An undecidable case of lineability in R-R

Abstract

Recently, it has been proved that, assuming that there is an almost disjoint family of cardinality 2(c) in c (which is assured, for instance, by either Martin's Axiom, or the Continuum Hypothesis, or even 2(<c) = c) one has that the set of Sierpinski-Zygmund functions is 2(c)-strongly algebrable (and, thus, 2(c)-lineable). Here we prove that these two statements are actually equivalent and, moreover, that they both are undecidable. This would be the first time in which one encounters an undecidable proposition in the recently coined theory of lineability and spaceability.