They are appealing to the law of large numbers, which is a valid form of probabilistic reasoning, not to the gambler's fallacy, which is not valid. In the quoted passage, you describe the gambler's fallacy. No one is advocating that.
This is the point of the proof. If you assume LHV and the QM correlations, you get absurd results like negative probabilities. They cannot both be right, as you have just shown.
It's true the predictions are for three disjoint sets. However, if the expected fraction of disagreeing pairs measured at (-30,0) is 1/4, the LHV theory must either say that 1/4 of all photon pairs would disagree if measured in that direction, or that the photons we measure are systematically...
Ok, how about this. Bill, if you think an LHV theory can match the predictions of QM, let's see a set of particles that does it.
QM predicts that 75% of pairs will differ when set at (-30,30), 25% will differ when measured at (-30,0) and 25% will differ when measured at (0,30). We want three...
Bell's inequality is about three separate expectation values, not three separate pairs. The results you actually measure for three pairs need not match the expectation values.
If I understand, the independence lugita15 is talking about is independence between what properties a particle has when it becomes entangled and what measurement we decide to make. Since the LHV theory is that properties of the particles determine measurement outcomes, we can use measurement...
I see. I was thinking of the question as asking whether backward determinism is true in QM even if forward determinism is not. I now realize this interpretation of the question does not make sense given the analogy to Schrodinger's cat, so your answer is better.
I am not sure about this answer. In two slit experiments, we have one wave function evolving deterministically until measurement at the screen. According to many QM interpretations, this is just one history: the particle does not follow a determinate path and does not have a determinate...
This is the reference I was talking about: http://arxiv.org/abs/0811.0164
I don't know if there is any notational convention for specifying that you are using the hyperreal number system.
There is no real number that is infinitely close to 1 yet not identical to 1. Thus, one must use a different structure, such as the hyperreal numbers. The interesting thing about switching to hyperreals is that using the same decimal numbering convention, where
.999... = 9/10 + 9/100 +...
Sad this is the first thing I find to comment on after joining. I heard about his cancer shortly after he announced it. I hoped he would live longer. The Algebraist and The Hydrogen Sonata are amazing, but every book of his I have read has been at least great. Sad.