 where \(\ket{z,i}_1\), with \(i = -1\) or 0 or 1 is the spin state of particle 1 with component of spin \(i\) along the axis \(z\), and \(\ket{z,i}_2\) has an analogous meaning for particle 2. If \(x,y,z\) is a triple of orthogonal axes in 3-space then the components \(s_x, s_y, s_z\) of the spin operator along these axes do not pairwise commute. However, the squares of these operators — \(s_x^2, s_y^2, s_z^2\) — do commute, and therefore, in view of the considerations of  Section 1,  any two of them can constitute a context in the measurement of the third. If the operator of interest is \(s_z^2\), the axes \(x\) and \(y\) can be any pair of orthogonal axes in the plane perpendicular to \(z\), thus offering an infinite family of contexts for the measurement of \(s_{z}^2\). As noted in  Section 1  Bell exhibited the possibility of a contextual hidden variables theory for a quantum system whose Hilbert space has dimension 3 or greater even though the Bell-Kochen-Specker theorem showed the impossibility of a non-contextual hidden variables theory for such a system. The strategy of the argument is to use the entangled state of Eq. (27) to predict the outcome of measuring \(s_{z}^2\) for particle 2 (for any choice of \(z)\) by measuring its counterpart on particle 1. A specific complete state \(\lambda\) would determine whether \(s_{z}^2\) of 1, measured together with a context in 1, is 0 or 1. Agreement with the quantum mechanical prediction of the entangled state of Eq. (27) implies that \(s_z^2\) of 2 has the same value 0 or 1. But if the factorizability condition (F) is assumed, then the result of measuring \(s_{z}^2\) on 2 must be independent of the remote context, that is, independent of the choice of \(s_{x}^2\) and \(s_y^2\) of 1, hence of 2 because of correlation, for any pair of orthogonal directions \(x\) and \(y\) in the plane perpendicular to \(z\). It follows that the hypothetical theory which supplies the complete state \(\lambda\) is not contextual after all, but maps the set of operators \(s_z^2\) of 2, for any direction \(z\), noncontextually into the pair of values (0, 1). But that is impossible in view of the Bell-Kochen-Specker theorem. The conclusion is that no theory satisfying the factorizability condition (F) is consistent with the quantum mechanical predictions of the entangled state (29).