Well, the Realism condition is the part just after (14) where it says: "It follows that c is another unit vector [in addition to a and b already referenced]". That's when Bell sets up the relationship between 3 observables. Those 3 can't simultaneously exhibit the Quantum Mechanical expectation value.

I'm still not getting the conclusions that seem to be drawn from EPR and Bell and I think its down to the meaning of "hidden variable theories". In my opinion a hidden variable theory would be, at the very least, just as powerful as any currently accepted quantum theory in that, for example, it would make exactly the same predictions. If it doesn't conform to the observations it shouldn't be called a theory. But I think if any hidden variable theories are developed they would have some extra feature(s), the hidden variables, which quantum theory doesn't have at the present time. Those hidden variables will account for any perceived apparent discrepancies or weirdness in quantum theories.

I think Einstein said that QM is not complete and I think that is as true today as when he first said it. I think quantum theorists are still hard at work. And there is a possibility that a hidden variable theory will be developed. But until that happens we have no knowledge of what that theory is. And that illustrates one of my sticking points because the impression is often given that Bell, and the subsequent testing of his theory ruled out the possibility of a certain type of hidden variable theory. But to rule out a theory you've got to know the full details of that theory not just one assumption you think is made in developing that theory.

Well, that's the power of mathematics. If you have a real number in mind, I can tell you that its square is not equal to -1. I don't need to know all the decimal places of your number to reach that conclusion. In Newtonian mechanics, if there is a force [itex]F(\vec{x})[/itex] acting on a particle that is constant in time and independent of the velocity of the particle, then I can tell you that the combination of potential energy and kinetic energy is constant. I don't need to know the details of the force.

Mathematics allows you to prove facts about a huge class of situations. You don't need to know precise details about your situation---it's enough to know that it's of a particular type.

Your analogies might be OK for the situations you describe but I don't think they're necessarily relevant to the point I'm trying to make.

A hidden variables theory, should one be developed, will accept all experimental results and will be just as good, if not better, than existing quantum theories at predicting those results. The theory will encompass the findings that have been made during the experimental testing of Bells paper and will encompass all other relevant experimental findings. In fact I can't imagine Einstein et al envisioning a theory that did not conform to observations because such a theory would be a failure.

I think hidden variables is an aspect of a theory, which has not yet been realised and which accounts for any perceived apparent weirdness or discrepancies in the currently known theories. It may be something complicated or it may be something very simple, perhaps an additional sentence or two pointing out something that has been overlooked. Exactly what the hidden variable are remains to be seen. Or perhaps will never be seen. I like to keep an open mind.

While EPR is talking about reality Bell's argument talks about theories. It says: "This [non-local structure] is charateristic, according to the results to be proved here, of any such theory which reproduces exactly quantum mechanical predictions."
And predictions of a theory we can recalculate as many times as we want. So as long as we can take description of initial state the same for different test parameters (description of initial state does not depend on later test parameters i.e. theory is not superdeterministic) we can redo the calculation for different test parameters and have all these results at our disposal at the same time. And this is exactly what is expected of that hypothetical theory when it says: "It follows that c is another unit vector" i.e. we get prediction for different test parameter but with the same ##\lambda##.

Right. That's the type of theory that Bell was interested in---one that made exactly the same predictions as QM (at least for experiments where QM proved to be correct). That's the type of theory that his proof is about.

So you agree that sometimes we can say something can't be true even when we don't know all the details about the situation, right? Say if I claim that I have made many small purchases and the money spent together is more than I had initially, would you say I got it wrong somewhere even without asking what exactly where those small purchases and how much I spent on every purchase?

Then what type of argument would convince you that particular assumptions give us enough information to conclude that any such a theory is impossible? Well, we have to make some inferences from assumptions that we make. How many steps would be acceptable for you? Say we can examine every step for some length so that you can be certain there are no holes in that particular step.

Sorry I don't understand this. You say that Bell was interested in a theory that made the same predictions as QM but the theory that was tested had observations that did not make the same predictions as QM. It cannot be described as a theory. I think what Bell and the experimenters did was to show that a theory, as they interpret(ed) it, was a failure because it predicted results that were not borne out by experiment. As I said before, to be classified as a theory a hidden variables theory must, among other things, conform to the observations. Bell and others did not analyse and test a hidden variable theory .They tested what they thought might be a hidden variable theory and then went on to prove it wasn't a theory at all because it did not meet the necessary criteria. None of that means to say that a proper hidden variable theory will not appear in the future.

Bell used EPR's elements of reality as the basis for his paper. No, he did not label it as such except by the title of the paper.

And the statement you quote simple is a restatement of the idea that the only hidden variable theories that are viable are ones in which the setting of Alice influences the outcome for Bob, however remote. Clearly, the realism assumption can be dropped and then that is not an issue. With the current evidence, I am not sure how it makes sense to say the non-commuting observables have definite values at all times. Which was essentially the assertion of EPR (that a more "compete" theory was possible).

Well, yes Bell is using EPR argument to conclude there can be more complete theory if we assume locality.
"Since we can predict in advance [assuming locality] the result of measuring any chosen component of σ2, by previously measuring the same component of σ1, it follows that the result of any such measurement must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state."
Your objection as I understand is that this "any" is applied to the same initial configuration in the quoted text from Bell paper.
So if we drop "any" from above have we got rid of that EPR assumption? Is this modified inference correct without relaying on that EPR assumption:
"Since we can predict in advance [assuming locality] the result of measuring chosen component of σ2, by previously measuring the same component of σ1, it follows that the result of that measurement must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state for that component of σ2."

Bell considered two predictions of QM:
#1 perfect correlations when measurement angles are the same
#2 imperfect correlations when measurement angles are not the same
He considered all the theories of certain type that satisfied QM prediction #1 and then demonstrated that there is no way how these theories could make QM prediction #2.