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

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

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.

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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