\(\Sigma ^{1}_{1}\)-definability at uncountable regular cardinals (Q2915903)

Let \(\kappa\) be an uncountable regular cardinal. The set of all functions \(f : \kappa \rightarrow \kappa\) is called the generalized Baire space for \(\kappa\). Let \(\kappa\) be an uncountable regular cardinal with \(\kappa = \kappa^{<\kappa}\), \(A \subseteq {}^{\kappa}\kappa\). The author shows that there is a \(<\kappa\)-closed forcing \(\mathbb{P}\) that satisfies the \(\kappa^+\)-chain condition and forces \(A\) to be a \(\Delta_1^1\)-subset of \({}^{\kappa}\kappa\) in every \(\mathbb{P}\)-generic extension of \(V\).NEWLINENEWLINE The author gives some applications of this result: (1) Given a set \(x\), there is a partial order with the above properties that forces \(x\) to be an element of \(L(\mathcal{P}(\kappa))\). (2) There is a partial order with the above properties forcing the existence of a well-ordering of \({}^{\kappa}\kappa\) whose graph is a \(\Delta_2^1\)-subset of \({}^{\kappa}\kappa \times {}^{\kappa}\kappa\). (3) The author shows that generic absoluteness for \(\Sigma_3^1({}^{\kappa}\kappa)\)-formulae under \(<\kappa\)-closed forcing that satisfy the \(\kappa^+\)-chain condition is inconsistent.