Abstract not found

Toposes and quasi-toposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to

The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this equation implies that Σ classifies some class of monos, and the Frobenius law ∃x.(φ(x) ∧ ψ) =(∃x.φ(x)) ∧ ψ) for the existential quantifier. In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory. The natural definitions of discrete and Hausdorff spaces correspond to equality and inequality, whilst the quantifiers considered as adjoints characterise open (or, as we call them, overt) and compact spaces. Our treatment of overt discrete spaces and open maps is precisely dual to that of compact Hausdorff spaces and proper maps. The category of overt discrete spaces forms a pretopos and the paper concludes with a converse of Paré’s theorem (that the contravariant powerset functor is monadic) that characterises elementary toposes by means of the monadic and Euclidean properties together with all quantifiers, making no reference to subsets.

into two principal paradigms: model-theoretic syntax (MTS), which

Lax modalities occur in intuitionistic logics concerned with hardware verification, the computational lambda calculus, and access control in secure systems. They also encapsulate the logic of Lawvere-Tierney-Grothendieck topologies on topoi. This paper provides a complete semantics for quantified lax logic by combining the Beth-Kripke-Joyal cover semantics for first-order intuitionistic logic with the classical relational semantics for a “diamond ” modality. The main technique used is the lifting of a multiplicative closure operator (nucleus) from a Heyting algebra to its MacNeille completion, and the representation of an arbitrary locale as the lattice of “propositions ” of a suitable cover system. In addition, the theory is worked out for certain constructive versions of the classical logics K and S4. An alternative completeness proof is given for (non-modal) first-order intuitionistic logic itself with respect to the cover semantics, using a simple and explicit Henkin-style construction of a characteristic model whose points are principal theories rather than prime saturated ones. The paper provides further evidence that there is more to intuitionistic modal logic than the generalisation of properties of boxes and diamonds from Boolean modal logic.

Summary. We examine Gödel’s completeness and incompleteness theorems for higher order arithmetic from a categorical point of view. The former says that a proposition is provable if and only if it is true in all models, which we take to be local toposes, i.e. Lawvere’s elementary toposes in which the terminal object is a nontrivial indecomposable projective. The incompleteness theorem showed that, in the classical case, it is not enough to look only at those local toposes in which all the numerals are standard. Thus, for a classical mathematician, Hilbert’s formalist program is not compatible with the belief in a Platonic standard model. However, for pure intuitionistic type theory, a single model suffices, the linguistically constructed free topos, which is the initial object in the category of all elementary toposes and logical functors. Hence, for a moderate intuitionist, formalism and Platonism can be reconciled after all. The completeness theorem can be sharpened to represent any topos by continuous sections of a sheaf of local toposes. 1.1

Abstract. We show how quantum mechanics can be understood as a space-time theory provided that its spatial continuum is modelled by a variable real number (qrumber) continuum. Such a continuum can be constructed using only standard Hilbert space entities. The geometry of atoms and subatomic objects differs from that of classical objects. The systems that are non-local when measured in the classical space-time continuum may be localized in the quantum continuum. We compare this new description of space-time with the Bohmian picture of quantum mechanics. 1. What is quantum space-time? Both modern mathematics and modern physics underwent serious foundational crises during the 20th century. The crisis in mathematics occured at the beginning of the century and the main problem was to deal with certain infinities that are directly related to the concept of real number. Poincaré [31] explained this crisis in terms of different attitudes to infinity, related to Aristotle’s actual infinity and the potential infinity (the first attitude believes that the actual infinity exists, we begin with the collection in which we find the pre-existing objects, the second holds that a collection is formed by successively adding new members, it is infinite because we can see no reason why this process should stop). It led finally to the emergence of new, non-standard definitions of real numbers. The crisis in physics concerns the interpretation of the quantum theory, the measurement problem and the question of non-locality. In previous works we showed how in principle certain paradoxes of the quantum theory can be explained provided we enlarge our conception of number [10].Our goal was to show how the basic axioms of quantum mechanics can be reformulated in terms of non-standard real numbers that we call qrumbers. It is our goal in the present paper to analyze non-locality and the concept of space-time at the light of the new conceptual tools that we developed in the past.