
 T. Jech, Set theory, Academic Press, New York, 1978. MR 506523 (80a:03062)
 [J2]
 , Multiple forcing, Cambridge Univ. Press, 1986. MR 895139 (89h:03001)
 [K]
 K. Kunen, Random and Cohen reals, Handbook of SetTheoretic Topology (K. Kunen and J. Vaughn, eds.), NorthHolland, 1984. MR 776619 (85k:54001)
 [Ku]
 K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966. MR 0217751 (36:840)
 [La]
 R. Laver, On the consistency of Borel's conjecture, Acta Math. 137, 151169. MR 0422027 (54:10019)
 [M1]
 A. W. Miller, Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), 93114. MR 613787 (84e:03058a)
 [M2]
 , Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), 575584. MR 716618 (84k:03125)
 [M3]
 , Special subsets of the real line, Handbook of SetTheoretic Topology (K. Kunen and J. Vaughn, eds.), NorthHolland, 1984. MR 776624 (86i:54037)
 [Ma]
 E. Szpilrajn (Marczewski), On absolutely measurable sets and functions, C. R. Soc. Sci. Varsovie (3) 30 (1937), 3968. (Polish)
 [P]
 J. Pawlikowski, Why Solovay real produces Cohen real, J. Symbolic Logic 51 (1986), 957968. MR 865922 (88e:03076)
 [R]
 F. Rothberger, Eine Verscharfung dei Eigenschaft , Fund. Math. 30 (1938), 5055.
 [S]
 G. E. Sacks, Forcing with perfect closed sets, Axiomatic Set Theory (D. Scott, ed.), Proc. Sympos. Pure Math., vol. 13, part 2, Amer. Math. Soc., Providence, R.I., 1971, pp. 331355. MR 0276079 (43:1827)
 [Sh]
 S. Shelah, Proper forcing, Lecture Notes in Math., vol. 940, SpringerVerlag, 1982. MR 675955 (84h:03002)
 [Si]
 W. Sierpiński, Sur la nonvariance topologique de la propriété , Fund. Math. 33 (1945), 264268. MR 0017332 (8:140c)
 [W]
 J. Walsh, Marczewski sets, measure and the Baire property, Dissertation, Auburn University, 1984.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719890982239X
PII:
S 00029947(1989)0982239X
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
