Borel Liftings of Borel Sets (eBook)

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 $(/star)$: "e;Any compact subset of $Y$ is the projection of some compact subset of $X$"e;. If moreover $X$ is $/mathbf{/Pi}^0_2$ then $(/star/star)$: "e;The restriction of $/pi$ to some relatively closed subset of $X$ is perfect onto $Y$"e; 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 "e;$/forall /alpha/in /omega^/omega, /,/aleph_1$ is inaccessible in $L(/alpha)$"e;. 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 "e;$/aleph_/xi^L</aleph_1$"e;. 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$(X, Y)$: "e;If any compact subset of $Y$ admits a continuous lifting in $X$, then $Y$ admits a continuous lifting in $X$"e;, 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 "e;continuous liftings"e; are replaced by "e;Borel liftings"e;, 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 "e;abstract algebra"e;. This representation was initially developed for the purposes of this proof, but has several other applications.