Abstract
Abstract
Erdős [7] proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family
$\mathcal {F}$
of (real or complex) analytic functions, such that
$\big \{ f(x) \ : \ f \in \mathcal {F} \big \}$
is countable for every x. We strengthen Erdős’ result by proving that CH is equivalent to the existence of what we call sparse analytic systems of functions. We use such systems to construct, assuming CH, an equivalence relation
$\sim $
on
$\mathbb {R}$
such that any ‘analytic-anonymous’ attempt to predict the map
$x \mapsto [x]_\sim $
must fail almost everywhere. This provides a consistently negative answer to a question of Bajpai-Velleman [2].
Publisher
Cambridge University Press (CUP)
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Theoretical Computer Science,Analysis
Cited by
1 articles.
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1. Wetzel families and the continuum;Journal of the London Mathematical Society;2024-05-13