Logic, Physics and Geometry

The first of the 10 problems that Hilbert posed in his famous conference the 1900 in Paris; the problem of the Continuum Hypothesis (CH for short), which asserts that every subset of the real numbers is either enumerable or has the cardinality of the Continuum; was the origin of probably one of the greatest achievements in mathematics of the second half of the XX century, that is the forcing technique created by P. Cohen during the sixties. Using this technique Cohen showed that the CH is independent (i.e. it cannot be proved or disproved) from the axioms of Zermelo-Fraenkel plus the Axiom of Choice (ZFC). A couple of years later, in 1965, Scott and Solovay gave a more constructive version of Cohen’s technique based on models ruled by a boolean-valued logic. Based on this latter approach, in 1975, Takeuti showed that in a boolean valued model of set theory constructed over a complete boolean algebra of projector operators acting over a Hilbert space, there is a correspondence between the continuum of the model and the self-adjoint operators that can be expressed by the projectors of the algebra. The connection of this surprising result with interpretational issues in quantum theory was explored by Davis  hypothesizing the possibility to introduce a relativistic principle in quantum theory. However, no conclusive results were obtained in this direction and maybe due to the fact that the mathematics of these constructions were based on advanced methods of set theory and model theory, these results did not reach the attention of the physicists community.

Meanwhile topos theory had emerged through the work of Lawvere, Tierney and others. In 1972, Tierney found that in this context the notion of sheaf allowed to explain Cohen’s Forcing and Solovay-Scott formulation in terms of topoi. The boolean approach to Cohen’s forcing then played an important role in the consolidation of many features of topos theory, particularly its relation with logic. Independently, Cohen’s technique was further improved by set theorists and alternative simple formulations were developed. Despite that all these different formulations of forcing found a unified topos perspective, set theorists have always preferred their own approaches, mainly because within the abstract categorical tools of topoi, the elegance and simplicity of Cantor’s formalism is lost. For this reason, during many years both theories evolved almost independently; their few interactions being mainly focused on the formulation of independence results in terms of topoi, and the possibility of deeper interactions remained for long unexplored.

In 1995 X. Caicedo  gave a unified perspective of the different formulations of the forcing technique based on an alternative approach to the logic of sheaves outside the intrinsic difficulties linked to the categorical language of topoi. In this approach the topos perspective and the set theoretic approaches merge in a smooth way revealing finally the potentiality of this implicit relation. In this context, classical results of model theory find a natural more simple derivations as well as classical independence results. Furthermore, interesting relations between different logics are revealed and the relations between logic and geometry implicit in the topos constructions find a remarkable simple and clear formulation.

Surprisingly, on this particular formulation of Cohen’s forcing, Deutsch- Everett ideas seem to find close analogies, this summed to Takeuti’s results brought us to think that these methods would probably provide the tools towards a possible formulation of the quantum multiverse. Based on Deutsch-Everett ideas and on the sheaf-logic approach to set theoretic forcing as developed by Caicedo, we have introduced a hierarchy of quantum variable sets constructed over a space of quantum histories. This construction generalizes and simplifies the analogous construction developed by Takeuti  on Boolean valued models of set theory. Over this model we have given two alternative proofs of Takeuti’s correspondence between self-adjoint operators and the generalized real numbers of the model. This approach results to be more constructive showing a direct relation with results of the theory of operators algebras, and revealing interesting connections with the Deutsch-Everett interpretation of quantum theory. We have shown also how the collapse via generic models of this structure of quantum variable sets, can help to give a sound mathematical formulation of quantization processes and of the emergence of classicality, in close relation with the Deutsch-Everett perspective.

Nowadays, we are studying possible methods to introduce time on this model, trying to show how the intuition of Davis about the relation of Takeuti’s results with a possible relativistic principle in quantum theory find a strong mathematical support on the constructions here developed. We are also looking to understand how these later results provide a new perspective to affront the quantum gravity program.


Sheaf Logic, Quantum Set Theory and the Interpretation of Quantum Mechanics.

The logic of sheaves, sheaf forcing and the independence of the continuum hypothesis.

Leaving Cantor's paradise through Paul Cohen's golden door.