Infinitary real-valued logic and a refinement of the Kochen-Specker theorem
Abstract
For nearly sixty years, refinements of the Bell-Kochen-Specker theorem have concentrated on finite configurations, combinatorics and robustness; once a configuration is fixed, contextuality is witnessed by the inconsistency of a finite family of Boolean constraints. We study the theorem instead with infinitary real-valued logic and a model existence game. We axiomatize, in real-valued Lω1ω, the lattice of closed subspaces of a separable infinite-dimensional complex Hilbert space together with a quantum probability measure. In the standard models the theory recovers Gleason's theorem. Once the axiom that the measure is {0,1}-valued is adjoined, the theory has no models, standard or nonstandard, although every finite set of its axioms has one; a model existence game converts this infinitary inconsistency into finite inconsistent sets of interval commitments on the values of sentences. Over the standard models, the subsets of [0,1] that formulas define through the values of the measure are exactly the σ-projective sets, the members of the least class containing the Borel sets and closed under complements, countable unions and projections along ℕN; the classification is proved in ZFC. For a specified strategy in the game, the plays that complete the parity argument for the eighteen-ray configuration of Cabello, Estebaranz and García-Alcaine record the geometric identities between the occurrences of a ray in different contexts, and the use that the measure makes of them. Probabilities are tracked up to rational tolerances, and σ-additivity along a named basis selects the four-dimensional block in which the argument takes place. The refinement is thus not a smaller Kochen-Specker set but a finer analysis of the logical form of the obstruction.
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BibTeX
@misc{aubone2026infinitary,
title = {{Infinitary real-valued logic and a refinement of the Kochen-Specker theorem}},
author = {Olivia Aubone and José Iovino},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05656},
eprint = {2611.05656},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05656},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Infinitary real-valued logic and a refinement of the Kochen-Specker theorem
AU - Olivia Aubone
AU - José Iovino
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05656
UR - https://arxiv.org/abs/2611.05656
AB - For nearly sixty years, refinements of the Bell-Kochen-Specker theorem have concentrated on finite configurations, combinatorics and robustness; once a configuration is fixed, contextuality is witnessed by the inconsistency of a finite family of Boolean constraints. We study the theorem instead with infinitary real-valued logic and a model existence game. We axiomatize, in real-valued L_ω_1ω, the lattice of closed subspaces of a separable infinite-dimensional complex Hilbert space together with a quantum probability measure. In the standard models the theory recovers Gleason's theorem. Once the axiom that the measure is {0,1}-valued is adjoined, the theory has no models, standard or nonstandard, although every finite set of its axioms has one; a model existence game converts this infinitary inconsistency into finite inconsistent sets of interval commitments on the values of sentences. Over the standard models, the subsets of [0,1] that formulas define through the values of the measure are exactly the σ-projective sets, the members of the least class containing the Borel sets and closed under complements, countable unions and projections along ℕ^N; the classification is proved in ZFC. For a specified strategy in the game, the plays that complete the parity argument for the eighteen-ray configuration of Cabello, Estebaranz and García-Alcaine record the geometric identities between the occurrences of a ray in different contexts, and the use that the measure makes of them. Probabilities are tracked up to rational tolerances, and σ-additivity along a named basis selects the four-dimensional block in which the argument takes place. The refinement is thus not a smaller Kochen-Specker set but a finer analysis of the logical form of the obstruction.
ER -