An undecidable case of lineability in R^R

TL;DR

This paper explores the undecidability of lineability properties in the context of Sierpiński-Zygmund functions, establishing equivalence with set-theoretic assumptions and marking a novel intersection of logic and functional analysis.

Contribution

It proves the equivalence and undecidability of certain lineability statements related to Sierpiński-Zygmund functions under set-theoretic assumptions.

Findings

Lineability of Sierpiński-Zygmund functions is undecidable.

Equivalence between existence of large algebraic structures and set-theoretic axioms.

First demonstration of undecidability in lineability theory.

Abstract

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.