Evasion and prediction. III: Constant prediction and dominating reals
The paper is part of a series of papers by the author and/or his collaborators related to evasion and prediction, some already published [the author, Forum Math. 7, 513-541 (1995; Zbl 0830.03026); the author and \textit{S. Shelah}, J. Lond. Math. Soc., II. Ser. 53, 19-27 (1996; Zbl 0854.03054)], yet others to be published soon. The constant prediction number \(\mathfrak v_2^{\text{const}}\) was introduced by Kamo in 2000/1 (with notation given by Kada), after the general concept of prediction was introduced by \textit{A. Blass} [J. Algebra 169, 512-540 (1994; Zbl 0816.20047)]; this number represents the size of the least family \(\Pi\) of predictors \(\pi:2^{<\omega}\leftrightarrow 2\), such that, for each \(x\in 2^\omega\) there are \(\pi\in\Pi\) and \(k\) such that for almost all intervals \(I\) of length \(k\), one has \(\pi(x|i)=x(i)\), for some \(i\in I\). The main result is that the unbounding number \(\mathfrak b\leq\mathfrak v_2^{\text{const}}\), which answers a question of Kamo. Kamo's consistency is dualized into consistency of \(\mathfrak e_2^{\text{const}}>\mathfrak b\), and some results of the former are proved in this way. There are a few unsolved problems stated in the paper. The paper perhaps also demonstrates that it can be hard (in the sense of descriptive set theory) to prove ZFC-inequalities between cardinal invariants of the continuum.
scientific article; zbMATH DE number 1943301