We show that there is a model of ZF in which the Borel hierarchy on the reals has length ω2. This implies that ω1 has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has exactly λ + 1 levels for any given limit ordinal λ less than ω2. We also show that assuming a large cardinal hypothesis there are models (...) of ZF in which the Borel hierarchy is arbitrarily long. (shrink)

In intuitionistic analysis, "Brouwer's Continuity Principle" implies, together with an "Axiom of Countable Choice", that the positively Borel sets form a genuinely growing hierarchy: every level of the hierarchy contains sets that do not occur at any lower level.

We analyze the set-theoretic strength of determinacy for levels of the Borel hierarchy of the form$\Sigma _{1 + \alpha + 3}^0 $, forα<ω1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to requireα+ 1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of weak reflection principles, Π1-RAPα, whose consistency strength corresponds exactly to the logical strength of${\rm{\Sigma }}_{1 + (...) \alpha + 3}^0 $determinacy, for$\alpha < \omega _1^{CK} $. This yields a characterization of the levels ofLby or at which winning strategies in these games must be constructed. Whenα= 0, we have the following concise result: The leastθso that all winning strategies in${\rm{\Sigma }}_4^0 $games belong toLθ+1is the least so that$L_\theta \models {\rm{``}}{\cal P}\left$exists, and all wellfounded trees are ranked”. (shrink)

The thesis focuses on a quite recent research field lying in between Descriptive Set Theory and Automata Theory (for infinite objects). In both areas, one is often concerned with subsets of the Cantor space or of its homeomorphic copies. In Descriptive Set Theory, such subsets are usually stratified in topological hierarchies, like the Borel hierarchy, the Wadge hierarchy and the difference hierarchy; in Automata Theory, such sets are studied in terms of regularity, that is, the property (...) of being recognised by an automaton or, equivalently, of being expressible in Monadic Second-Order Logic. This double point of view leads to many interesting questions about the interplay and relationship between topological complexity and regularity. While we have a complete picture of what happens in the case of automata on words, the case of automata on trees is still a terra incognita. Some results have already been obtained for particular classes of languages, like Buchi languages, deterministic languages, ¨ and unambiguous languages. In this thesis we instead drop any restriction and prove some new results concerning arbitrary regular tree languages which belong to low levels of the Borel hierarchy and of the Wadge hierarchy. (shrink)

In intuitionistic analysis, a subset of a Polish space like or is called positively Borel if and only if it is an open subset of the space or a closed subset of the space or the result of forming either the countable union or the countable intersection of an infinite sequence of (earlier constructed) positively Borel subsets of the space. The operation of taking the complement is absent from this inductive definition, and, in fact, the complement of a (...) positively Borel set is not always positively Borel itself (see Veldman, 2008a). The main result of Veldman (2008a) is that, assuming Brouwer's Continuity Principle and an Axiom of Countable Choice, one may prove that the hierarchy formed by the positively Borel sets is genuinely growing: every level of the hierarchy contains sets that do not occur at any lower level. The purpose of the present paper is a different one: we want to explore the truly remarkable fine structure of the hierarchy. Brouwer's Continuity Principle again is our main tool. A second axiom proposed by Brouwer, his Thesis on Bars is also used, but only incidentally. (shrink)

We consider Borel sets of finite rank $A \subseteq\Lambda^\omega$ where cardinality of Λ is less than some uncountable regular cardinal K. We obtain a "normal form" of A, by finding a Borel set Ω, such that A and Ω continuously reduce to each other. In more technical terms: we define simple Borel operations which are homomorphic to ordinal sum, to multiplication by a countable ordinal, and to ordinal exponentiation of base K, under the map which sends every (...) Borel set A of finite rank to its Wadge degree. (shrink)

We prove that if f is a partial Borel function from one Polish space to another, then either f can be decomposed into countably many partial continuous functions, or else f contains the countable infinite power of a bijection that maps a convergent sequence together with its limit onto a discrete space. This is a generalization of a dichotomy discovered by Solecki for Baire class 1 functions. As an application, we provide a characterization of functions which are countable unions (...) of continuous functions with domains of type Πn0, for a fixed n<ω. For Baire class 1 functions, this generalizes analogous characterizations proved by Jayne and Rogers for n=1 and Semmes for n=2. (shrink)

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and (...) corresponding complete functions. We use this classification and an effective version of a Selection Theorem of Bhattacharya-Srivastava in order to prove a generalization of the Representation Theorem of Kreitz-Weihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the Banach-Hausdorff-Lebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level. (shrink)

We establish some results on the Borel and difference hierarchies in φ-spaces. Such spaces are the topological counterpart of the algebraic directed-complete partial orderings. E.g., we prove analogs of the Hausdorff Theorem relating the difference and Borel hierarchies and of the Lavrentyev Theorem on the non-collapse of the difference hierarchy. Some of our results generalize results of A. Tang for the space Pω. We also sketch some older applications of these hierarchies and present a new application to (...) the question of characterizing the ω-ary Boolean operations generating a given level of the Wadge hierarchy from the open sets. (shrink)

The focus of this paper is the incomputability of some topological functions using the tools of Borel computability theory, as introduced by V. Brattka in [3] and [4]. First, we analyze some basic topological functions on closed subsets of ℝn, like closure, border, intersection, and derivative, and we prove for such functions results of Σ02-completeness and Σ03-completeness in the effective Borel hierarchy. Then, following [13], we re-consider two well-known topological results: the lemmas of Urysohn and Urysohn-Tietze for (...) generic metric spaces . Both lemmas define Σ02-computable functions which in some cases are even Σ02-complete. (shrink)

We show that if Ƒ is any "well-behaved" subset of the Borei functions and we assume the Axiom of Determinacy then the hierarchy of degrees on $P(^\omega \omega )$ induced by Ƒ turns out to look like the Wadge hierarchy (which is the special case where Ƒ is the set of continuous functions).

We prove two sets of results concerning computational complexity classes. First, we propose a new variation of the random oracle hypothesis, originally posed by Bennett and Gill after they showed that relative to a randomly chosen oracle, with probability 1. Their original hypothesis was quickly disproven in several ways, most famously in 1992 with the result that, in spite of the classes being shown unequal with probability 1. Here we propose a variation of what it means to be “large” using (...) the Ellentuck topology. In this new context, we demonstrate that the set of oracles separating and is not small, and obtain similar results for the separation of from along with the separation of from. We also show that the set of oracles equating with is large in this new sense. We demonstrate that this version of the hypothesis provides a sufficient condition for unrelativized relationships, at least in the cases considered here. Second, we examine the descriptive complexity of the classes of oracles providing the separations for these various classes, and determine their exact placement in the Borel hierarchy. (shrink)

In [9] we have considered a wide class of "well-behaved" reducibilities for sets of reals. In this paper we continue with the study of Borel reducibilities by proving a dichotomy theorem for the degree-structures induced by good Borel reducibilities. This extends and improves the results of [9] allowing to deal with a larger class of notions of reduction (including, among others, the Baire class ξ functions).

The author proved in his Ph.D. Thesis [W. Veldman, Investigations in intuitionistic hierarchy theory, Ph.D. Thesis, Katholieke Universiteit Nijmegen, 1981] that, in intuitionistic analysis, the positively Borel subsets of Baire space form a genuinely growing hierarchy: every level of the hierarchy contains sets that do not occur at any lower level. It follows from this result that there are natural examples of analytic and also of co-analytic sets that are not positively Borel. It turns out, (...) however, that, in intuitionistic analysis, one may give surprisingly different and, in some sense, much more simple examples of analytic and co-analytic sets that fail to be positively Borel. In the paper, two such examples are given. In proving them correct, one obtains new proofs of the Borel Hierarchy Theorem. Brouwer’s Continuity Principle plays a crucial role in arguments. (shrink)

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish space and E a Borel equivalence relation on X, a classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c = c. To be of any value we would expect I and c (...) to be “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ fFf. Then if is an embedding of X/E into Y/F, which is “Borel”.Intuitively, E ≤BF might be interpreted in any one of the following ways: The classi.cation problem for E is simpler than that of F: any invariants for F work as well for E. One can classify E by using as invariants F-equivalence classes. The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F. (shrink)

We investigate diagonal actions of Polish groups and the related intersection operator on closed subgroups of the acting group. The Borelness of the diagonal orbit equivalence relation is characterized and is shown to be connected with the Borelness of the intersection operator. We also consider relatively tame Polish groups and give a characterization of them in the class of countable products of countable abelian groups. Finally an example of a logic action is considered and its complexity in the Borel (...) reducbility hierarchy determined. (shrink)

The topological arithmetical hierarchy is the effective version of the Borel hierarchy. Its class Δta 2 is just large enough to include several types of pointsets in Euclidean spaces ℝ k which are fundamental in computable analysis. As a crossbreed of Hausdorff's difference hierarchy in the Borel class ΔB 2 and Ershov's hierarchy in the class Δ0 2 of the arithmetical hierarchy, the Hausdorff-Ershov hierarchy introduced in this paper gives a powerful classification (...) within Δta 2. This is based on suitable characterizations of the sets from Δta 2 which are obtained in a close analogy to those of the ΔB 2 sets as well as those of the Δ0 2 sets. A helpful tool in dealing with resolvable sets is contributed by the technique of depth analysis. On this basis, the hierarchy properties, in particular the strict inclusions between classes of different levels, can be shown by direct constructions of witness sets. The Hausdorff-Ershov hierarchy runs properly over all constructive ordinals, in contrast to Ershov's hierarchy whose denotation-independent version collapses at level ω 2. Also, some new characterizations of concepts of decidability for pointsets in Euclidean spaces are presented. (shrink)

The Wadge hierarchy was originally defined and studied only in the Baire space. Here we extend the Wadge hierarchy of Borel sets to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g., several Hausdorff–Kuratowski -type theorems in quasi-Polish spaces. In fact, many (...) results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q. (shrink)

The wedge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). Here we extend the Wadge hierarchy of Borel sets to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g. several Hausdorff-Kuratowski-type theorems in quasi-Polish (...) spaces. In fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q. (shrink)

We define a notion of reducibility for subsets of a second countable T 0 topological space based on relatively continuous relations and admissible representations. This notion of reducibility induces a hierarchy that refines the Baire classes and the Hausdorff–Kuratowski classes of differences. It coincides with Wadge reducibility on zero dimensional spaces. However in virtually every second countable T 0 space, it yields a hierarchy on Borel sets, namely it is well founded and antichains are of length at (...) most 2. It thus differs from the Wadge reducibility in many important cases, for example on the real line $${\mathbb{R}}$$ or the Scott Domain $${\mathcal{P}\omega}$$. (shrink)

Résumé En 1975, la loi inscrit l’IVG dans la juridiction de la médecine, investie comme instrument du contrôle social dans la régulation des naissances. La demande d’IVG demeurait pensée comme phénomène relevant de la pathologie. Cependant, une normalisation de cette activité s’opère dans le système de soins avec l’évolution de l’encadrement réglementaire et des techniques de prise en charge. Notamment, les sages-femmes sont enrôlées dans le réagencement des formes de l’action publique en la matière. Leur champ de compétence est élargi. (...) Ces nouvelles définitions de l’expertise médicale en santé sexuelle et reproductive donnent lieu à des délibérations sur les conceptions de la pratique abortive. L’association nationale des sages-femmes orthogénistes (ANSFO) se saisit de cette opportunité pour renforcer la légitimité du groupe professionnel, en opérant un travail cognitif de « physiologisation » de l’IVG. (shrink)

Early scientific investigation of the reproductive process was neither a cause nor a direct result of changing social attitudes toward sex. It was instead part of the continuing search, initiated in the 1890s, to discover internal secretions that might be isolated and prove useful in therapy. Laboratory scientists, nonetheless, were among the many groups altering understanding of human sexual physiology in the first quarter of this century. The new data they generated regarding the dependence of human sexuality and fertility on (...) chemical substances elaborated by the ovaries and testes became part of the evidence which hastened that quite tortuous transformation called the sexual revolution.Increasingly, as social reformers demanded more information about human reproduction, scientists were asked to provide it, and scientific concerns were shaped in the process. In the 1920s, especially in the United States, studies of the physiology of reproduction began to receive financial backing from social reformers. This was because for many, contraception had become a fact of life. Expression rather than repression of sexuality was accepted as being necessary to the stability of the fundamental social unit, the family. Theoretical and practical incentives for continued study of the reproductive process could thus be joined in common effort. This convergence of goals occurred just as the gonadal hormones were being isolated. The 1920s, therefore, mark an important juncture in the congruence of interests represented by theoretical science on the one hand and expressed social needs on the other.As a result of this convergence of interests and goals, physiologists gained a new realm of influence, claiming authoritative knowledge over a subject which once had been allocated to the priest and then, by the mid-nineteenth century, transferred to the physician. Since that formative period other research scientists, such as Alfred Kinsey and, more recently, William Masters and Virginia Johnson, have explored that niche and applied the scientific method to study of behavioral and physiological aspects of the human sexual response. They have been beset by many of the same tactical issues that confronted the earliest investigators of reproductive physiology. The definition, specification, and measurement of phenomena relating to sexual and reproductive activity have been as inherently difficult as they have been taboo.79Traditional resistance to scientific study of the reproductive process was overcome in the early twentieth century by the expectation that this particular line of inquiry would lead to understanding and therefore medical control of the complex of disorders—physiological, psychological, and social — associated with the generative glands. Early investigators, particularly in Britain, did not wish to be social reformers. Especially in the 1910s such a stance would have threatened their credibility in a field determined to establish itself as a theoretical science. Once scientific progress was assured, as it was from about 1920, new problems opened for investigation and began to be motivated by social needs as well as by the concerns of a rapidly developing science. Especially in the United States new resources encouraged this work, allowing development of the specialized focus on gonadal hormones represented by the emerging field of reproductive endocrinology. (shrink)

This paper argues that stoicism as a central element of whiteness shapes, controls, and ultimately limits the experience and expression of emotion in public space. I explore how this may play out in particular medical settings like hospitals in Aotearoa New Zealand. I argue that working in conjunction with other values of whiteness identified by Myser —hyper-individualism, a contractual view of relationships, and an emphasis on personal control and autonomy—this makes hospitals emotionally unsafe spaces for Māori and other groups who (...) place high importance in the collective sharing of emotion. Using death and bereavement as an example, I suggest that challenging and addressing stoicism in the structure and performance of whiteness in hospital settings may provide an important point of entry for anti-racism measures and health equity. (shrink)

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish space and E a Borel equivalence relation on X, a classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c = c. To be of any value we would expect I and c (...) to be “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ fFf. Then if is an embedding of X/E into Y/F, which is “Borel”.Intuitively, E ≤BF might be interpreted in any one of the following ways: The classi.cation problem for E is simpler than that of F: any invariants for F work as well for E. One can classify E by using as invariants F-equivalence classes. The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F. (shrink)

In spite of these efforts in the 1920s and 1930s to initiate ongoing research on contraception, the subject of birth control remained a problem of concern primarily to the social activist rather than to the research scientist or practicing physician.80 In the 1930s, as has been shown, American scientists turned to the study of other aspects of reproductive physiology, while American physicians, anxious to eliminate the moral and medical dangers of contraception, only reluctantly accepted birth control as falling within their (...) professional domain.As a result, the problem of cheap, effective, and safe contraception was not solved by these earliest attempts. Consideration of the subject was initiated afresh by private philanthropy after World War II, sparked by a new wave of interest in population studies.81 Summarizing such efforts to support research in the reproductive sciences, a recent Ford Foundation study has noted: “To initiate and sustain serious research in the reproductive sciences has required for more than half a century concerted effort by interested individuals and private organizations, mainly from outside the mainstreams of the biomedical research community.”82The early laboratory research on chemical contraception described in this paper was but one important outcome of the concerted effort made by reformers in the 1920s to eliminate a variety of social problems thought to derive from excessive fertility. Scientific arguments and expertise were employed to advocate reform as well as to define the appropriate solution to such social problems. Scientists were recruited as advocates for the movement, but they were also employed as researchers in laboratory investigations sponsored by these same reformers.Sponsors of these early laboratory studies noted the difficulty of obtaining first-class investigators.83 The routine analyses necessary for such research, as well as the traditional scientific aversion to applied problems, provide only a partial explanation for this response. The real difficulty lay in recruiting investigators to a field that had previously been taboo. Once opened up — first as socially relevant, and finally as scientifically sound — there was much interest in this area, and the appeal to researchers of the scientific issues surrounding fertility and reproduction soon surpassed that of the reforming value of birth control.A survey of the kinds of experimental investigations sponsored by birth control advocates indicates the range of physiological problems explored by contraceptive research. The most definitive work was done on the efficacy and safety of spermicides, but the potential of other contraceptive methods was also examined. Investigators attempted to develop spermatoxins that would effectively immunize women against sperm, and they also tried to elucidate the mechanism of hormonal control of reproduction. In fact, speculations about the possible hormonal manipulation of fertility were expressed at the Seventh International Birth Control Conference held in Zurich in 1930.84In the 1920s, clinical studies were undertaken to assess the effectiveness of the various birth control methods. Laboratory investigators complemented this work by screening spermicides for safety and testing for their ability to kill sperm. There were a variety of birth control preparations on the market, but no one really knew whether these were effective or even safe.85 Although the physiology of other major organ systems was well advanced, the scientific study of reproductive physiology and contraceptive technology was clearly in its infancy in this period. Routine analyses simply could not be conducted, because the fundamental research establishing baselines had not yet been done. Scientists used this fact to redirect attention to basic research on reproduction.Laboratory research on contraception indicated important unexplored areas for physiological investigation. Social activists, who had encouraged prominent scientists to become interested in both the social value and the genetic implications of birth control, found these investigators revising the goals of their research. The biologists had formed their own network and had begun to seek out funding, reformulating the justification for sponsorship of further investigations. The eugenic motivations underlying these studies, which had initially made them theoretically attractive to biologists, were gradually eroded. Concern with “human evolution” ceded its place to interest in physiological mechanisms. Crew and others began to note that the use of biological theory to justify essentially political decisions had serious limitations. Biologists had become uncomfortable with those very arguments which had originally captured their interest. Recognition of the potential political abuse wrought by applying scientific principles to society was expressed by Crew just one year after the Zurich meeting. Referring to previous assessments of the role of sex in reproduction, he generalized: “In the past the biologist has justified feudalism, Manchester Liberalism, socialism and every other type of social organization and political programme by reference to selected biological phenomena.”86 By 1932 Crew had also begun to question the biological logic of regulated breeding, and had made it clear to his American sponsor that there was no simple correspondence between the practice of birth control and the genetic improvement of the human race.87Biologists further began to recognize, however, that although the hopeful genetic solution to human problems was probably an illusion, contraception still remained one tangible means to alleviate, human misery. Some laboratory scientists, like Crew, acknowledged the applicability of their own particular skills to this problem. For a few brief years, social needs and scientific goals were mutually supportive and closely intertwined. But as laboratory researchers gained interest in the study of reproduction and established their own priorities in this field, they temporarily withdrew from the arena of debate over birth control as an important mechanism for social reform.With the rise of Hitler, the genetic arguments for birth control rapidly lost their appeal. But by that time the scientific problem of how to achieve effective contraception had entered the professional consciousness. Both physicians and scientists began to be aware of birth control as a subject within their domain of expertise, although outside the principal focus of their research. Scientific discussion of birth control permanently altered from a question of justification to a problem of method: How could one achieve reliable and safe contraception? This had been Sanger's and Dickinson's goal from the beginning. Laboratory scientists had indeed been persuaded to undertake this work; this research had in turn affected biologists' perceptions of the whole field of reproductive physiology, encouraging further study of reproductive mechanisms. The promise of new knowledge provided for continued funding of this research, despite the caution by scientists that the social benefits would not be as immediate or as far-reaching as advocates and they themselves had first argued.The activities of birth control activists and their supporting agencies, and the financial backing of private contributors and foundations, notably the Rockefeller philanthropies, provided an important new stimulus to the development of research on the biology of reproduction in the late 1920s and early 1930s. Biologists were able to claim an enlarged realm of issues for scientific study through their activities as advocates and as investigators for the birth control movement. At the same time, they promised as-yet-undiscovered possibilities for regulating human reproduction once its physiology was understood. The knowledge and control that they promised lay in understanding the whole reproductive cycle — not merely in evaluating the toxic effects of presumptive spermicides.Chemical spermicides never summoned the interest of scientists as the contraceptive pill was to do, yet that research did reinforce the widespread perceptions of scientific research as essential to social reform. Spermicide investigations focused research efforts in reproductive biology by challenging traditional taboos, defining problems for further study, and providing laboratory investigators the opportunity to assert the social and scientific value of their own skills. Crew echoed this attitude as he observed in 1934:Man has turned from the adventurous conquest of his environment to the conquest of himself. To-day is the day of biological invention, eagerly used for the control of the undesirable and the unwanted. Sex and reproduction are no longer hedged around by myth and taboo; they are no longer accepted as mysteries that defy understanding. They are matters inviting examination and explanation; they are regarded as expressions of physico-chemical forces, the nature of which is to be displayed. It is accepted that when knowledge is sufficient, control will be absolute, and, though knowledge is not yet sufficient, readers... must be persuaded to the view that this will not always be so.88The synergism between reproductive biology and social needs has been temporary and sporadic, but recurring, since the 1920s. Scientific research programs have clearly been influenced by issues raised by public debate. Nevertheless, reproductive biologists have continued to assert their own professional goals. For the most part, they have rejected problems without inherent scientific interest and have spurned applied research except as it has had a direct bearing on current research themes. This attitude, apparent among American and British investigators in the 1930s, created the intellectual context for the invention and acceptance of the technically sophisticated oral contraceptive pill. It did not foster the production and improvement of the simple, safe, effective, and cheap vaginal contraceptive desired by early birth control advocates. (shrink)

S'il est vrai qu'aucune passion, aucun souci ne résistent à la sérénité qu'apporte à l'esprit la discipline mathématique, c'est faire une cure de sérénité que lire le nouvel ouvrage d'Émile Borel, qui s'est proposé de raconter, dans ses traits essentiels, l'histoire des relations entre les mathématiciens et la notion d'infini. La première partie du livre relate les circonstances où s'est produite la rencontre des mathématiciens et de l'infini, en Grèce, quelques siècles avant l'ère chrétienne (Zénon, cruel Zénon!). La deuxième (...) partie expose la conquête et l'organisation de l'infini par les mathématiciens modernes, du XVIe au XIXe siècle. La troisième partie, enfin, étudie les plus importantes des difficultés et les divergences de vues qui se sont présentées dans les temps actuels lorsqu'on a cherché à préciser et à approfondir les résultats acquis sur l'infini mathématique. (shrink)

[688/2197] In many economic matters there arises a paradox that may be related to what in logic courses is called the “sophism of the heap of wheat”. Among the sophisms bequeathed to us by the Greeks, none is worthier to have come down through the centuries than this “sophism of the heap of wheat”; indeed this no mere puzzle, but a topical example of a frequent difficulty, as much in practical life as in pure speculation.One grain of wheat does not (...) make a heap; neither do two grains; nor do three grains; \(\ldots \) on the other hand, if a million grains are gathered, we can agree that they constitute a heap. What is the exact limit? Should we say that 2,342 grains, for instance, do not make a heap, whereas 2,343 grains do? This is obviously ridiculous. There appears no logical way out of this dead-end; it is therefore not possible to know what a heap of wheat is.The expression of the sophism can be varied in many ways; a classification of these expressions will not be unhelpful. (shrink)

Book Reviews in this ArticlePaul GochetMaurice BoudotKlaus‐Henrik JacobsenLèon BrunschvicgAndrè Bièler.

Ever since Kant, the possibility of having objects of knowledge has been one of the most basic anthropological questions (“what can I know?”). For the logician, the linguist, or the semiologist who studies natural language, negation is one of these objects. However, as an operation and as a symbol, it has the paradoxical property of not being able to be objectivized in the discourse that treats it without being used in this construction. Of course, it is an entirely general problem (...) that it is not possible to have a “distant view” of symbolism without using a symbolism of some sort or another. Anthropology is essentially concerned with such objects. The descriptions of ethnologists are thus fertile ground for the epistemologist who believes, like Aristotle, that non-being is not and therefore that, even though in fact we speak only of things (otherwise, what good is speaking?) our discourse cannot be a copy of them. (shrink)