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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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 $ 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.

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Additional Information

Keywords: Borel's conjecture, strong measure zero, universal measure zero, iterated forcing Sacks order, uniformly continuous maps, maps onto $ [0,1]$, proper orders, $ {\omega ^\omega }$-bounding orders
Article copyright: © Copyright 1989 American Mathematical Society

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