The generalized Borel conjecture and strongly proper orders

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.