Full-splitting Miller trees and infinitely often equal reals
For every tree \(T \subseteq\,^{< \omega}\omega\), \([T]\) denote, as usual, the set of all branches of \(T\), i.e. \([T] = \{s \in\,^\omega \omega: \forall n < \omega\,[s \upharpoonright n \in T]\}\). A tree \(T \subseteq\,^{< \omega}\omega\) is said to be a \textit{full-splitting Miller tree} if every \(t \in T\) has an extension \(s \in T\) which is \textit{full-splitting}, i.e., \(s^\frown\langle n \rangle \in T\) for every \(n < \omega\), and it is said to be an \textit{infinitely often equal tree}, or simply \textit{ioe-tree}, if for every \(t \in T\) there exists some \(N > |t|\) such that for every \(k < \omega\) there exists \(s \in T\) which extends \(t\) and satisfies \(s(N) = k\). Letting \(\mathbb{FM}\) and \(\mathbb{IE}\) denote the partial orders of full-splitting Miller trees and ioe-trees (both ordered by inclusion), respectively, one has \(\mathbb{FM} \subsetneq \mathbb{IE}\). Two reals \(x,y \in\,^\omega \omega\) are \textit{infinitely often equal (ioe)} if \(\exists^\infty n\, [x(n) = y(n)]\), and are \textit{eventually different (evd)} if \(\forall^\infty n\, [x(n) \neq y(n)]\). \(\mathfrak{D}_\omega\) denotes the \(\sigma\)-ideal generated by \(\{D_f: f \in\,^{(^{<\omega} \omega)} \omega\}\), where for a given function \(f:\, ^{<\omega} \omega \to \omega\) one has \(D_f = \{x \in\,^\omega \omega: \forall^\infty n\, [x(n) \neq f(x\upharpoonright n)]\}\), and \(\mathfrak{J}_{\mathrm{ioe}}\) denotes the \(\sigma\)-ideal generated by \(\{K_x: x \in\,^\omega \omega\}\), where for a given real \(x: \omega \to \omega\) one has \(K_x = \{y \in\,^\omega \omega: \forall^\infty n\, [x(n) \neq y(n)]\}\). One has \(\mathfrak{J}_{\mathrm{ioe}} \subsetneq \mathfrak{D}_\omega \subsetneq \mathcal{M}\), where \(\mathcal{M}\) denotes the ideal of meager subsets of \(^\omega \omega\). \(\mathfrak{J}_{\mathrm{ioe}}\)-positive sets are said to be \textit{countably infinitely often equal families}. In the paper under review, the authors investigate dichotomy theorems asserting that every Borel (or analytic) set is either a member of some \(\sigma\)-ideal \(\mathfrak{J}\) on \(^\omega \omega\) or else it contains the branches of a certain kind of tree. Such theorems imply the existence of certain dense embeddings in forcing theory. \textit{L. Newelski} and \textit{A. Rosłanowski} [Proc. Am. Math. Soc. 117, No. 3, 823--831 (1993; Zbl 0778.03016)] have shown that analytic sets are either \(\mathfrak{D}_\omega\)-small or contain \([T]\) for some \(T \in \mathbb{FM}\), and the proof of a wrongly stated theorem of \textit{O. Spinas} [Fundam. Math. 201, No. 2, 179--195 (2008; Zbl 1159.03033)] establishes that every analytic set is either \(\mathfrak{J}_{\mathrm{ioe}}\)-small or contains \([T]\) for some \(T \in \mathbb{IE}\). Such theorems are investigated from various points of view throughout the paper. For instance, it is a folklore statement that if \(V_0 \subseteq V_1 \subseteq V_2\) are models of set theory such that in \(V_1\) there is an ioe real over \(V_0\) and in \(V_2\) there is an ioe real over \(V_1\), then in \(V_2\) there is a Cohen real over \(V_0\) -- and, consequently, \(\mathbb{IE}*\mathbb{IE}\) adds a Cohen real; for that reason, an ioe real is sometimes called ``half a Cohen real''\,. Only recently \textit{J. Zapletal} [Topology Appl. 167, 31--35 (2014; Zbl 1349.03057)] has answered (in the positive) a question posed by Fremlin in the 90's: is it possible to add an ioe real without adding a Cohen real? However, the methods used by Zapletal in his proof (which uses the notion of infinite topological dimension) are regarded as \textit{rather unorthodox} by the authors of the paper under review. Accordingly, the authors present some partial results towards a purely combinatorial description of a forcing which adds ioe reals but does not add Cohen reals.
scientific article; zbMATH DE number 6720136
scientific article