Electronic ISBN:  9781470404802 
Product Code:  MEMO/187/876.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 187; 2007; 118 ppMSC: Primary 03; Secondary 54;
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 nontrivial 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

Chapters

Introduction

1. A tree representation for Borel sets

2. A doubletree representation for Borel sets

3. Two applications of the tree representation

4. Borel liftings of Borel sets

5. More consequences and reverse results

6. Proof of the main result


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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 nontrivial 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.

Chapters

Introduction

1. A tree representation for Borel sets

2. A doubletree representation for Borel sets

3. Two applications of the tree representation

4. Borel liftings of Borel sets

5. More consequences and reverse results

6. Proof of the main result