Forcing the \(\Pi_3^1\)-reduction property and a failure of \(\Pi_3^1\)-uniformization (Q6156423)

The reduction property is one of the three regularity properties of subsets of the reals which were extensively studied by descriptive set theorists, along with the separation and the uniformization property. We say that the \(\Pi^1_n\)-reduction property holds if every pair \(A_0, A_1\) of \(\Pi^1_n\)-subsets of the reals can be reduced by a pair of \(\Pi^1_n\)-sets \(D_0, D_1\), which means that \(D_0\subseteq A_0,\) \(D_1\subseteq A_1,\) \(D_0 \cap D_1 = \emptyset\) and \(D_0\cup D_1 = A_0\cup A_1.\) We say that \(\Pi^1_n\)-uniformization property is true if every \(\Pi^1_n\)-set \(A\subseteq 2^\omega\times 2^\omega\) has a uniformizing function \(f_A\) (i.e. \(f_A: \pi_1[A]\to 2^\omega\) and a graph of \(f_A\) is a subset of \(A\)), whose graph is \(\Pi^1_n\). It is known that the \(\Pi^1_n\)-uniformization property implies the \(\Pi^1_n\)-reduction property. Kondo's theorem states that the \(\Pi^1_1\)-uniformization property is true. However, there was no known example of a universe of set theory where the reduction property holds for some pointclass, yet the corresponding uniformization property fails. In the paper under review, the author constructs such a model (being a generic extension of Gödel's constructible universe \(L\)) in which the \(\Pi^1_3\)-reduction property is true and the \(\Pi^1_3\)-uniformization property is false.