Does $\mathcal{P}(\ddot{I}‰)/\mathrm{fin}$ know its right hand from its left?

Will Brian

Let $\ddot{I}ƒ$ denote the shift automorphism on $\mathcal{P}(\ddot{I}‰)/\mathrm{fin}$, defined by setting $\ddot{I}ƒ([A])=[A+1]$ for all $A\subseteq \ddot{I}‰$. We show that the Continuum Hypothesis imp…