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Generalizations of Kochen and Specker’s theorem and the effectiveness of Gleason’s theorem
 Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 35, 177194
, 2004
"... Abstract. Kochen and Specker’s theorem can be seen as a consequence of Gleason’s theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason’s th ..."
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Abstract. Kochen and Specker’s theorem can be seen as a consequence of Gleason’s theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason’s theorem itself has a constructive proof, based on a generic, finite, effectively generated set of rays, on which every quantum state can be approximated. 1. Gleason’s Theorem and Logical Compactness Kochen and Specker’s (1967) theorem (KS) puts a severe constraint on possible hiddenvariable interpretations of quantum mechanics. Often it is considered an improvement on a similar argument derived from Gleason (1957) theorem (see, for example, Held. 2000). This is true in the sense that KS provide an explicit construction of a finite set of rays on which no twovalued homomorphism exists. However, the fact that there is such a finite set follows from Gleason’s theorem using a simple logical compactness argument (Pitowsky 1998, a similar point is made in Bell 1996). The existence of finite sets of rays with other interesting features
INDEPENDENCE OF THE EXISTENCE OF PITOWSKY SPIN MODELS
"... In the last several decades the study of the foundations of Mathematics is dominated by the impact of the independence phenomena in Set Theory. In many fields of Mathematics like analysis, topology, algebra etc. basic problems were shown to be independent of the accepted axiom system for Set Theory, ..."
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In the last several decades the study of the foundations of Mathematics is dominated by the impact of the independence phenomena in Set Theory. In many fields of Mathematics like analysis, topology, algebra etc. basic problems were shown to be independent of the accepted axiom system for Set Theory, known as ZermelloFrankel Set Theory or ZFC. The fact that undecided problems exist in any recursively stated axiom system which has some minimal strength was well known since Gödel’s famous incompleteness theorem of 1931. But the undecided statement produced by Gödel’s proof was considered to be somewhat esoteric and unrelated to main stream of the subject. The new aspect that came up after Cohen independence proofs of 1963 is that mathematical problems that were considered to be central to the particular discipline were shown to be undecided. So the very notion of mathematical truth was shaken. The possibility of having a multitude of mathematical universes with different properties became a definite option for the foundation of Mathematics. To what extent these developments are relevant to Physics? When a physical theory is stated in terms of mathematical concepts like real numbers, Hilbert spaces, manifolds etc. it implicitly adapts all the mathematical facts which are accepted by the Mathematicians to be valid for these concepts. If the mathematical “truths ” may depend of the foundation of Set Theory then it is possible, at least in principle, that whether a given physical theory implies a particular physically meaningful statement may depend on the foundational framework in which the implicitly assumed Mathematics is embedded. This may seem far fetched and it is very likely that physically consequences of a physical theory will never depend on the set theoretical foundation of the mathematical reasoning that accompanied the theory but the point of this paper is that this is still a definite possibility. Let us admit from the outset that we do not have an example of a physically meaningful statement that its truth depends on the set theoretical foundation. We shall instead demonstrate that there are independent