The Borel Conjecture is the statement that
C
=
[
R
]
>
ω
1
C = {[\mathbb {R}]^{ > {\omega _1}}}
, where
C
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
=
[
R
]
>
c
C = {[\mathbb {R}]^{ > {\mathbf {c}}}}
. We show that this statement is also independent. The construction involves forcing with an
ω
2
{\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
ω
\omega
-proper,
ω
ω
{\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
]
[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
(
U
0
)
({{\text {U}}_0})
sets (as it is by the presence of large
C
C
sets). We also investigate the
σ
\sigma
-ideal
J
=
{
X
⊂
R
:
X
cannot be mapped uniformly continuously onto
[
0
,
1
]
}
\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
J
,
U
0
\mathcal {J},\;{{\text {U}}_0}
, and AFC (where
AFC
=
{
X
⊂
R
:
X
is always first category}
\operatorname {AFC} = \{ X \subset \mathbb {R}:X\;{\text {is always first category\} }}
). These latter results partially answer two questions of J. Brown.