On the existence of overcomplete sets in some classical nonseparable Banach spaces

For a Banach space X its subset Y⊆X is called overcomplete if |Y|=dens(X) and Z linearly dense in for every Z⊆Y with |Z|=|Y|. In the context of nonseparable spaces this notion was introduced recently by T. Russo J. Somaglia but sets have been considered separable since 1950ties. We prove some absolute consistency results concerning existence nonexistence classical spaces. example: c0(ω1), C([0,ω1]), L1({0,1}ω1), ℓp(ω1), Lp({0,1}ω1) p∈(1,∞) or general WLD density ω1 admit (in ZFC). The ℓ∞, ℓ∞/c0, form C(K) K extremally disconnected, superspaces ℓ1(ω1) do not Whether Johnson-Lindenstrauss generated ℓ∞ c0 characteristic functions elements an almost disjoint family subsets N cardinality admits set undecidable. same refers to all dual balls which are weak⁎ topology. proved refer wider classes several natural open questions remain open.