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