## The Bell-Kochen-Specker Theorem

Citations: | 6 - 0 self |

### BibTeX

@MISC{Appleby_thebell-kochen-specker,

author = {D M Appleby},

title = {The Bell-Kochen-Specker Theorem},

year = {}

}

### OpenURL

### Abstract

Meyer, Kent and Clifton (MKC) claim to have nullified the Bell-Kochen-Specker (Bell-KS) theorem. It is true that they invalidate KS’s account of the theorem’s physical implications. However, they do not invalidate Bell’s point, that quantum mechanics is inconsistent with the classical assumption, that a measurement tells us about a property previously possessed by the system. This failure of classical ideas about measurement is, perhaps, the single most important implication of quantum mechanics. In a conventional colouring there are some remaining patches of white. MKC fill in these patches, but only at the price of introducing patches where the colouring becomes “pathologically” discontinuous. The discontinuities mean that the colours in these patches are empirically unknowable. We prove a general theorem which shows that their extent is at least as great as the patches of white in a conventional approach. The theorem applies, not only to the MKC colourings, but also to any other such attempt to circumvent the Bell-KS theorem (Pitowsky’s colourings, for example). We go on to discuss the implications. MKC do not nullify the Bell-KS theorem. They do, however, show that we did not, hitherto, properly understand the theorem. For that reason their results (and Pitowsky’s earlier results) are of major importance. 1 1.

### Citations

719 |
Quantum Theory: Concepts and Methods
- Peres
- 1998
(Show Context)
Citation Context ...n integral defined in terms of a finite KS-uncolourable set. For the sets which have been described in the literature this integral is rather small. For instance, in the case of the Conway-Kochen set =-=[22, 23]-=- the integral is � 0.01 (see Appendix A). So the possibility is not excluded that ∼ 99% of the total solid angle can be covered with a KS-colourable Borel set (in terms of the Earth this would corresp... |

602 |
Speakable and unspeakable in quantum mechanics
- BELL
- 1987
(Show Context)
Citation Context ...dinary language the word “measurement” very strongly suggests the ascertaining of some pre-existing property of some thing, any instrument involved playing a purely passive role (in the words of Bell =-=[1]-=-, p.166). The Bell-Kochen-Specker (Bell-KS) theorem [2, 3] shows that quantum mechanics is inconsistent with that natural idea. Or so Bell thought. His conclusion has, however, been challenged: first ... |

196 |
The Undivided Universe
- Bohm, Hiley
- 1993
(Show Context)
Citation Context ...coveries. When Bell wrote the above passage he had in mind the way that spin measurements work in the de Broglie-Bohm theory (Bell [1] pp. 35 and 163; Dewdney et al [19]; Holland [20]; Bohm and Hiley =-=[33]-=-). In that theory the apparatus interacts with the system, so as to manufacture a value which did not previously exist. The unobservability of the pre-existing values is a consequence of this. So it a... |

135 |
An Introduction to Chaotic Dynamical Systems, Addison-Wesley Studies in Nonlinearity
- Devaney
- 1989
(Show Context)
Citation Context ... However, in the general case it might happen that µ(Di) < µ(D) (it might, for example, happen that D has an infinitely complex lace-like structure, something like what one sees in the Mandelbrot set =-=[27]-=-). We therefore need to examine the nature of the discontinuities outside Di. For that we can appeal to the following lemma. Lemma 3. Let f : K → {0, 1} be a pseudo-KS-colouring with discontinuity reg... |

119 |
The Quantum Theory of Motion (Cambridge
- Holland
- 1993
(Show Context)
Citation Context ... system, but about something which2 has come into being in the combination of system and apparatus. [Bell [1], p. 35] (for a more detailed discussion of this point see Dewdney et al [19] and Holland =-=[20]-=-). PMKC have discovered a completely different mechanism for concealing the values. This is an important discovery. It means, in particular, that Bell’s emphasis on the active role of the apparatus ne... |

26 |
Integral, Measure and Derivative: A Unified Approach. Englewood Cliffs
- Shilov, Gurevich
- 1966
(Show Context)
Citation Context ... require further investigation to tell whether that is actually the case. Appendix B. Proof of Lemma 3 The lemma is a consequence of the Lebesgue-Vitali theorem (see, for example, Shilov and Gurevich =-=[50]-=-, Chapter 10). Let φ be an integrable function defined on a measure space (X, S, µ) with σ-ring S and countably additive measure µ. Then it can be shown (Shilov and Gurevich [50], pp. 220–1) that, for... |

12 |
Measure Theory, Graduate texts in mathematics 18
- Halmos
- 1975
(Show Context)
Citation Context ... a colouring of maximal extent. In Appendix A we prove Lemma 2. dB is strictly > 0. We know that dR ≥ dO ≥ dB and dC ≥ dB (because R ⊆ O ⊆ B and C ⊆ B). It can also be shown (see, for example, Halmos =-=[21]-=-, Chapter 10) that each Borel set B contains a sequence of closed subsets Cn such that µ(B) = limn→∞ (µ(Cn)). Consequently dC ≤ dB. Putting these facts together we deduce dR ≥ dO ≥ dC = dB > 0 (6) It ... |

12 |
Interpreting the Quantum World (Cambridge
- Bub
- 1998
(Show Context)
Citation Context ...n integral defined in terms of a finite KS-uncolourable set. For the sets which have been described in the literature this integral is rather small. For instance, in the case of the Conway-Kochen set =-=[22, 23]-=- the integral is � 0.01 (see Appendix A). So the possibility is not excluded that ∼ 99% of the total solid angle can be covered with a KS-colourable Borel set (in terms of the Earth this would corresp... |

4 |
Rapport au V’ieme Congres de Physique Solvay,Gauthier-Villars, Paris de Broglie,L.(1959), L’interprétation de la méchanique ondulataire,Jorunal de Physique et le Radium, 20,963-979 de Broglie,L
- Broglie, in
- 1960
(Show Context)
Citation Context ...anics is just plain inconsistent with the classical picture, of particles moving along sharply-defined, objective trajectories. However, Bohm’s 1952 rediscovery [28] of de Broglie’s pilot wave theory =-=[29]-=- showed that that is incorrect. As Bell puts it: “in 1952 I saw the impossible done” (Bell [1], p.160). The Bell-KS theorem was conceived in response to that event. Bell and KS, in their different way... |

3 |
e-prints quant-ph/0105039
- Fuchs
(Show Context)
Citation Context ..., that the distinction between classical and quantum is partly epistemological in character, acccords with the current interest in quantum information 11 . In particular, it accords with Fuchs’s idea =-=[46, 47]-=-, that quantum mechanics can partly (and perhaps even mostly) be seen as a “law of thought”. In this paper we have tried to avoid taking sides in the interpretational dispute. But, now that we have re... |

1 |
and e-print quant-ph/0205053
- A
- 1995
(Show Context)
Citation Context ...it is only the finite precision of real laboratory instruments which prevents us giving a non-hidden variables interpretation of quantum 8 It seems that a similar phenomenon occurs in Palmer’s models =-=[34]-=-. Palmer’s models are of some independent interest, and should be examined by anyone concerned with these questions.15 mechanics (what Bell calls an exposed variables interpretation—see below). Howev... |