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

Based on the Sheaf Logic approach to set theoretic forcing, a hierarchy of Quantum Variable Sets is constructed which generalizes and simplifies the analogous construction developed by Takeuti on boolean valued models of set theory. Over this model two alternative proofs of Takeuti's correspondence, between self adjoint operators and the real numbers of the model, are given. This approach results to be more constructive showing a direct relation with the Gelfand representation theorem, revealing also the importance of these results with respect to the interpretation of Quantum Mechanics in close connection with the Deutsch-Everett multiversal interpretation. Finally, it is shown how in this context the notion of genericity and the corresponding generic model theorem can help to explain the emergence of classicality also in connection with the Deutsch- Everett perspective.Comment: 34 pages, 2 figure

Category forcings, MM+++MM^{+++}, and generic absoluteness for the theory of strong forcing axioms

We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial segment of the universe of height a super compact which is a limit of super compact cardinals is a stationary set preserving partial order which forces $MM^{++}$ and collapses its size to become the second uncountable cardinal. Next we argue that any of the known methods to produce a model of $MM^{++}$ collapsing a superhuge cardinal to become the second uncountable cardinal produces a model in which the cutoff of the category of stationary set preserving forcings at any rank initial segment of the universe of large enough height is forcing equivalent to a presaturated tower of normal filters. We let $MM^{+++}$ denote this statement and we prove that the theory of $L(Ord^{\omega_1})$ with parameters in $P(\omega_1)$ is generically invariant for stationary set preserving forcings that preserve $MM^{+++}$. Finally we argue that the work of Larson and Asper\'o shows that this is a next to optimal generalization to the Chang model $L(Ord^{\omega_1})$ of Woodin's generic absoluteness results for the Chang model $L(Ord^{\omega})$. It remains open whether $MM^{+++}$ and $MM^{++}$ are equivalent axioms modulo large cardinals and whether $MM^{++}$ suffices to prove the same generic absoluteness results for the Chang model $L(Ord^{\omega_1})$.Comment: - to appear on the Journal of the American Mathemtical Societ

Normalization for planar string diagrams and a quadratic equivalence algorithm

In the graphical calculus of planar string diagrams, equality is generated by exchange moves, which swap the heights of adjacent vertices. We show that left- and right-handed exchanges each give strongly normalizing rewrite strategies for connected string diagrams. We use this result to give a linear-time solution to the equivalence problem in the connected case, and a quadratic solution in the general case. We also give a stronger proof of the Joyal-Street coherence theorem, settling Selinger's conjecture on recumbent isotopy

SEPARATING FRAGMENTS OF WLEM, LPO, AND MP

Abstract. We separate many of the basic fragments of classical logic which are used in reverse constructive mathematics. A group of related Kripke and topological models is used to show that various fragments of the Weak Law of the Excluded Middle, the Limited Principle of Omniscience, and Markov's Principle, including Weak Markov's Principle, do not imply each other. §1. Introduction. At the beginning of the twentieth century, Brouwer identified a number of constructively dubious principles, which Bishop later, in his 1967 monograph Omniscience principles are commonly used to show the independence of more subject specific theorems: if a (classical) result constructively implies an omniscience principle, then it cannot be proved using constructive techniques. By separating different omniscience principles over IZF we make this task easier: if under the assumption of a classical result together with an omniscience principle we can derive a stronger omniscience principle, then we can still conclude that the classical theorem is nonconstructive. More generally, implications among these principles and theorems of mainstream mathematics have been studied for a long time. Often this is the motivation for introducing these principles (some references being provided with the principles below), and often this study is done for foundational reasons after the principles are already established (as, for instance, in In this paper we present many models, often related to each other, that separate a large number of the omniscience principles defined in terms of binary sequences and related principles. The genesis of this work was the first author's question to the second of whether Richman's LLPO n hierarchy [23] could be separated, a question about results. Since then, much interest has shifted to technique: could an argumen

Indeterminacy and the law of the excluded middle

This thesis is an investigation into indeterminacy in the foundations of mathematics and its possible consequences for the applicability of the law of the excluded middle (LEM). It characterises different ways in which the natural numbers as well as the sets may be understood to be indeterminate, and asks in what sense this would cease to support applicability of LEM to reasoning with them. The first part of the thesis reviews the indeterminacy phenomena on which the argument is based and argues for a distinction between two notions of indeterminacy: a) indeterminacy as applied to domains and b) indefiniteness as applied to concepts. It then addresses possible attempts to secure determinacy in both cases. The second part of the thesis discusses the advantages that an argument from indeterminacy has over traditional intuitionistic arguments against LEM, and it provides the framework in which conditions for the applicability of LEM can be explicated in the setting of indeterminacy. The final part of the thesis then applies these findings to concrete cases of indeterminacy. With respect to indeterminacy of domains, I note some problems for establishing a rejection of LEM based on the indeterminacy of the height of the set theoretic hierarchy. I show that a coherent argument can be made for the rejection of LEM based on the indeterminacy of its width, and assess its philosophical commitments. A final chapter addresses the notion of indefiniteness of our concepts of set and number and asks how this might affect the applicability of LEM