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We know from QM that if we measure the spin of a photon along the x axis, and categorize it as either x+ or x-, we cannot with precision do a similar measurement along the y and z axes. This is a fact well supported by experiments and observations (or so I'm told, and believe).

The EPR paradox shows that while we cannot do these multiple measurements on a single photon, we can determine two of the values using entangled pairs of particles for which the values must be opposite in order to preserve angular momentum.

So, by measuring x axis spin on particle A, and y axis spin on particle B, we get precise measurements for both axes for both particles. We know this can be done, and we know that it needs explanation to allow QM to still be correct, and we know QM is correct, so what is the explanation? Is it a collapse of a wave function? Is it an instantaneous transmission of information across space?

Bell’s paper tries to prove that, whatever the explanation is, it involves the observation on particle A having some mysterious instantaneous effect on particle B. He claims to prove, mathematically, that any possible theory involving only local variables can never explain the results observed by Quantum Mechanics.

To do this he starts with a seemingly obvious assumption. There is some function A on a given axis spin measurement a, and any imaginable set of variables λ, such that A(a, λ) = ±1. That is, the result of the observation, with all variables considered, will be either +1 or -1.

What Bell fails to do is take into account the probabilities that QM requires, and that local variable explanations MUST not violate. The fact is, there is a simple way around Bell’s first assumption, and that is that the order of measurement makes a difference. QM predictions are accurate, the probabilities hold out, but there is something deeper going on.

I would suggest that if we set up an "entangled" system and measure x spin, then y spin, and finally z spin, we will get a different answer depending on the order and manner of the observations made. If this is the case, Bell's initial assumption is wrong, and his equations fall apart. His formulation assumes all values along the z axis are accounted for when a particular result of x and y are found, and that pairs of x and y measurements must be distinct for a particular particle being measured. But in fact this is not so, QM requires that it not be so, and there is no reason to believe it is so. Why? Because of the theory of relativity.

What QM really says is, you cannot measure definitely along x, y and z axes in a single measurement. You need multiple measurements to get multiple definite values, and the type and order of the measurements makes a difference. We know from relativity that absolute simultaneity of events at a distance is not meaningful, and so not possible. And so saying that one set of measurements includes all instances of any other set of measurements is not true. Perhaps this is a clue as to WHY relativity is true.

Going back to the math, the following statements related to sets of observations are all assumed to be true by Bell, but are ALL inaccurate:

(x+, y+) = (x+, y+, z+) + (x+, y+, z-)

(x+, y-) = (x+, y-, z+) + (x+, y-, z-)

(x+) = (x+, y+) + (x+, y-)

A(a, λ) = ±1 (as in Bell's paper)

They are inaccurate because some of (x+, y-) may be the same as the (x+, y+), depending on how the measurements were performed. A(a, λ) is undefined and meaningless. p(λ) is also meaningless. Bell's paper, as you can see, is wrong from assumption 1. The actual observations of (x+, y-) could just as well have been (x+, y+), depending on how the observations were performed.

By adding order of measurement, we invalidate Bell's Theorem, maintain the reality of local variables perfectly, while also agreeing with all predictions of QM. The EPR Paradox holds up, which is to say, QM is correct but incomplete. Bell's Theorem, when it comes right down to it, is all a silly misunderstanding.

One final thought. The real shame is that Einstein did not live long enough to put down this silly "inequality" forty years ago.