Let X be a norm-closed and not norm-separable subspace of \(\ell^{\infty}({\mathbb{N}})\). The authors study the question of whether X contains a biorthogonal system of cardinality \(2^{\aleph_ 0}\). The answer depends on the set theory which is used. For example, if X is in the projective hierarchy with respect to the \(w^*\)-topology [i.e. in \(\Sigma^ 1_{n+1}\), cf. \textit{Y. N. Moschovakis}, Descriptive Set Theory, Amsterdam (1980; Zbl 0433.03025)] then the answer is in the affirmative provided one assumes that all \(\Pi^ 1_ n\) games on the integers are determined.