Eventually different functions and inaccessible cardinals
The conditions of eventually different forcing, denoted by \textbf{E}, are pairs \(\langle s,F \rangle\) where \(s\) is a finite sequence of natural numbers and \(F\) is a finite set of reals. The order is given by \(\langle s,F \rangle \leq \langle t, G \rangle\) iff \(t \subseteq s\), \(G \subseteq F\) and for all \(i \in \text{dom}(s \setminus t)\) and all \(g \in G\), we have \(s(i) \neq g(i)\). This forcing generates a topology \(\mathcal E\), the eventually different topology, and refines the standard topology on Baire spaces. The authors investigate the Baire property of the eventually different topology. They obtain a characterization of the \(\mathcal{E}\)-Baire property of the \({\pmb \Sigma}^1_2\) sets by showing (Theorem 7) that the following are equivalent: (i) every \({\pmb \Sigma}^1_2\) set has the \(\mathcal{E}\)-Baire property (i.e., \({\pmb \Sigma}^1_2({\mathbf E}))\), (ii) for every \(x\), the set of \({\mathbf E}\)-generics over \(L[x]\) is \(\mathcal{E}\)-comeager, and (iii) \(\omega_1\) is inaccessible by reals. Further on, the authors show: Let \(G\) be \(L\)-generic for the \(\omega_1\)-stage finite support iteration of Hechler forcing. Then, in \(L[G]\), \({\pmb \Delta}^1_2({\mathbf D})\) holds while \({\pmb \Delta}^1_2({\mathbf E})\) fails.
scientific article; zbMATH DE number 5871033