Sparse analytic systems

Abstract Erdős [7] proved that the Continuum Hypothesis (CH) is equivalent to existence of an uncountable family $\mathcal {F}$ (real or complex) analytic functions, such $\big \{ f(x) \ : f \in \mathcal {F} \big \}$ countable for every x . We strengthen Erdős’ result by proving CH what we call sparse systems functions. use construct, assuming CH, equivalence relation $\sim $ on $\mathbb {R}$ any ‘analytic-anonymous’ attempt predict map $x \mapsto [x]_\sim must fail almost everywhere. This provides a consistently negative answer question Bajpai-Velleman [2].