One of the aims of this work is to investigate some
natural properties of Borel sets which are undecidable in $ZFC$. The
authors' starting point is the following elementary, though non-trivial
result: Consider $X \subset 2^\omega\times2^\omega$, set
$Y=\pi(X)$, where $\pi$ denotes the canonical projection of
$2^\omega\times2^\omega$ onto the first factor, and suppose that
$(\star)$: “Any compact subset of $Y$
is the projection of some compact subset of $X$”.
If moreover $X$ is $\mathbf{\Pi}^0_2$ then
$(\star\star)$: “The restriction of
$\pi$ to some relatively closed subset of $X$ is perfect onto
$Y$”
it follows that in the present case $Y$
is also $\mathbf{\Pi}^0_2$. Notice that the reverse implication
$(\star\star)\Rightarrow(\star)$ holds trivially for any $X$
and $Y$.
But the implication $(\star)\Rightarrow (\star\star)$ for an
arbitrary Borel set $X \subset 2^\omega\times2^\omega$ is equivalent
to the statement “$\forall \alpha\in \omega^\omega,
\,\aleph_1$ is inaccessible in $L(\alpha)$”. More
precisely The authors prove that the validity of
$(\star)\Rightarrow(\star\star)$ for all $X \in
\varSigma^0_{1+\xi+1}$, is equivalent to
“$\aleph_\xi^L<\aleph_1$”. However we shall show
independently, that when $X$ is Borel one can, in $ZFC$,
derive from $(\star)$ the weaker conclusion that $Y$ is also
Borel and of the same Baire class as $X$. This last result solves an
old problem about compact covering mappings.
In fact these results are closely related to the following general
boundedness principle Lift$(X, Y)$: “If any
compact subset of $Y$ admits a continuous lifting in $X$,
then $Y$ admits a continuous lifting in $X$”,
where by a lifting of $Z\subset \pi(X)$ in $X$ we mean a
mapping on $Z$ whose graph is contained in $X$. The main
result of this work will give the exact set theoretical strength of this
principle depending on the descriptive complexity of $X$ and
$Y$. The authors also prove a similar result for a variation of
Lift$(X, Y)$ in which “continuous liftings” are
replaced by “Borel liftings”, and which answers a question of H.
Friedman.
Among other applications the authors obtain a complete solution to
a problem which goes back to Lusin concerning the existence of
$\mathbf{\Pi}^1_1$ sets with all constituents in some given
class $\mathbf{\Gamma}$ of Borel sets, improving earlier
results by J. Stern and R. Sami.
The proof of the main result
will rely on a nontrivial representation of Borel sets (in
$ZFC$) of a new type, involving a large amount of
“abstract algebra”. This representation was initially
developed for the purposes of this proof, but has several other
applications.