Borel Liftings of Borel Sets : Some Decidable and Undecidable Statements

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 $X subset 2omega imes2omega$, set $Y=pi(X)$, where $pi$ denotes the canonical projection of $2omega imes2omega$ 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 $(starstar)$: 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 $(starstar)Rightarrow(star)$ holds trivially for any $X$ and $Y$. But the implication $(star)Rightarrow (starstar)$ for an arbitrary Borel set $X subset 2omega imes2omega$ is equivalent to the statement $forall alphain omegaomega, , aleph 1$ is inaccessible in $L(alpha)$. More precisely The authors prove that the validity of $(star)Rightarrow(starstar)$ for all $X in varSigma0 {1+xi+1 $, is equivalent to $aleph xi aleph 1$. $ZFC$, 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 $Zsubset pi(X)$ 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. 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.