# Experimental Non-Violation of the Bell Inequality
## 3 The Sequential Stern-Gerlach Experiment  
As a warm up to the Bell Theorem, we discuss in this Section one of the classic experiments designed to introduce students to non-commutativity of spin observables in quantum theory (e.g., [11]). Consider an ensemble of spin-1/2 particles prepared by the first of three Stern-Gerlach apparatuses (Figure 4a) with spins oriented in the direction \(\hat{\mathbf{a}}\) in physical 3-space. The particles that are prepared spin-up by this first apparatus pass through a second Stern-Gerlach apparatus oriented in the direction \(\hat{\mathbf{b}}\). The particles that are output along the spin-up channel of the second apparatus are then passed into a third Stern-Gerlach apparatus oriented in the direction \(\hat{\mathbf{c}}\). The directions \(\hat{\mathbf{a}}\), \(\hat{\mathbf{b}}\) and \(\hat{\mathbf{c}}\) correspond to points \(A\), \(B\) and \(C\) on the celestial sphere \(\hat{\mathbb{S}}^{2}\) (Figure 4b). Typically, the directions \(\hat{\mathbf{a}}\), \(\hat{\mathbf{b}}\) and \(\hat{\mathbf{c}}\) are designed to be coplanar, i.e. \(A\), \(B\) and \(C\) lie on a great circle. However, this is impossible to achieve _precisely._ as a matter of principle, one cannot shield the experiment from the distorting effects of gravitational waves. Hence, as in Figure 4b, we assume that \(A\), \(B\) and \(C\) are the vertices of some non-degenerate triangle \(\triangle ABC\), where the angle \(\gamma\) is not equal to \(180^{\circ}\) precisely. We now show that if a particle was measured by the apparatus in the order A-B-C, then it could not have been measured in the order A-C-B. That is, the measurements in directions \(\hat{\mathbf{b}}\) and \(\hat{\mathbf{c}}\) cannot be performed simultaneously. In Invariant Set Theory, this result is derived by number theory. Consider a particle sent through the sequential Stern-Gerlach apparatus in the configuration A-B-C and where the detector corresponding to either \(c\) or \(\
ot{c}\) was triggered. Then, in Invariant Set Theory, we require that all of \(\cos\theta_{AB}\), \(\cos\theta_{BC}\) and \(\gamma\) must be rational for the experiment to lie on \(I_{U}\) and hence correspond to _reality_. We now ask the question: what would the outcome have been for that particle had the configuration been A-C-B. For there to have been a definite outcome, we require, in addition, that \(\cos\theta_{AC}\in\mathbb{Q}\). However, by the cosine rule for spherical triangles  
Figure 4: (**a**) a sequential Stern-Gerlach experiment where a particle is sent through three Stern-Gerlach devices, A, B and C; (**b**) A, B and C shown as directions on the celestial sphere. Although to experimental accuracy A, B and C may be coplanar, they are not coplanar precisely. In invariant set theory, we demonstrate the non-commutativity of spin observables by number theory. \[\cos\theta_{AC}=\cos\theta_{AB}\cos\theta_{BC}+\sin\theta_{AB}\sin\theta_{BC}\cos\gamma. \tag{4}\]  
The right-hand side is the sum of two terms. The first is rational since it is the product of two terms each of which, by construction, is rational. The second is the product of three terms the last of which, \(\cos\gamma\), is irrational, except for the eight exceptions listed in the Theorem above. Since \(\gamma\) is only approximately equal to \(180^{\circ}\), \(\cos\gamma\) is irrational. Since \(\theta_{AB}\), \(\theta_{BC}\) and \(\gamma\) are independent degrees of freedom defining the triangle \(\triangle ABC\), there is no reason why \(\sin\theta_{AB}\) and \(\sin\theta_{BC}\) should conspire with \(\cos\gamma\) to make the product \(\sin\theta_{AB}\sin\theta_{BC}\cos\gamma\) rational. Hence, \(\cos\theta_{AC}\) is the sum of a rational and an irrational and is therefore irrational. Hence, for the particle where A-B-C was measured, the counterfactual A-C-B is undefined and could not be an element of _reality_.