We study Dedekind cuts on ordered Abelian groups. We introduce a monoid structure on them, and we characterise, via a suitable representation theorem, the universal part of the theory of such structures.

Let M be a polynomially bounded, o-minimal structure with archimedean prime model, for example if M is a real closed field. Let C be a convex and unbounded subset of M. We determine the first order theory of the structure M expanded by the set C. We do this also over any given set of parameters from M, which yields a description of all subsets of Mn, definable in the expanded structure.

In this chapter, we will provide an overview of Richard Dedekind's work on continuity, both foundational and mathematical. His seminal contribution to the foundations of analysis is the well-known 1872 booklet Stetigkeit und irrationale Zahlen (Continuity and irrational numbers), which is based on Dedekind's insight into the essence of continuity that he arrived at in the fall of 1858. After analysing the intuitive understanding of the continuity of the geometric line, Dedekind characterized the property of continuity for (...) the real numbers in terms of what are nowadays called 'Dedekind cuts' on the rational numbers. This treatment, which can be characterized as being 'arithmetical' as well as 'axiomatic', will be presented in detail. To better position Dedekind's contributions in their historical context, we will also consider some of his more mathematical treatments of continuity in addition to his foundational work. Of particular interest is the definition of the Riemann surface in his joint work with Heinrich Weber (1882). Moreover, Dedekind's reflections on space and continuity in his unpublished papers 'Allgemeine Sätze über Räume' (General theorems about spaces; before 1870) and 'Beweis und Anwendungen eines allgemeinen Satzes über mehrfach ausgedehnte stetige Gebiete' (Proof and applications of a general theorem about multiply extended continuous domains; 1892) illustrate the wide range and general coherency of his thoughts. By discussing Dedekind's works in which the notion of continuity plays a central role, we will show how Dedekind's approaches became increasingly abstract, while at the same time retaining a common methodology. (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 paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of rational numbers. Let P be a property (...) (of rational numbers) and r a rational number. Say that r is an upper bound of P, written P≤r, if for any rational number s, if Ps then either s<r or s=r. In other words, P≤r if r is greater than or equal to any rational number that P applies to. Consider the Cut Abstraction Principle: (CP) ∀P∀Q(C(P)=C(Q) ≡ ∀r(P≤r ≡ Q≤r)). In other words, the cut of P is identical to the cut of Q if and only if P and Q share all of their upper bounds. The axioms of second-order real analysis can be derived from (CP), just as the axioms of second-order Peano arithmetic can be derived from Hume’s principle. The paper raises some of the philosophical issues connected with the neo-Fregean program, using the above abstraction principles as case studies. (shrink)

It is a commonly held view that Dedekind's construction of the real numbers is impredicative. This naturally raises the question of whether this impredicativity is justified by some kind of Platonism about sets. But when we look more closely at Dedekind's philosophical views, his ontology does not look Platonist at all. So how is his construction justified? There are two aspects of the solution: one is to look more closely at his methodological views, and in particular, the places (...) in which predicativity restrictions ought to be applied; another is to take seriously his remarks about the reals as things created by the cuts, instead of considering them to be the cuts themselves. This can lead us to make finer-grained distinctions about the extent to which impredicative definitions are problematic, since we find that Dedekind's use of impredicative definitions in analysis can be justified by his philosophical views. (shrink)

In the present note, we study a generalization of Dedekind cuts in the context of constructive Zermelo–Fraenkel set theory CZF. For this purpose, we single out an equivalent of CZF's axiom of fullness and show that it is sufficient to derive that the Dedekind cuts in this generalized sense form a set. We also discuss the instance of this equivalent of fullness that is tantamount to the assertion that the class of Dedekind cuts in the rational numbers, (...) in the customary constructive sense including locatedness, is a set. This is to be compared with the situation for the partial reals, a generalization in a different direction: if one drops locatedness from the notion of a Dedekind real, then it becomes inherently impredicative. We make this precise, and further present the curiosity that the irrational Dedekind reals form a set already with exponentiation. (shrink)

The computability of reals was introduced by Alan Turing [20] by means of decimal representations. But the equivalent notion can also be introduced accordingly if the binary expansion, Dedekind cut or Cauchy sequence representations are considered instead. In other words, the computability of reals is independent of their representations. However, as it is shown by Specker [19] and Ko [9], the primitive recursiveness and polynomial time computability of the reals do depend on the representation. In this paper, we explore (...) how the weak computability of reals depends on the representation. To this end, we introduce three notions of weak computability in a way similar to the Ershov's hierarchy of Δ02-sets of natural numbers based on the binary expansion, Dedekind cut and Cauchy sequence, respectively. This leads to a series of classes of reals with different levels of computability. We investigate systematically questions as on which level these notions are equivalent. We also compare them with other known classes of reals like c. e. and d-c. e. reals. (shrink)

The Dedekind cuts in an ordered set form a set in the sense of constructive Zermelo—Fraenkel set theory. We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness. Together with exponentiation, refinement is equivalent to fullness. None of the defining properties of an ordering is needed, and only refinement for two—element coverings is used. In particular, the Dedekind reals form a set; whence we have also refined an earlier result by (...) Aczel and Rathjen, who invoked the full form of fullness. To further generalise this, we look at Richman's method to complete an arbitrary metric space without sequences, which he designed to avoid countable choice. The completion of a separable metric space turns out to be a set even if the original space is a proper class; in particular, every complete separable metric space automatically is a set. (shrink)

Dedicated to Dana Scott on his sixtieth birthday.It is common knowledge that for a short while Hermann Weyl joined Brouwer in his pursuit of a revision of mathematics according to intuitionistic principles. There is, however, little in the literature that sheds light on Weyl's role and in particular on Brouwer's reaction to Weyl's allegiance to the cause of intuitionism. This short episode certainly raises a number of questions: what made Weyl give up his own program, spelled out in “Das Kontinuum”, (...) how did Weyl come to be so well-informed about Brouwer's new intuitionism, in what respect did Weyl's intuitionism differ from Brouwer's intuitionism, what did Brouwer think of Weyl's views,…? To some of these questions at least partial answers can be put forward on the basis of some of the available correspondence and notes. The present paper will concentrate mostly on the historical issues of the intuitionistic episode in Weyl's career.Weyl entered the foundational controversy with a bang in 1920 with his sensational paper “On the new foundational crisis in mathematics”. He had already made a name for himself in the foundations of mathematics in 1918 with his monograph “The Continuum” [18]; this contained in addition to a technical logical-mathematical construction of the continuum, a fairly extensive discussion of the shortcomings of the traditional construction of the continuum on the basis of arbitrary—and hence also impredicative—Dedekind cuts. (shrink)

Pursuing the proof-theoretic program of Friedman and Simpson, we begin the study of the metamathematics of countable linear orderings by proving two main results. Over the weak base system consisting of arithmetic comprehension, II 1 1 -CA0 is equivalent to Hausdorff's theorem concerning the canonical decomposition of countable linear orderings into a sum over a dense or singleton set of scattered linear orderings. Over the same base system, ATR0 is equivalent to a version of the Continuum Hypothesis for linear orderings, (...) which states that every countable linear ordering either has countably many or continuum many Dedekind cuts. (shrink)

This paper introduces the notion of cW10-creative set, which strengthens that of semicreative set in a similar way as complete creativity strengthens creativity. Two results are proven, both of which imply that not all semicreative sets are cW10-creative. First, it is shown that semicreative Dedekind cuts cannot be cW10-creative; the existence of semicreative Dedekind cuts was shown by Soare. Secondly, it is shown that if A ⊕ B, the join of A and B, is cW10-creative, then either A (...) or B is cW10-creative, and the same is not true with ‘cW10-creative’ replaced by ‘semicreative’. Moreover, sets A, B which provide a counterexample for can be constructed within any given nonrecursive r.e. T-degrees a, b. MSC: 03D30. (shrink)

Jan Krajíček posed the following problem: Is there is a generalization result in the theory of real closed fields of the form: If A is provable in length k for all n ϵ ω , then A is provable? It is argued that the answer to this question depends on the particular formulation of the “theory of real closed fields.” Four distinct formulations are investigated with respect to their generalization behavior. It is shown that there is a positive answer to (...) Krajíček's question for 1. the axiom system RCF of Artin-Schreier with Gentzen's LK as underlying logical calculus, 2. RCF with the variant LK B of LK allowing introduction of several quantifiers of the same type in one step, 3. LK B and the first-order schemata corresponding to Dedekind cuts and the supremum principle. A negative answer is given for 4. any system containing the schema of extensionality. (shrink)

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends (...) to teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms. (shrink)

We present correct and natural development of fundamental analysis in a predicative set theory we call PZFU\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {PZF}^{\mathsf {U}}}$$\end{document}. This is done by using a delicate and careful choice of those Dedekind cuts that are adopted as real numbers. PZFU\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {PZF}^{\mathsf {U}}}$$\end{document} is based on ancestral logic rather than on first-order logic. Its key feature is that it is definitional in the (...) sense that every object which is shown in it to exist is defined by some closed term of the theory. This allows for a very concrete, computationally-oriented model of it, and makes it very suitable for MKM and ITP. The development of analysis in PZFU\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {PZF}^{\mathsf {U}}}$$\end{document} does not involve coding, and the definitions it provides for the basic notions are the natural ones, almost the same as one can find in any standard analysis book. (shrink)

The aim of this paper is to present a conception of the triple nature of mathematics. It is argued that the nature of mathematics is best served by distinguishing deep ideas (of concepts or propositions), their surface representations (signs which can be perceived by senses) and their formal models (in axiomatic theories). For instance, the concept „number π” has several different models in set theory (those based on Dedekind cuts and on Cantor's equivalence classes of Cauchy sequences) and yet (...) all working mathematicians in the world have the same object π in mind. They have a common deep idea of π. Generally, the deep idea of a concept X is a well-formed mental construction of X which controls reasoning. It manifests itself in a characteristic, definite feeling of purpose, in firm certainty of the meaning of X in various contexts, and in robustness of understanding of X in cases of typical cognitive conflicts. Epistemic deep ideas are intersubjective and have been formed in phylogeny whereas individual deep ideas (or deep intuitions) are formed in ontogeny. In certain situations a deep idea may be described in terms of intuition, of meaning or sense, or of understanding, but none of these terms can provide a satisfactory description fitting all cases. (shrink)

Several researchers have recently established that for every Turing degree \, the real closed field of all \-computable real numbers has spectrum \. We investigate the spectra of real closed fields further, focusing first on subfields of the field \ of computable real numbers, then on archimedean real closed fields more generally, and finally on non-archimedean real closed fields. For each noncomputable, computably enumerable set C, we produce a real closed C-computable subfield of \ with no computable copy. Then we (...) build an archimedean real closed field with no computable copy but with a computable enumeration of the Dedekind cuts it realizes, and a computably presentable nonarchimedean real closed field whose residue field has no computable presentation. (shrink)

In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure - Dedekind-complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if computable is replaced by primitive recursive (p. r., for short), these definitions lead to a number of (...) different concepts, which we compare in this article. We summarize the known results and add new ones. In particular we show that there is a proper hierarchy among p. r. real numbers by nested interval representation, Cauchy representation, b -adic expansion representation, Dedekind cut representation, and continued fraction expansion representation. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers. (shrink)

In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure – Dedekind-complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if “computable” is replaced by “primitive recursive” , these definitions lead to a number of different concepts, which (...) we compare in this article. We summarize the known results and add new ones. In particular we show that there is a proper hierarchy among p. r. real numbers by nested interval representation, Cauchy representation, b -adic expansion representation, Dedekind cut representation, and continued fraction expansion representation. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers. (shrink)

We give arguments explaining why, when adopting a minimalist approach to constructive mathematics as that formalized in our two-level minimalist foundation, the choice for a pointfree approach to topology is not just a matter of convenience or mathematical elegance, but becomes compulsory. The main reason is that in our foundation real numbers, either as Dedekind cuts or as Cauchy sequences, do not form a set.

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the (...) geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction. (shrink)

Strong completeness of S. Titani's system for lattice valued logic is shown by means of Dedekind cuts.

We study some representations of real numbers. We compare these representations, on the one hand from the viewpoint of recursive functionals, and of complexity on the other hand.The impossibility of obtaining some functions as recursive functionals is, in general, easy. This impossibility may often be explicited in terms of complexity: - existence of a sequence of low complexity whose image is not a recursive sequence, - existence of objects of low complexity but whose images have arbitrarily high time- complexity .Moreover, (...) some representations of real numbers that are equivalent from the viewpoint of recursive functionals, are very distinct from the viewpoint of complexity.We make a particular study of representations via continued fractions . We precise exactly what part of information available in the x's dfc is equivalent to the information available in its Dedekinds cut. We show that the sum of two reals whose dfcs are polynomial-time computable may be a real whose dfc has time complexity arbitrarily high.This work confirms that the unique representation of real numbers suitable for the ordinary calculus is via explicit Cauchy sequences of rationals.RésuméNous étudions différentes manières de présenter les nombres réels. Nous comparons ces présentations du point de vue des fonctionnelles récursives d'une part, et de celui des classes de complexité d'autre part.L'impossibilité d'obtenir certaines fonctions sous forme de fonctionnelles récursives est en général facile à établir. Cette impossiblité peut souvent être explicitée en termes de complexité: - il existe une suite de faible complexité dont l'image est une suite non récursive, - il existe des objets de faible complexité mais dont les images sont des objets de complexité arbitrairement grande .En outre, certaines présentations des réels équivalentes du point de vue des fonctionnelles récursives se distinguent nettement du point de vue de la complexité.Nous faisons une étude particulière concernant les développements en fraction continue . Nous précisions exactement quelle est la partie de l'information disponible dans le dfc d'un réel x qui équivaut à l'information disponible dans sa coupure de Dedekind. Nous montrons également que la somme de deux réels dont le dfc est calculable en temps polynomial peut être un réel dont le dfc est de complexité arbitrairement grande.Ce travail confirme que seule une présentation des réels via des suites de rationnels explicitement de Cauchy est adaptée aux calculs avec les réels. (shrink)

A central claim of Deleuze's reading of Bergson is that Bergson's distinction between space as an extensive multiplicity and duration as an intensive multiplicity is inspired by the distinction between discrete and continuous manifolds found in Bernhard Riemann's 1854 thesis on the foundations of geometry. Yet there is no evidence from Bergson that Riemann influences his division, and the distinction between the discrete and continuous is hardly a Riemannian invention. Claiming Riemann's influence, however, allows Deleuze to argue that quantity, in (...) the form of ‘virtual number’, still pertains to continuous multiplicities. This not only supports Deleuze's attempt to redeem Bergson's argument against Einstein in Duration and Simultaneity, but also allows Deleuze to position Bergson against Hegelian dialectics. The use of Riemann is thereby an important element of the incorporation of Bergson into Deleuze's larger early project of developing an anti-Hegelian philosophy of difference. This article first reviews the role of discrete and continuous multiplicities or manifolds in Riemann's Habilitationsschrift, and how Riemann uses them to establish the foundations of an intrinsic geometry. It then outlines how Deleuze reinterprets Riemann's thesis to make it a credible resource for Deleuze's Bergsonism. Finally, it explores the limits of this move, and how Deleuze's later move away from Bergson turns on the rejection of an assumption of Riemann's thesis, that of ‘flatness in smallest parts’, which Deleuze challenges with the idea, taken from Riemann's contemporary, Richard Dedekind, of the irrational cut. (shrink)

Deleuze's theory of time set out in Difference and Repetition is a complex structure of three different syntheses of time – the passive synthesis of the living present, the passive synthesis of the pure past and the static synthesis of the future. This article focuses on Deleuze's third synthesis of time, which seems to be the most obscure part of his tripartite theory, as Deleuze mixes different theoretical concepts drawn from philosophy, Greek drama theory and mathematics. Of central importance is (...) the notion of the cut, which is constitutive of the third synthesis of time defined as an a priori ordered temporal series separated unequally into a before and an after. This article argues that Deleuze develops his ordinal definition of time with recourse to Kant's definition of time as pure and empty form, Hölderlin's notion of ‘caesura’ drawn from his ‘Remarks on Oedipus’ (1803) and Dedekind's method of cuts as developed in his pioneering essay ‘Continuity and Irrational Numbers’ (1872). Deleuze then ties together the conceptions of the Kantian empty form of time and the Nietzschean eternal return, both of which are essentially related to a fractured I or dissolved self. This article aims to assemble the different heterogeneous elements that Deleuze picks up on and to show how the third synthesis of time emerges from this differential multiplicity. (shrink)

A box type is an n-type of an o-minimal structure which is uniquely determined by the projections to the coordinate axes. We characterize heirs of box types of a polynomially bounded o-minimal structure M. From this, we deduce various structure theorems for subsets of $M^k $ , definable in the expansion M of M by all convex subsets of the line. We show that M after naming constants, is model complete provided M is model complete.

We now know of a number of ways of developing real analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals byion" program differs by placing additional emphasis upon what I here term Frege's (...) Constraint, that a satisfactory foundation for any branch of mathematics should somehow so explain its basic concepts that their applications are immediate. This paper is concerned with the meaning of and motivation for this constraint. Structuralism has to represent the application of a mathematical theory as always posterior to the understanding of it, turning upon the appreciation of structural affinities between the structure it concerns and a domain to which it is to be applied. There is, therefore, a case that Frege's Constraint has bite whenever there is a standing body of informal mathematical knowledge grounded in direct reflection upon sample, or schematic, applications of the concepts of the theory in question. It is argued that this condition is satisfied by simple arithmetic and geometry, but that in view of the gap between its basic concepts (of continuity and of the nature of the distinctions among the individual reals) and their empirical applications, it is doubtful that Frege's Constraint should be imposed on a neo-Fregean construction of analysis. (shrink)

El objetivo de este artículo es presentar una demostración de un teorema clásico sobre álgebras booleanas y ordenes parciales de relevancia actual en teoría de conjuntos, como por ejemplo, para aplicaciones del método de construcción de modelos llamado “forcing” (con álgebras booleanas completas o con órdenes parciales). El teorema que se prueba es el siguiente: “Todo orden parcial se puede extender a una única álgebra booleana completa (salvo isomorfismo)”. Donde extender significa “sumergir densamente”. Tal demostración se realiza utilizando cortaduras de (...) Dedekind siguiendo el texto “Set Theory” de Jech, y otras ideas propias del autor de este artículo. Adicionalmente, se formulan algunas versiones débiles del axioma de elección relacionadas con las álgebras booleanas, las cuales son también de gran importancia para la investigación en teoría de conjuntos y teoría de modelos, pues estas son poderosas técnicas de construcción de modelos, como por ejemplo, el teorema de compacidad (permite construir modelos no estándar, etc) y el teorema del ultrafiltro, que permite construir ultraproductos (pueden ser usados para investigar problemas de cardinales grandes, etc). Se presentan algunas referencias de problemas abiertos sobre el tema. -/- The objective of this paper is to present a demonstration of a classical theorem on boolean algebras and partial orders of current relevance in set theory, as for example, for applications of model construction method called forcing" (with boolean algebras complete or with partial orders). The theorem to be proved is as follows: Any partial order can be extended to a single complete boolean algebra (up to isomorphism)". Where to extend means embed densely". Such a demonstration is done using Dedekind's cuts following the text Set Theory" of Jech, and other ideas of the author of this article. In addition, some weak versions of the axiom of choice related to boolean algebras are formulated, which are also of great importance for the research in set theory and model theory, since this are powerful model construction techniques, such as the compactness theorem (allows the construction of non-standard models, etc.) and the ultralter theorem, which allows the construction of ultraproducts (can be used to investigate problems of large cardinals, etc). Some references of open problems on the subject are presented. (shrink)

Psychopathology is the study of the signs and symptoms of psychiatric disorders - delusions, hallucinations, phobias, depression, for example. This book gives an account of the terms currently in use and attempts an in-depth analysis of the nature of each. The matter is examined both from a philosophical perspective and from the point of view of what is known about the function of the hemispheres of the brain.

In lieu of an abstract, here is a brief excerpt of the content:Joanne Cutting-Gray HANNAH ARENDT'S RAHEL VARNHAGEN Hannah Arendt fled Nazi Germany in 1933, a year she called the end of Jewish history. She was 27 years old at the time and carried with her a manuscript that was later to become the peculiar biography of an eighteenth-century German-Jewish "pariah," Rahel Varnhagen (1771-1833). The Life of a fewish Woman, subtitle of the biography by Arendt, distills the largely unpublished Varnhagen (...) correspondence and marks the impact of that life upon German-Jewish history.1 Published more than "half a lifetime" later in 1957, it assays the politics of form in a highly unusual manner. By expunging all detail in order to analyze Varnhagen's introspection, Arendt implicitly questions selfhood as the form for identity and as the appropriate form for biography. She reveals that seeking to preserve the self determines the blank dumbness of an unnarratable life. Rahel Levin (Antonie Friederike Robert) Varnhagen, or simply "Rahel " as Arendt calls her, lived a life as complicated as her many names. The tortured relation she had to her identity, her insatiable preoccupation with self, reflects an effort to eradicate herJewishness by baptism and marriage and becomes a paradigm of a story about a woman lost to a sense of others and herself. Neither rich, cultivated, nor beautiful, Rahel lived her unhappiness fervently, with an ecstasy that made her appear to others as startlingly original. This helps to explain how she came to preside over the most important intellectual salon in Germany during the late eighteenth century. Princes, intellectuals, and artists flocked to her attic apartment, attracted by the impenetrable spectacle of her despair. Rahel's intensity pulled others into her orbit—including, one-hundred years later, the intense and self-absorbed Arendt at the beginning Philosophy and Literature, © 1991, 15: 229-245 230Philosophy and Literature of her university education.2 Each woman suffered the private failure of a love affair complicated by the broader, historical issue of racial identity that connected their lives. Prewar Nazi Germany taught Arendt what Varnhagen learned during the period of German-Jewish assimilation in the nineteenth century—"one does not escape Jewishness."3 It is no wonder that, in analogous moments of personal and civic upheaval, Rahel Varnhagen and Hannah Arendt "corresponded," the one by writing letters about her life, the other by giving that life to a world. Like Arendt, Varnhagen was unclassifiably exotic, an unattached, Jewish female seemingly without a history, aligned with no one and nothing. If"Jewishness" meant letting life "rain upon" her, Rahel Varnhagen let it strike her "like a storm without an umbrella." And like Arendt, she refused to hide the traits and talents of her personality under cover of morality and convention. Her letters are not self-congratulatory, for she never wittingly chose notoriety. Indeed, she spent much of her life trying to escape the stigma ofJewishness that marked her destiny. Notwithstanding the milieu of nineteenth-century German romanticism that drew the biographer toward her subject, various events surrounding the text make more compelling narratives than Varnhagen 's life: how the manuscript survived World War II, Arendt's emigration to the United States, and, especially, the public and private influence of Martin Heidegger.4 At the age of 18 (1925), Arendt had discovered Varnhagen during her brief love-affair with the 35-year-old philosophy professor at the University of Marburg. Though Heidegger would later call Arendt the "passion of his life" and inspiration for his work—he was writing Being and Time during that period—his brush with Nazism precipitated their breach. Since the Heidegger-Arendt correspondence remains closed, much about their youthful relationship and life-long intellectual affair remains a mystery open only to speculation. From Arendt's biographer, Elisabeth Young-Bruehl, we know that Varnhagen's story became a consolation to her in her sorrow, a means ofgiving significance to the ambiguities of a private relationship as it was caught up in public history.5 Certainly Heidegger's retreat from political events for the sake of thinking was as strong an influence on Arendt's political development as was Varnhagen's retreat from her Jewish destiny. Nonetheless, it would be a mistake to expect... (shrink)

Hannah Arendt's early biography of Rahel Varnhagen, an eighteenth-century German-Jew, provides a revolutionary feminist component to her political theory. In it, Arendt grapples with the theoretical constitution of a female subject and relates Jewish alterity, identity, and history to feminist politics. Because she understood the "female condition" of difference as belonging to the political subject rather than an autonomous self, her theory entails a "politics of alterity" with applications for feminist practice.

Hannah Arendt's early biography of Rahel Varnhagen, an eighteenth-century German-Jew, provides a revolutionary feminist component to her political theory. In it, Arendt grapples with the theoretical constitution of a female subject and relates Jewish alterity, identity, and history to feminist politics. Because she understood the "female condition" of difference as belonging to the political subject rather than an autonomous self, her theory entails a "politics of alterity" with applications for feminist practice.

At least 177 scales are available to researchers who want to measure religiosity, but questions exist as to exactly what these scales are measuring and in whom they are measuring it. A review of these scales found a lack items designed to measure ethical action in society or the world as a prophetic response to the experience of the divine. Instead, the vast majority of scales focus on internal experiences and beliefs or institutional relationships. A review of scale norm groups (...) found that norm groups often are not fully described, particularly in the area of race/ethnicity, and when they are described, they reveal an over-reliance on convenience samples of college students and an under-representation of racial/ethnic minority groups. Examples of scales with more fully described and more representative norm groups are given, and recommendations are offered for researchers using and developing religiosity scales. (shrink)

At least 177 scales are available to researchers who want to measure religiosity, but questions exist as to exactly what these scales are measuring and in whom they are measuring it. A review of these scales found a lack items designed to measure ethical action in society or the world as a prophetic response to the experience of the divine. Instead, the vast majority of scales focus on internal experiences and beliefs or institutional relationships. A review of scale norm groups (...) found that norm groups often are not fully described, particularly in the area of race/ethnicity, and when they are described, they reveal an over-reliance on convenience samples of college students and an under-representation of racial/ethnic minority groups. Examples of scales with more fully described and more representative norm groups are given, and recommendations are offered for researchers using and developing religiosity scales. (shrink)

Popular movies present chunk-like events that promote episodic, serial updating of viewers’ representations of the ongoing narrative. Event-indexing theory would suggest that the beginnings of new scenes trigger these updates, which in turn require more cognitive processing. Typically, a new movie event is signaled by an establishing shot, one providing more background information and a longer look than the average shot. Our analysis of 24 films reconfirms this. More important, we show that, when returning to a previously shown location, the (...) re-establishing shot reduces both context and duration while remaining greater than the average shot. In general, location shifts dominate character and time shifts in event segmentation of movies. In addition, over the last 70 years re-establishing shots have become more like the noninitial shots of a scene. Establishing shots have also approached noninitial shot scales, but not their durations. Such results suggest that film form is evolving, perhaps to suit more rapid encoding of narrative events. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy, Psychiatry, & Psychology 8.4 (2001) 337-338 [Access article in PDF] On Kimura's Écrits de psychopathologie phénomenologique John Cutting This book is a French translation of six articles that the Japanese psychiatrist, Kimura Bin, wrote in the 1970s and 1980s. There is the usual long introduction in such books by the translator. There is also the mandatory explanation of the whole matter as a postface by philosopher Henry Maldiney (...) as if the meaning of anything these days has to be in the form of a sandwich.The first article by Kimura Bin deals with the meaning of certain Japanese words pertaining to self and inter-subjectivity. The critical Japanese word is aida, which means something like interpersonal bonding and is that primordial state of the human being that founds both everyday sociality and the individual self, the latter both in its external and internal composition and orientation: "From time immemorial, the Japanese understood interpersonal social existence, the being between, life in the aida of itself and of others, as the essential trait of an individual human. The interpersonal aida is in some sense the quintessence of known individuality." Schizophrenia, for Kimura, stems from a disturbance in this aida at the most fundamental level, and all its manifestations follow from this.The second article is entitled "Time and Anxiety (Angoisse)." This sets out his major psychopathological thesis—judging from these six articles—which is that schizophrenia and depressive illness differ fundamentally in their temporality. In schizophrenia the subject is geared toward the future: "Subjects with a schizophrenic tendency give sufficient proof in their daily life of a marked inclination to imagine future possibilities or ideas whose feasibility is impossible or implausible." In depressive illness there is a preoccupation with the past: "The lived time of the melancholic type of person has a preponderance towards the past, or, more precisely, towards lived being up to the present."The third article, entitled "Schizophrenic Temporality," amplifies this thesis and introduces his terminology for these, respectively, as the Latin terms, ante festum and post festum. The French translation gives the meaning of the latter as "apres la fête" and "trop tard"—after the holiday, too late. (I shall deal with the potpourri of languages involved below.) This article is the most elaborate, and in it Kimura attempts to derive the core characteristics of the two nosological entities from these two underlying anomalous temporalities.The fourth article is about borderline states, a nosological entity that I have no sympathy for, and I shall not discuss it.The fifth article is entitled "Reflection and Self in Schizophrenia." Again, it concerns a fundamental difference between the mode of reflection in schizophrenia and that in depressive illness, but I cannot say, despite numerous readings, [End Page 337] that I understand it. The reason for this may be my poor French, an inadequate translation, or a confused formulation in the first place, or all three. The gist of it seems to be that in schizophrenia the "other" is a "stranger" projected by the thinking self, whereas in depressive illness the "other" is a "familiar" inhabitant of the sociality adherent to this condition.The sixth article, "Pathology of Immediateness," is an attempt to seek philosophical, theoretical, psychopathological, and even nosological and biological backing for what Kimura has set out earlier, particularly in the third article. Hegel, Lacan, Bergson, Heidegger, Derrida, and Husserl are brought in as philosophical support. Blankenburg and Binswanger provide psychopathological backing. A third prong is the nosological tradition, prevalent in Germany and Japan in the twentieth century, of promoting the concept of an atypical or third type of psychosis with a biological resemblance to epileptic psychosis. All this is then brought together and complicated by a discussion of language and 'mediation' to support the notion that there is a third psychopathological principle—in addition to the ante festum pertaining to schizophrenia and the post festum pertaining to depressive illness—which is that of an intra festum: a preoccupation with the now, characteristic of atypical psychoses with an epileptic basis.These articles tax one's linguistic and conceptual repertoire to the... (shrink)