Topos perspective on the Kochen--Specker theorem. IV: Interval valuations (Q1610694)

In previous papers [Int. J. Theor. Phys. 38, 827--859 (1999; Zbl 1007.81009); in collaboration with \textit{J. Hamilton}, ibid. 39, 1413--1436 (2000; Zbl 1055.81004); ibid. 37, 2669--2733 (1998; Zbl 0979.81018)] the authors gave a topos-theoretical treatment of assigning values to quantities in quantum theory. Their idea was ``to assign a sieve as a partial and contextual truth value to the proposition that the value of a quantity lies in a certain set'' of real numbers. In this paper this perspective is extended in the sense that they ``relate such sieve-valued valuations to valuations that assign to quantities subsets, rather than single elements of their spectra.'' They achieve this by using a base category of commutative von Neumann algebras. This kind of interval valuations, closely related to the previous works mentioned above, allows a simple way to secure some properties of valuations like monotonicity. It is an interesting point of view to see the Kochen-Specker theorem, which is translated into this topos-theoretical language.