Evasion numbers via zero-prediction

Abstract

Cruz Chapital, Goto, Hayashi and the author showed that the game-theoretic variants ${\mathfrak{s}}_{{\mathrm{game}}^{\ast }}^{\mathrm{I}}$ and ${\mathfrak{s}}_{{\mathrm{game}}^{\ast \ast }}^{\mathrm{I}}$ of the splitting number $\mathfrak{s}$ are consistently different, although the corresponding two games differ only in a minor case. This result suggests that even if two relational systems $\mathbf{R}=⟨X,Y,⊏⟩$, ${\mathbf{R}}^{\prime }=⟨X,Y,{⊏}^{\prime }⟩$ are the same modulo a countable set $C\subseteq X$, the associated cardinal invariants might be different. We study this phenomenon for the standard relational system of evasion and prediction and for a variation of it. We show that such a difference occurs for the standard one, but not for the variation.

To appear in Proceedings of RIMS Set Theory Workshop 2025