This book is written for those who are in sympathy with its spirit. This spirit is different from the one which informs the vast stream of European and American civilization in which all of us stand. That spirit expresses itself in an onwards movement, in building ever larger and more complicated structures; the other in striving in clarity and perspicuity in no matter what structure. The first tries to grasp the world by way of its periphery—in its variety; the second (...) at its centre—in its essence. And so the first adds one construction to another, moving on and up, as it were, from one thing to the next, while the other remains where it is and what it tries to grasp is always the same. (shrink)

We consider all combinatorially possible systems corresponding to subsets of finite set theory (i.e., Zermelo-Fraenkel set theory without the axiom of infinity) and for each of them either provide a well-founded locally finite graph that is a model of that theory or show that this is impossible. To that end, we develop the technique of _axiom closure of graphs_.

We study constructive set theories, which deal with operations applying both to sets and operations themselves. Our starting point is a fully explicit, finitely axiomatized system ESTE of constructive sets and operations, which was shown in 10 to be as strong as PA. In this paper we consider extensions with operations, which internally represent description operators, unbounded set quantifiers and local fixed point operators. We investigate the proof theoretic strength of the resulting systems, which turn out to be impredicative (...) . © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

Given a topological Ramsey space math formula, we extend the notion of semiselective coideal to sets math formula and study conditions for math formula that will enable us to make the structure math formula a Ramsey space and also study forcing notions related to math formula which will satisfy abstract versions of interesting properties of the corresponding forcing notions in the realm of Ellentuck's space. This extends results from to the most general context of topological Ramsey spaces. As applications, we (...) prove that for every topological Ramsey space math formula, under suitable large cardinal hypotheses every semiselective ultrafilter math formula is generic over math formula; and that given a semiselective coideal math formula, every definable subset of math formula is math formula-Ramsey. This generalizes the corresponding results for the case when math formula is equal to Ellentuck's space. (shrink)

It is proved that in a suitable intuitionistic, locally classical, version of the theory ZFC deprived of the axiom of infinity, the requirement that every set be finite is equivalent to the assertion that every ordinal is a natural number. Moreover, the theory obtained with the addition of these finiteness assumptions is equivalent to a theory of hereditarily finite sets, developed by Previale in "Induction and foundation in the theory of hereditarily finite sets." This solves some (...) problems stated there. The analysis is undertaken using for each of these results a limited fragment of the relevant theory. (shrink)

Working in constructive set theory we formulate notions of constructive topological space and set-generated locale so as to get a good constructive general version of the classical Galois adjunction between topological spaces and locales. Our notion of constructive topological space allows for the space to have a class of points that need not be a set. Also our notion of locale allows the locale to have a class of elements that need not be a set. Class sized mathematical structures (...) need to be allowed for in constructive set theory because the powerset axiom and the full separation scheme are necessarily missing from constructive set theory. We also consider the notion of a formal topology, usually treated in Intuitionistic type theory, and show that the category of set-generated locales is equivalent to the category of formal topologies. We exploit ideas of Palmgren and Curi to obtain versions of their results about when the class of formal points of a set-presentable formal topology form a set. (shrink)

If X is a locally compact Polish space, then LSC denotes the compact Polish space of lower semi-continuous real-valued functions on X equipped with the topology of epi-convergence.Our purpose in this article is to prove the following: if –∞ < α < β < ∞ and –∞ < a < b < ∞, while r ∈ ℕ \ {0}, then the set CV of all f ∈ LSC for which there is u ∈ Cr such that for any v ∈ (...) Cr we have that ∫αβf , v ′)dx ≥ ∫αβf , v )dx is not Borel. (shrink)

Weak Quantum Theory (WQT) and the Model of Pragmatic Information (MPI) are two psychophysical concepts developed on the basis of quantum physics. The present study contributes to their empirical examination. The issue of the study is whether WQT and MPI can not only explain ‘psi’-phenomena theoretically but also prove to be consistent with the empirical phenomenology of extrasensory perception (ESP). From the main statements of both models, 33 deductions for psychic readings are derived. Psychic readings are defined as settings, (...) in which psychics support or counsel clients by using information not mediated through the five senses. A qualitative approach is chosen to explore how the psychics experience extrasensory perceptions. Eight psychics are interviewed with a half-structured method. The reports are examined regarding deductive and inductive aspects, using a multi-level structured content analysis. The vast majority of deductions is clearly confirmed by the reports. Even though the study has to be seen as an explorative attempt with many aspects to be specified, WQT and MPI prove to be coherent and helpful concepts to explain ESP in psychic readings. (shrink)

The Lovász local lemma (LLL) is a technique from combinatorics for proving existential results. There are many different versions of the LLL. One of them, the lefthanded local lemma, is particularly well suited for applications to two player games. There are also constructive and computable versions of the LLL. The chief object of this thesis is to prove an effective version of the lefthanded local lemma and to apply it to effectivise constructions of non-repetitive sequences.The second goal (...) of this thesis is to categorize some classes of cohesive powers. We completely describe both the isomorphism types of cohesive powers of equivalence structures and injection structures, as well as clarify the relationship between these cohesive powers and their original structures. We also describe the finite condensation of cohesive powers of computable copies of the integers as a linear order by cohesive sets whose complement are computably enumerable.Finally, we investigate the possibility of decomposing problems in the Weihrauch degrees into a product of first order part and second order part. We give a preliminary result in this direction.Abstract prepared by Daniel Mourad.E-mail: [email protected]: https://daniel-mourad.scholar.uconn.edu/wp-content/uploads/sites/2530/2023/06/ThesisFinalWithRevisio ns-2.pdf. (shrink)

In this paper we study local induction w.r.t. Σ1‐formulas over the weak arithmetic. The local induction scheme, which was introduced in, says roughly this: for any virtual class that is progressive, i.e., is closed under zero and successor, and for any non‐empty virtual class that is definable by a Σ1‐formula without parameters, the intersection of and is non‐empty. In other words, we have, for all Σ1‐sentences S, that S implies, whenever is progressive. Since, in the weak context, we (...) have (at least) two definitions of Σ1, we obtain two minimal theories of local induction w.r.t. Σ1‐formulas, which we call Peano Corto and Peano Basso.In the paper we give careful definitions of Peano Corto and Peano Basso. We establish their naturalness both by giving a model theoretic characterization and by providing an equivalent formulation in terms of a sentential reflection scheme.The theories Peano Corto and Peano Basso occupy a salient place among the sequential theories on the boundary between weak and strong theories. They bring together a powerful collection of principles that is locally interpretable in. Moreover, they have an important role as examples of various phenomena in the metamathematics of arithmetical (and, more generally, sequential) theories. We illustrate this by studying their behavior w.r.t. interpretability, model interpretability and local interpretability. In many ways the theories are more like Peano arithmetic or Zermelo Fraenkel set theory, than like finitely axiomatized theories as Elementary Arithmetic, and. On the one hand, Peano Corto and Peano Basso are very weak: they are locally cut‐interpretable in. On the other hand, they behave as if they were strong: they are not contained in any consistent finitely axiomatized arithmetical theory, however strong. Moreover, they extend, the theory of parameter‐free Π1‐induction. (shrink)

Necessity and sufficiency are the building blocks of all successful explanations. Yet despite their importance, these notions have been conceptually underdeveloped and inconsistently applied in explainable artificial intelligence, a fast-growing research area that is so far lacking in firm theoretical foundations. In this article, an expanded version of a paper originally presented at the 37th Conference on Uncertainty in Artificial Intelligence, we attempt to fill this gap. Building on work in logic, probability, and causality, we establish the central role of (...) necessity and sufficiency in XAI, unifying seemingly disparate methods in a single formal framework. We propose a novel formulation of these concepts, and demonstrate its advantages over leading alternatives. We present a sound and complete algorithm for computing explanatory factors with respect to a given context and set of agentive preferences, allowing users to identify necessary and sufficient conditions for desired outcomes at minimal cost. Experiments on real and simulated data confirm our method’s competitive performance against state of the art XAI tools on a diverse array of tasks. (shrink)

Volume II of Classical Recursion Theory describes the universe from a local (bottom-up or synthetical) point of view, and covers the whole spectrum, from the recursive to the arithmetical sets. The first half of the book provides a detailed picture of the computable sets from the perspective of Theoretical Computer Science. Besides giving a detailed description of the theories of abstract Complexity Theory and of Inductive Inference, it contributes a uniform picture of the most basic complexity classes, (...) ranging from small time and space bounds to the elementary functions, with a particular attention to polynomial time and space computability. It also deals with primitive recursive functions and larger classes, which are of interest to the proof theorist. The second half of the book starts with the classical theory of recursively enumerable sets and degrees, which constitutes the core of Recursion or Computability Theory. Unlike other texts, usually confined to the Turing degrees, the book covers a variety of other strong reducibilities, studying both their individual structures and their mutual relationships. The last chapters extend the theory to limit sets and arithmetical sets. The volume ends with the first textbook treatment of the enumeration degrees, which admit a number of applications from algebra to the Lambda Calculus. The book is a valuable source of information for anyone interested in Complexity and Computability Theory. The student will appreciate the detailed but informal account of a wide variety of basic topics, while the specialist will find a wealth of material sketched in exercises and asides. A massive bibliography of more than a thousand titles completes the treatment on the historical side. (shrink)

Branching theories are popular frameworks for modeling objective indeterminism in the form of a future of open possibilities. In such theories, the notion of a history plays a crucial role: it is both a basic ingredient in the axiomatic definition of the framework, and it is used as a parameter of truth in semantics for languages with a future tense. Furthermore, histories—complete possible courses of events—ground the notion of modal consistency: a set of events is modally consistent iff there is (...) a history containing that set. We will explain these roles of histories and highlight some critical aspects having to do with the fact that histories are global and, in a relevant sense, “big” objects. The notion of modal consistency, on the other hand, has both local and global aspects. We ask in how far a local notion of modal consistency can serve as an alternative to the common uses of histories, and work out two recent approaches to alternatives to histories. Combining these approaches, we develop a novel semantics for branching time. (shrink)

The class of all Artinian local rings of length at most l is ∀ 2 -elementary, axiomatised by a finite set of axioms Art l . We show that its existentially closed models are Gorenstein, of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory Got l of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and (...) the theory Art l is companionable, with model-companion Got l. (shrink)

A recent analysis by de Barros and Suppes of experimentally realizable GHZ correlations supports the conclusion that these correlations cannot be explained by introducing local hidden variables. We show, nevertheless, that their analysis does not exclude local hidden variable models in which the inefficiency in the experiment is an effect not only of random errors in the detector equipment, but is also the manifestation of a pre-set, hidden property of the particles ("prism models"). Indeed, we present an explicit (...) prism model for the GHZ scenario; that is, a local hidden variable model entirely compatible with recent GHZ experiments. (shrink)

Let D be a strongly minimal set in the language L, and $D' \supset D$ an elementary extension with infinite dimension over D. Add to L a unary predicate symbol D and let T' be the theory of the structure (D', D), where D interprets the predicate D. It is known that T' is ω-stable. We prove Theorem A. If D is not locally modular, then T' has Morley rank ω. We say that a strongly minimal set D is (...) pseudoprojective if it is nontrivial and there is a $k < \omega$ such that, for all a, b ∈ D and closed $X \subset D, a \in \mathrm{cl}(Xb) \Rightarrow$ there is a $Y \subset X$ with a ∈ cl(Yb) and |Y| ≤ k. Using Theorem A, we prove Theorem B. If a strongly minimal set D is pseudoprojective, then D is locally projective. The following result of Hrushovski's (proved in $\S4$ ) plays a part in the proof of Theorem B. Theorem C. Suppose that D is strongly minimal, and there is some proper elementary extension D1 of D such that the theory of the pair (D1, D) is ω1-categorical. Then D is locally modular. (shrink)

Since the l960s, archaeology has become increasingly taught in universities and practiced on a growing scale by national and local heritage agencies throughout the world. This book addresses the criticisms of postmodernist writers about archaeology's social role, and asserts its intellectual importance and achievements in discovering real facts about the human past. It looks forward to the creation of a truly global consciousness of the origins of human societies and civilizations.

Algebraic Set Theory, a reconsideration of Zermelo-Fraenkel set theory in category-theoretic terms, has been built up in the mid-nineties by André Joyal and Ieke Moerdijk. Since then, it has developed into a whole research program. This paper gets back to the original formulation by Joyal and Moerdijk, and more specifically to its first three axioms. It explains in detail that these axioms set up a framework directly linked to descent theory, a theory having to do with (...) the shift from local data to a global item in modern algebraic geometry. Fibered categories, introduced by Grothendieck, provide a powerful framework for descent theory: They formalize in a very general way the consideration of local data of different kinds over the objects of some base category. The paper shows that fibered categories fit Joyal and Moerdijk’s axiomatization in a natural way, since the latter actually aims to secure a descent condition. As a result, Algebraic Set Theory is shown to accomplish, not only an original and fruitful combination of set theory with category theory, but the genuine graft of a deeply geometric idea onto the usual setting of zfc. (shrink)

I will give a broad overview of what has become the standard paradigm in cosmology. I will describe the relational notion of time that is often used in cosmological calculations and discuss how the local nature of Einstein's equations allows us to translate this notion into statements about `initial' data. Classically this relates our local definition of time to a quasi-local region of a particular spatial slice, however incorporating quantum theory comes at the expense of losing (...) this locality entirely. This occurs due to the presence of two, apparently distinct, issues: Seemingly classical issues to do with the infinite spatial volume of the universe and Quantum field theory issues, which revolve around trying to apply renormalization in cosmology. Following the ‘cosmological principle’ - an extension of the ‘Copernicus principle’ - that physics at every point in our universe should look the same, we are lead to the modern view of cosmology. This procedure is reasonably well understood for an exactly homogeneous universe, however the inclusions of small perturbations over this homogeneity leads to many interpretational/ conceptual difficulties. For example, in an infinite universe perturbations can be arbitrarily close to homogeneous. To any observer, such a perturbation would appear to be a simple rescaling of the homogenous background and hence, physically, would not be considered an inhomogeneous perturbation at all. However, any attempt to choose the physically relevant scale at which perturbations should be considered homogeneous will break the cosmological principle i.e. it will make the resulting physics observer dependent. It amounts to `putting the perturbations in a box' and a delicate practical issue is that the universe is not static, hence the scale of the box will be time dependent. Thus what appears ‘physically homogeneous’ to an observer at one time will not appear so at another. This issue is brought to the forefront by considering the canonical version of the theory. The phase space formulation of General Relativity, just as for any other theory, contains the shadow of the underlying quantum theory. This means that, although the formulation is still classical, many of the subtleties that are present in the quantum theory are already apparent. In the cosmological context the infinite spatial volume renders almost all expressions formal or ill-defined. In order to proceed, we must restrict our attention to a cosmology that has some finite spatial extent, on which our relational notion of time is everywhere definable. In particular, this would constrain the permissible data outside our `observable universe'. This difficulty is an IR or large scale issues in cosmology, however in addition there are UV or short scale problems that need to be tackled. There are the usual problems of renormalization, which are further complicated by the fact that the universe is not static. In the cosmological setting this leads to new IR problems which again prevent one from taking the spatial extent of the universe to infinity. The physical relevance of this problem, the consequence for defining a time variable, and the distinction of homogeneous and inhomogeneous IR issues will be discussed. (shrink)

Suppose T is a weakly minimal theory and p a strong 1-type having locally finite but nontrivial geometry. That is, for any M [boxvR] T and finite Fp, there is a finite Gp such that acl∩p = gεGacl∩pM; however, we cannot always choose G = F. Then there are formulas θ and E so that θεp and for any M[boxvR]T, E defines an equivalence relation with finite classes on θ/E definably inherits the structure of either a projective or affine (...) space over some finite field. We then specify what other structure θ/E may inherit: there is some collection of definable subspaces of finite codimension and some set of algebraic points, which in the affine case may be in the canonically associated vector space, Up to acl, no further structure is possible. If we assume T is weakly minimal and has a strong type p as above, and also that T is unidimensional, we obtain a global description of any model of T in terms of those structures mentioned in the previous paragraph. (shrink)

Colman shows that normative theories of rational decision-making fail to produce rational decisions in simple interactive games. I suggest that well-formed theories are possible in local settings, keeping in mind that a good part of each game is the generation of a rational approach appropriate for that game. The key is rationality defined in terms of the game, not individual decisions.

A local hidden variable theory of quantum mechanics is formulated by adapting Gell-Mann and Hartle’s many-histories formulation. The resulting theory solves the measurement problem by exploiting the independence loophole in Bell’s theorem; it violates the independence of hidden variable values and measuring device settings. Although the theory is problematic in some respects, it provides a concrete example via which the tenability of this approach can be better evaluated.

This paper presents several independence results concerning the topos-valid and the intuitionistic predicative theory of locales. In particular, certain consequences of the consistency of a general form of Troelstra's uniformity principle with constructive set theory and type theory are examined.

In this paper, I reconstruct an argument of Aristidis Arageorgis against empirical underdetermination of the state of a physical system in a C*-algebraic setting and explore its soundness. The argument, aiming against algebraic imperialism, the operationalist attitude which characterized the first steps of Algebraic Quantum Field Theory, is based on two topological properties of the state space: being T1 and being first countable in the weak*-topology. The first property is possessed trivially by the state space while the latter is (...) highly non-trivial, and it can be derived from the assumption of the algebra of observables’ separability. I present some cases of classical and of quantum systems which satisfy the separability condition, and others which do not, and relate these facts to the dimension of the algebra and to whether it is a von Neumann algebra. Namely, I show that while in the case of finite-dimensional algebras of observables the argument is conclusive, in the case of infinite-dimensional von Neumann algebras it is not. In addition, there are cases of infinite-dimensional quasilocal algebras in which the argument is conclusive. Finally, I discuss Porrmann's construction of a net of local separable algebras in Minkowski spacetime which satisfies the basic postulates of Algebraic Quantum Field Theory. (shrink)

Locally finite omega languages were introduced by Ressayre [Formal languages defined by the underlying structure of their words. J Symb Log 53(4):1009–1026, 1988]. These languages are defined by local sentences and extend ω-languages accepted by Büchi automata or defined by monadic second order sentences. We investigate their topological complexity. All locally finite ω-languages are analytic sets, the class LOC ω of locally finite ω-languages meets all finite levels of the Borel hierarchy and there exist some locally finite ω-languages which (...) are Borel sets of infinite rank or even analytic but non-Borel sets. This gives partial answers to questions of Simonnet (Automates et Théorie Descriptive, Ph.D. Thesis. Université Paris 7, March 1992) and of Duparc et al. [Computer science and the fine structure of Borel sets. Theor Comput Sci 257(1–2):85–105, 2001]. (shrink)

This article elaborates local selective realism in view of the shifting from classical to quantum electrodynamics. After some introductory remarks, we critically address what we call global selective realism, hence setting forth the background for outlining local selective realism. When examining the transition from classical to quantum electrodynamics, we evaluate both continuities and discontinuities in observational features, mathematical structures, and ontological presuppositions. Our argument leads us to criticise the narrow understanding of limiting-case strategies, and to reject the claim (...) that we need a fully coherent theoretical framework to account for the transition from one theory to its successor in the case of electrodynamics. We close with a few remarks on the scope of local selective realism. (shrink)

In 1996, W. Veldman and F. Waaldijk present a constructive (intuitionistic) proof for the homogeneity of the ordered structure of the Cauchy real numbers, and so this result holds in any topos with natural number object. However, it is well known that the real numbers objects obtained by the traditional constructions of Cauchy sequences and Dedekind cuts are not necessarily isomorphic in an arbitrary topos with natural numbers object. Consequently, Veldman and Waaldijk's result does not apply to the ordered structure (...) of Dedekind real numbers in toposes. The main result to be proved in the present paper is that the ordered structure of the Dedekind real numbers object is homogeneous, in any topos with natural numbers object. This result is obtained within the framework of local set theory. (shrink)

This Element is an exposition of second- and higher-order logic and type theory. It begins with a presentation of the syntax and semantics of classical second-order logic, pointing up the contrasts with first-order logic. This leads to a discussion of higher-order logic based on the concept of a type. The second Section contains an account of the origins and nature of type theory, and its relationship to set theory. Section 3 introduces Local Set Theory, an (...) important form of type theory based on intuitionistic logic. In Section 4 number of contemporary forms of type theory are described, all of which are based on the so-called 'doctrine of propositions as types'. We conclude with an Appendix in which the semantics for Local Set Theory - based on category theory - is outlined. (shrink)

We develop an untyped framework for the multiverse of set theory. $\mathsf {ZF}$ is extended with semantically motivated axioms utilizing the new symbols $\mathsf {Uni}(\mathcal {U})$ and $\mathsf {Mod}(\mathcal {U, \sigma })$, expressing that $\mathcal {U}$ is a universe and that $\sigma $ is true in the universe $\mathcal {U}$, respectively. Here $\sigma $ ranges over the augmented language, leading to liar-style phenomena that are analyzed. The framework is both compatible with a broad range of multiverse conceptions and suggests (...) its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle that the background theory does not hold a privileged position over the theories of its internal universes. Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. We apply this to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory $\mathsf {ZF}$, and thus not seriously limiting to the diversity of the set-theoretic multiverse. We end with case studies applying the framework to two multiverse conceptions of set theory: arithmetic absoluteness and Joel D. Hamkins’ multiverse theory. (shrink)

We propose a sheaf-theoretic framework for the representation of a quantum observable structure in terms of Boolean information sieves. The algebraic representation of a quantum observable structure in the relational local terms of sheaf theory effectuates a semantic transition from the axiomatic set-theoretic context of orthocomplemented partially ordered sets, la Birkhoff and Von Neumann, to the categorical topos-theoretic context of Boolean information sieves, la Grothendieck. The representation schema is based on the existence of a categorical adjunction, which is (...) used as a theoretical platform for the development of a functorial formulation of information transfer, between quantum observables and Boolean localisation devices in typical quantum measurement situations. We also establish precise criteria of integrability and invariance of quantum information transfer by cohomological means. (shrink)

In Vague Language, Elasticity Theory and the Use of 'Some', Nguyet Nhu Le and Grace Qiao Zhang present the first comprehensive study of the word 'some', focusing on its elasticity. In particular, they consider how 'some' is both a quantifier and a qualifier, has positive or negative meanings, and has local and global interpretations. They show that the word is used across a meaning continuum and can be used to convey a range of states, including approximation, uncertainty, politeness, (...) and evasion. Finally, they demonstrate that the functions of 'some' are also multi-directional and non-categorical, consisting of four major functions (right amount of information, mitigation, withholding information, and discourse management). Based on naturally-occurring classroom data of L1 (American English) and L2 (Chinese- and Vietnamese-speaking learners of English) speakers, Vague Language shows that L2 speakers used 'some' more than L1 speakers and explores the significance of this, particularly taking account of speakers' language ability and cultural backgrounds. While this book focuses on the single word 'some', the authors' discussion has important implications for language studies more generally, as they call for a rethinking of our approaches to language study and more attention to its elasticity. (shrink)

This paper is a sequel to Tzouvaras :571–601, 2010), where a local version of ZFC, LZFC, was introduced and examined and transitive models of ZFC with properties that resemble large cardinal properties, namely Mahlo and Π11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^1}$$\end{document}-indescribable models, were considered. By analogy we refer to such models as “large models”, and the properties in question as “large model properties”. Continuing here in the same spirit we consider further large model properties, that (...) resemble stronger large cardinals, namely, “elementarily embeddable”, “extendible” and “strongly extendible”, “critical” and “strongly critical”, “self-critical” and “strongly self-critical”, the definitions of which involve elementary embeddings. Each large model property ϕ gives rise to a localization axiom Locϕ saying that every set belongs to a transitive model of ZFC satisfying ϕ. The theories LZFCϕ = LZFC + Locϕ are local analogues of the theories ZFC+“there is a proper class of large cardinals ψ”, where ψ is a large cardinal property. If sext is the property of strong extendibility, it is shown that LZFCsext proves Powerset and Σ1-Collection. In order to refute V = L over LZFC, we combine the existence of strongly critical models with an axiom of different flavor, the Tall Model Axiom. V = L can also be refuted by TMA plus the axiom GC saying that “there is a greatest cardinal”, although it is not known if TMA + GC is consistent over LZFC. Finally Vopěnka’s Principle and its impact on LZFC are examined. It is shown that LZFCsext + V P proves Powerset and Replacement, i.e., ZFC is fully recovered. The same is true for some weaker variants of LZFCsext. Moreover the theories LZFCsext+V P and ZFC+V P are shown to be identical. (shrink)

A local interpretation of quantum mechanics is presented. Its main ingredients are: first, a label attached to one of the “virtual” paths in the path integral formalism, determining the output for measurement of position or momentum; second, a mathematical model for spin states, equivalent to the path integral formalism for point particles in space time, with the corresponding label. The mathematical machinery of orthodox quantum mechanics is maintained, in particular amplitudes of probability and Born’s rule; therefore, Bell’s type inequalities (...) theorems do not apply. It is shown that statistical correlations for pairs of particles with entangled spins have a description completely equivalent to the two slit experiment, that is, interference instead of non locality gives account of the process. The interpretation is grounded in the experimental evidence of a point like character of electrons, and in the hypothetical existence of a wave like, the de Broglie, companion system. A correspondence between the extended Hilbert spaces of hidden physical states and the orthodox quantum mechanical Hilbert space shows the mathematical equivalence of both theories. Paradoxical behaviour with respect to the action reaction principle is analysed, and an experimental set up, modified two slit experiment, proposed to look for the companion system. (shrink)

In the present paper we use the theory of exact completions to study categorical properties of small setoids in Martin-Löf type theory and, more generally, of models of the Constructive Elementary Theory of the Category of Sets, in terms of properties of their subcategories of choice objects. Because of these intended applications, we deal with categories that lack equalisers and just have weak ones, but whose objects can be regarded as collections of global elements. In this context, (...) we study the internal logic of the categories involved, and employ this analysis to give a sufficient condition for the local cartesian closure of an exact completion. Finally, we apply this result to show when an exact completion produces a model of CETCS. (shrink)

We give a model of set theory in which the strongly non‐stationary ideal over is weakly presaturated below some canonical set. Here μ is a regular uncountable cardinal. The model is the forcing extension with the Lévy collapse of a Woodin cardinal to the successor of μ. This improves on results of Goldring and of the first author.

The empirical validity of the locality (LOC) principle of relativity is used to argue in favour of a local hidden variable theory (HVT) for individual quantum processes. It is shown that such a HVT may reproduce the statistical predictions of quantum mechanics (QM), provided the reproducibility of initial hidden variable states is limited. This means that in a HVT limits should be set to the validity of the notion of counterfactual definiteness (CFD). This is supported by the empirical (...) evidence that past, present, and future are basically distinct. Our argumentation is contrasted with a recent one by Stapp resulting in the opposite conclusion, i.e. nonlocality or the existence of faster-than-light influences. We argue that Stapp’s argumentation still depends in an implicit, but crucial, way on both the notions of hidden variables and of CFD. In addition, some implications of our results for the debate between Bohr and Einstein, Podolsky and Rosen are discussed. (shrink)

This work is a step toward the development of a logic for types and computation that includes not only the usual spaces of mathematics and constructions, but also spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes that we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we first study axiomatically, and then by deriving a modal calculus as its internal logic. (...) The resulting framework is intended as a setting for the logical and categorical study of relative computability. (shrink)

From the beginning, the implications of quantum theory for our most general understanding of the world have been a matter of intense debate. Einstein argues that the theory had to be regarded as fundamentally incomplete. Its inability, for example, to predict the exact time of decay of a single radioactive atom had to be due to a failure of the theory and not due to a permanent inability on our part or a fundamental indeterminism in nature itself. (...) In 1964, John Bell derived a theorem which showed that any deterministic theory which preserved locality would have certain consequences for measurements performed at a distance from one another. An experimental check seems to show that these consequences are not, in fact, realized. The correlation between the sets of events is much stronger than any local deterministic theory could allow. What is more, this stronger correlation is precisely that which is predicted by quantum theory. The astonishing result is that local deterministic theories of the classical sort seem to be permanently excluded. Not only can the individual decay not be predicted, but no future theory can ever predict it. The contributors in this volume wrestle with this conclusion. Some welcome it; others leave open a return to at lease some kind of deterministic world, one which must however allow something like action-at-a distance. How much lit it? And how can one avoid violating relativity theory, which excludes action-at-a-distance? How can a clash between the two fundamental theories of modern physics, relativity and quantum theory, be avoided? What are the consequences for the traditional philosophic issue of causality explanation and objectivity? One thing is certain; we can never return to the comfortable Newtonian world where everything that happened was, in principle, predictable and where what happened at one measurement site could not affect another set of measurements being performed light-years away, at a distance that a light-signal could not bridge. Contributors: James T. Cushing, Abner Shimony, N. David Mermin, Jon P. Jarrett, Linda Wessels, Bas C. van Fraassen, Jeremy Butterfield, Michael L. G. Redhead, Henry P. Stapp, Arthur Fine, R. I. G. Hughes, Paul Teller, Don Howard, Henry J. Folse, and Ernan McMullin. (shrink)

In this paper (a sequel to [4]) I put forward a "local" interpretation of mathematical concepts based on notions derived from category theory. The fundamental idea is to abandon the unique absolute universe of sets central to the orthodox set-theoretic account of the foundations of mathematics, replacing it by a plurality of local mathematical frameworks - elementary toposes - defined in category-theoretic terms.

Matter interacting classically with gravity in 3+1 dimensions usually gives rise to a continuum of degrees of freedom, so that, in any attempt to quantize the theory, ultraviolet divergences are nearly inevitable. Here, we investigate matter of a form that only displays a finite number of degrees of freedom in compact sections of space-time. In finite domains, one has only exact, analytic solutions. This is achieved by limiting ourselves to straight pieces of string, surrounded by locally flat sections of (...) space-time. Globally, however, the model is not finite, because solutions tend to generate infinite fractals. The model is not (yet) quantized, but could serve as an interesting setting for analytical approaches to classical general relativity, as well as a possible stepping stone for quantum models. Details of its properties are explained, but some problems remain unsolved, such as a complete description of the most violent interactions, which can become quite complex. (shrink)

Inspired by a framework of multi lingual sequent calculus, we introduce a formal logical system called locally continuous sequent calculus to represent _L_-domains. By considering the logic states defined on locally continuous sequent calculi, we show that the collection of all logic states of a locally continuous sequent calculus with respect to set inclusion forms an _L_-domain, and every _L_-domain can be obtained in this way. Moreover, we define conjunctive consequence relations as morphisms between our sequent calculi, and prove that (...) the category of locally continuous sequent calculi and conjunctive consequence relations is equivalent to that of _L_-domains and Scott-continuous functions. This result extends Abramsky’s “Domain theory in logical form” to a continuous setting. (shrink)

It is a classic result in lattice theory that a poset is a complete lattice iff it can be realized as fixpoints of a closure operator on a powerset. Dragalin [9,10] observed that a poset is a locale (complete Heyting algebra) iff it can be realized as fixpoints of a nucleus on the locale of upsets of a poset. He also showed how to generate a nucleus on upsets by adding a structure of “paths” to a poset, forming what (...) we call a Dragalin frame. This allowed Dragalin to introduce a semantics for intuitionistic logic that generalizes Beth and Kripke semantics. He proved that every spatial locale (locale of open sets of a topological space) can be realized as fixpoints of the nucleus generated by a Dragalin frame. In this paper, we strengthen Dragalin’s result and prove that every locale—not only spatial locales—can be realized as fixpoints of the nucleus generated by a Dragalin frame. In fact, we prove the stronger result that for every nucleus on the upsets of a poset, there is a Dragalin frame based on that poset that generates the given nucleus. We then compare Dragalin’s approach to generating nuclei with the relational approach of Fairtlough and Mendler [11], based on what we call FM-frames. Surprisingly, every Dragalin frame can be turned into an equivalent FM-frame, albeit on a different poset. Thus, every locale can be realized as fixpoints of the nucleus generated by an FM-frame. Finally, we consider the relational approach of Goldblatt [13] and characterize the locales that can be realized using Goldblatt frames. (shrink)

Philosophers have recently argued that we should revise our understanding of character. An individual's behaviour is governed not by a set of ‘global’ traits, each elicited by a certain kind of situational feature, they argue, but by a much larger array of ‘local’ traits, each elicited by a certain combination of situational features. But the data cited by these philosophers support their theory only if we conceive of traits purely in terms of stimulus and response, rather than in (...) the more traditional terms of inner mental items such as inclinations. We should not adopt the former conception, moreover, since doing so would impede pursuit of the ethical aims for which we need a theory of character, whereas retaining the latter conception will facilitate this pursuit. So we should not revise our understanding of character in the way proposed. (shrink)

This book uses the formal semantics of counterfactual conditionals to analyze the problem of non-locality in quantum mechanics. Counterfactual conditionals enter the analysis of quantum entangled systems in that they enable us to precisely formulate the locality condition that purports to exclude the existence of causal interactions between spatially separated parts of a system. They also make it possible to speak consistently about alternative measuring settings, and to explicate what is meant by quantum property attributions. The book develops the possible-world (...) semantics of quantum counterfactuals using David Lewis's famous approach as a starting point but modifying it significantly in order to achieve compatibility with the demands of the special theory of relativity as well as quantum mechanics. There have been several attempts to use counterfactuals semantics to strengthen Bell's theorem and its cognates such as the GHZ and Hardy theorems. These are critically evaluated in the book. Finally, a counterfactual reconstruction of the EPR argument and Bell's theorem is proposed that sheds a new light on their philosophical consequences regarding the relations between realism and local causation. (shrink)

Based on the new general framework for the probabilistic description of experiments, introduced in [E.R. Loubenets, Research Report No 8, MaPhySto, University of Aarhus, Denmark (2003); Proceedings Conference “Quantum Theory, Reconsideration of Foundations”, Ser. Math. Modeling, Vol. 10 (University Press, Vaxjo, 2004), pp. 365–385], we analyze in mathematical terms the link between the validity of Bell-type inequalities under joint experiments upon a system of any type and the physical concept of “local realism”. We prove that the violation of (...) Bell-type inequalities in the quantum case has no connection with the violation of “local realism”. In a general setting, we formulate in mathematical terms the condition on “local realism” under a joint experiment and consider examples of quantum “locally realistic” joint experiments. We, in particular, show that quantum joint experiments of the Alice/Bob type are “locally realistic”. For an arbitrary bipartite quantum state, we derive quantum analogs of the original Bell inequality. In view of our results, we argue that the violation of Bell-type inequalities in the quantum case cannot be a valid argument in the discussion on locality or non-locality of quantum interactions. (shrink)

When we ask a decision maker to express her preferences, it is typically assumed that we are eliciting a pre-existing set of preferences. However, empirical research has suggested that our preferences are often constructed on the fly for the decision problem at hand. This paper explores the ramifications of this empirical research for our understanding of instrumental rationality. First, I argue that these results pose serious challenges for the traditional decision-theoretic view of instrumental rationality, which demands global coherence amongst all (...) of one's beliefs and desires. To address these challenges, I first develop a minimal notion of instrumental rationality that issues in localized, goal-relative demands of coherence. This minimal conception of instrumental rationality is then used to offer a more sophisticated account of the global aspects of instrumental rationality. The resulting view abandons all-or-nothing assessments of rationality and allows us to evaluate decision makers as being rational to varying degrees. My aim is to propose a theory that is both psychologically and normatively plausible. (shrink)