# Borel Liftings of Borel Sets: Some Decidable and Undecidable Statements

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*Gabriel Debs; Jean Saint Raymond*

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.

#### Table of Contents

# Table of Contents

## Borel Liftings of Borel Sets: Some Decidable and Undecidable Statements

- Contents v6 free
- Introduction 110 free
- Chapter 1. A Tree Representation for Borel Sets 1120
- Chapter 2. A Double-Tree Representation for Borel Sets 3342
- Chapter 3. Two Applications of the Tree Representation 4958
- Chapter 4. Borel Liftings of Borel Sets 6372
- Chapter 5. More Consequences and Reverse Results 7584
- Chapter 6. Proof of The Main Result 91100
- Bibliography 115124
- Index 117126 free