We show that a first category homogeneous zero-dimensional Borel set X can be embedded in as an ideal on ω if and only if X is homeomorphic to X × X if and only if X is Wadge-equivalent to X × X. Furthermore, we determine the Wadge classes of such X, thus giving a complete picture of the possible descriptive complexity of Borel ideals on ω. We also discuss the connection with ideals of compact sets.

We study the Katětov order on Borel ideals. We prove two structural theorems, one for Borel ideals, the other for analytic P-ideals. We isolate nine important Borel ideals and study the Katětov order among them. We also present a list of fundamental open problems concerning the Katětov order on Borel ideals.

We study the possible values of the cofinality invariant for various Borel ideals on the natural numbers. We introduce the notions of a fragmented and gradually fragmented ideal and prove a dichotomy for fragmented ideals. We show that every gradually fragmented ideal has cofinality consistently strictly smaller than the cardinal invariant and produce a model where there are uncountably many pairwise distinct cofinalities of gradually fragmented ideals.

Minami–Sakai :883–898, 2016) investigated the cofinal types of the Katětov and the Katětov–Blass orders on the family of all \ ideals. In this paper we discuss these orders on analytic P-ideals and Borel ideals. We prove the following:The family of all analytic P-ideals has the largest element with respect to the Katětov and the Katětov–Blass orders.The family of all Borel ideals is countably upward directed with respect to the Katětov and the Katětov–Blass orders. (...) In the course of the proof of the latter result, we also prove that for any analytic ideal \ there is a Borel ideal \ with \. (shrink)

We present several naturally defined σ-ideals which have Borel bases but, unlike for the classical examples, these ideals are not of bounded Borel complexity. We investigate set-theoretic properties of such σ-ideals.

It is shown that the power set of $\kappa$ ordered by the subset relation modulo various versions of the non-stationary ideal can be embedded into the partial order of Borel equivalence relations on $2^\kappa$ under Borel reducibility. Here $\kappa$ is an uncountable regular cardinal with $\kappa^{<\kappa}=\kappa$.

Families of Borel equivalence relations and quasiorders that are cofinal with respect to the Borel reducibility ordering, ≤B, are constructed. There is an analytic ideal on ω generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal. Several Borel equivalence relations, among them Lipschitz isomorphism of compact metric spaces, are shown to be Kσ complete.

We study the κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}-Borel-reducibility of isomorphism relations of complete first order theories in a countable language and show the consistency of the following: For all such theories T and T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{\prime }$$\end{document}, if T is classifiable and T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{\prime }$$\end{document} is not, then the isomorphism of models of T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{\prime }$$\end{document} is strictly above the isomorphism of models of T with respect to κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}-Borel-reducibility. In fact, we can also ensure that a range of equivalence relations modulo various non-stationary ideals are strictly between those isomorphism relations. The isomorphism relations are considered on models of some fixed uncountable cardinality obeying certain restrictions. (shrink)

Elekes proved that any infinite-fold cover of a σ-finite measure space by a sequence of measurable sets has a subsequence with the same property such that the set of indices of this subsequence has density zero. Applying this theorem he gave a new proof for the random-indestructibility of the density zero ideal. He asked about other variants of this theorem concerning I-almost everywhere infinite-fold covers of Polish spaces where I is a σ-ideal on the space and the set of indices (...) of the required subsequence should be in a fixed ideal ${{\mathcal{J}}}$ on ω. We introduce the notion of the ${{\mathcal{J}}}$ -covering property of a pair ${({\mathcal{A}}, I)}$ where ${{\mathcal{A}}}$ is a σ-algebra on a set X and ${{I \subseteq \mathcal{P}(X)}}$ is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal ${\mathcal{N}}$ and the meager ideal ${\mathcal{M}}$ . We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals ${{\mathcal{J}}}$ on ω such that ${\mathcal{M}}$ has the ${{\mathcal{J}}}$ -covering property consists exactly of non weak Q-ideals. We also study the existence of smallest elements, with respect to Katětov–Blass order, in the family of those ideals ${\mathcal{J}}$ on ω such that ${\mathcal{N}}$ or ${\mathcal{M}}$ has the ${\mathcal{J}}$ -covering property. Furthermore, we prove a general result about the cases when the covering property “strongly” fails. (shrink)

We examine the question of how many Boolean algebras, distinct up to isomorphism, that are quotients of the powerset of the naturals by Borel ideals, can be proved to exist in ZFC alone. The maximum possible value is easily seen to be the cardinality of the continuum $2^{\aleph_{0}}$ ; earlier work by Ilijas Farah had shown that this was the value in models of Martin's Maximum or some similar forcing axiom, but it was open whether there could be (...) fewer in models of the Continuum Hypothesis. We develop and apply a new technique for constructing many ideals whose quotients must be nonisomorphic in any model of ZFC. The technique depends on isolating a kind of ideal, called shallow, that can be distinguished from the ideal of all finite sets even after any isomorphic embedding, and then piecing together various copies of the ideal of all finite sets using distinct shallow ideals. In this way we are able to demonstrate that there are continuum-many distinct quotients by Borel ideals, indeed by analytic P-ideals, and in fact that there is in an appropriate sense a Borel embedding of the Vitali equivalence relation into the equivalence relation of isomorphism of quotients by analytic P-ideals. We also show that there is an uncountable definable wellordered collection of Borel ideals with distinct quotients. (shrink)

§1. Introduction. Ideals and filters of subsets of natural numbers have been studied by set theorists and topologists for a long time. There is a vast literature concerning various kinds of ultrafilters. There is also a substantial interest in nicely definable ideals—these by old results of Sierpiński are very far from being maximal— and the structure of such ideals will concern us in this announcement. In addition to being interesting in their own right, Borel and analytic (...) ideals occur naturally in the investigations of compact subsets of the space of all Baire class 1 functions on a Polish space, see [12, 18]. Also, certain objects associated with such ideals are of considerable interest and were quite extensively studied by several authors. Let us list here three examples; in all three of them I stands for an analytic or Borel ideal.1. The partial order induced by I on P: X ≥I Y iff X \ Y ϵ I and the partial order.2. Boolean algebras of the form P/I and their automorphisms.3. The equivalence relation associated with I: XEI Y iff X Δ ϵ I.In Section 4, we will have an opportunity to state some consequences of our results for equivalence relations as in 3. (shrink)

Let $\mathcal {I}$ be an ideal on $\omega $. For $f,\,g\in \omega ^{\omega }$ we write $f \leq _{\mathcal {I}} g$ if $f(n) \leq g(n)$ for all $n\in \omega \setminus A$ with some $A\in \mathcal {I}$. Moreover, we denote $\mathcal {D}_{\mathcal {I}}=\{f\in \omega ^{\omega }: f^{-1}[\{n\}]\in \mathcal {I} \text { for every } n\in \omega \}$ (in particular, $\mathcal {D}_{\mathrm {Fin}}$ denotes the family of all finite-to-one functions).We examine cardinal numbers $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal (...) {I}}))$ and $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}}\times \mathcal {D}_{\mathrm {Fin}}))$ describing the smallest sizes of unbounded from below with respect to the order $\leq _{\mathcal {I}}$ sets in $\mathcal {D}_{\mathrm {Fin}}$ and $\mathcal {D}_{\mathcal {I}}$, respectively. For a maximal ideal $\mathcal {I}$, these cardinals were investigated by M. Canjar in connection with coinitial and cofinal subsets of the ultrapowers.We show that $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}} \times \mathcal {D}_{\mathrm {Fin}})) =\mathfrak {b}$ for all ideals $\mathcal {I}$ with the Baire property and that $\aleph _1 \leq \mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) \leq \mathfrak {b}$ for all coanalytic weak P-ideals (this class contains all $\bf {\Pi ^0_4}$ ideals). What is more, we give examples of Borel (even $\bf {\Sigma ^0_2}$ ) ideals $\mathcal {I}$ with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))=\mathfrak {b}$ as well as with $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) =\aleph _1$.We also study cardinals $\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {J}} \times \mathcal {D}_{\mathcal {K}}))$ describing the smallest sizes of sets in $\mathcal {D}_{\mathcal {K}}$ not bounded from below with respect to the preorder $\leq _{\mathcal {I}}$ by any member of $\mathcal {D}_{\mathcal {J}}\!$. Our research is partially motivated by the study of ideal-QN-spaces: those cardinals describe the smallest size of a space which is not ideal-QN. (shrink)

Given a family${\cal C}$of infinite subsets of${\Bbb N}$, we study when there is a Borel function$S:2^{\Bbb N} \to 2^{\Bbb N} $such that for every infinite$x \in 2^{\Bbb N} $,$S\left \in {\Cal C}$and$S\left \subseteq x$. We show that the family of homogeneous sets as given by the Nash-Williams’ theorem admits such a Borel selector. However, we also show that the analogous result for Galvin’s lemma is not true by proving that there is an$F_\sigma $tall ideal on${\Bbb N}$without a (...) class='Hi'>Borel selector. The proof is not constructive since it is based on complexity considerations. We construct a${\bf{\Pi }}_2^1 $tall ideal on${\Bbb N}$without a tall closed subset. (shrink)

We examine the question of how many Boolean algebras, distinct up to isomorphism, that are quotients of the powerset of the naturals by Borel ideals, can be proved to exist in ZFC alone. The maximum possible value is easily seen to be the cardinality of the continuum $2^{\aleph_{0}}$ ; earlier work by Ilijas Farah had shown that this was the value in models of Martin's Maximum or some similar forcing axiom, but it was open whether there could be (...) fewer in models of the Continuum Hypothesis. We develop and apply a new technique for constructing many ideals whose quotients must be nonisomorphic in any model of ZFC. The technique depends on isolating a kind of ideal, called shallow, that can be distinguished from the ideal of all finite sets even after any isomorphic embedding, and then piecing together various copies of the ideal of all finite sets using distinct shallow ideals. In this way we are able to demonstrate that there are continuum-many distinct quotients by Borel ideals, indeed by analytic P-ideals, and in fact that there is in an appropriate sense a Borel embedding of the Vitali equivalence relation into the equivalence relation of isomorphism of quotients by analytic P-ideals. We also show that there is an uncountable definable wellordered collection of Borel ideals with distinct quotients. (shrink)

Let J be any proper ideal of subsets of the real line R which contains all finite subsets of R. We define an ideal J * ∣B as follows: X ∈ J * ∣B if there exists a Borel set $B \subset R \times R$ such that $X \subset B$ and for any x ∈ R we have $\{y \in R: \langle x,y\rangle \in B\} \in \mathscr{J}$ . We show that there exists a family $\mathscr{A} \subset \mathscr{J}^\ast\mid\mathscr{B}$ of power (...) ω 1 such that $\bigcup\mathscr{A} \not\in \mathscr{J}^\ast\mid\mathscr{B}$ . In the last section we investigate properties of ideals of Lebesgue measure zero sets and meager sets in Cohen extensions of models of set theory. (shrink)

We study the structure of analytic ideals of subsets of the natural numbers. For example, we prove that for an analytic ideal I, either the ideal {X (Ω × Ω: En X ({0, 1,…,n} × Ω } is Rudin-Keisler below I, or I is very simply induced by a lower semicontinuous submeasure. Also, we show that the class of ideals induced in this manner by lsc submeasures coincides with Polishable ideals as well as analytic P-ideals. We (...) study this class of ideals and characterize, for example, when the ideals in it are Fσ or when they carry a locally compact group topology. We apply these results to Borel partial orders to rederive a theorem of Todorcevic and to Borel equivalence relations to answer a question of Kechris and Louveau. As another application we give a characterization of σ-ideals of μ-zero sets for Maharam submeasures μ on the Cantor set which is to a large extent analogous to a characterization of the meager ideal due to Kechris and the author. (shrink)

A \(\sigma \) -ideal \(\mathcal {I}\) on a Polish group \((X,+)\) has the Smital Property if for every dense set _D_ and a Borel \(\mathcal {I}\) -positive set _B_ the algebraic sum \(D+B\) is a complement of a set from \(\mathcal {I}\). We consider several variants of this property and study their connections with the countable chain condition, maximality and how well they are preserved via Fubini products. In particular we show that there are \(\mathfrak {c}\) many maximal invariant (...) \(\sigma \) -ideals with Borel bases on the Cantor space \(2^\omega \). (shrink)

The following will be shown: Let I be a σ-ideal on a Polish space X so that the associated forcing of I+${\bf{\Delta }}_1^1$ sets ordered by ⊆ is a proper forcing. Let E be a ${\bf{\Sigma }}_1^1$ or a ${\bf{\Pi }}_1^1$ equivalence relation on X with all equivalence classes ${\bf{\Delta }}_1^1$. If for all $z \in {H_{{{\left}^ + }}}$, z♯ exists, then there exists an I+${\bf{\Delta }}_1^1$ set C ⊆ X such that E ↾ C is a ${\bf{\Delta }}_1^1$ equivalence (...) relation. (shrink)

With every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ-ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal. We also study the (...) 1—1 or constant property of σ-ideals, i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1—1 or constant. We prove the following dichotomy: if I is a σ-ideal generated by closed sets, then either the forcing P I adds a Cohen real, or else I has the 1—1 or constant property. (shrink)

Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F $\subseteq$ P(X) of size c, consisting of Borel sets which are not in I. Condition (M) states that there is a Borel function f: X → X with $f^{-1}[\{x\}] \not\in$ I for (...) each x ∈ X. Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set B $\not\in$ I and a perfect set P $\subseteq$ X for which the family $\{B + x: x \in P\}$ is disjoint. The aim of the paper is to study whether the reverse implications in the chain (D) $\Rightarrow$ (M) $\Rightarrow$ (B) $\Rightarrow$ not-ccc can hold. We build a σ-ideal on the Cantor group witnessing (M) & ¬ (D) (Section 2). A modified version of that σ-ideal contains the whole space (Section 3). Some consistency results on deriving (M) from (B) for "nicely" defined ideals are established (Sections 4 and 5). We show that both ccc and (M) can fail (Theorems 1.3 and 5.6). Finally, some sharp versions of (M) for invariant ideals on Polish groups are investigated (Section 6). (shrink)

Answering in negative a question of M. Hrušák, we construct a Borel ideal not extendable to any \(F_\sigma \) ideal and such that it is not Katětov above the ideal \(\mathrm {conv}\).

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)

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)

Answering in negative a question of M. Hrušák, we construct a Borel ideal not extendable to any Fσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\sigma $$\end{document} ideal and such that it is not Katětov above the ideal conv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {conv}$$\end{document}.

[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)

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)

We address some phenomena about the interaction between lower semicontinuous submeasures on $${\mathbb {N}}$$ N and $$F_{\sigma }$$ F σ ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological $$F_{\sigma }$$ F σ ideals. We give a partial answers to the question of whether every nonpathological tall $$F_{\sigma }$$ F σ ideal is Katětov above the random ideal or at least has a Borel selector. Finally, we show a representation of (...) nonpathological $$F_{\sigma }$$ F σ ideals using sequences in Banach spaces. (shrink)

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)

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

It is shown that cardinals below a real-valued measurable cardinal can be split into finitely many intervals so that the powers of cardinals from the same interval are the same. This generalizes a theorem of Prikry [9]. Suppose that the forcing with a κ-complete ideal over κ is isomorphic to the forcing of λ-Cohen or random reals. Then for some τ<κ, λτ2κ and λ2<κ implies that 2κ=2τ= cov. In particular, if 2κ<κ+ω, then λ=2κ. This answers a question from [3]. If (...) A0, A1,..., An,… are sets of reals, then there are disjoint sets B0, B1,..., Bn,… such that BnAn and μ*=μ* for every n<ω, where μ* is the Lebes gue outer measure. For finitely many sets the result is due to N. Lusin. Let be a σ-centered forcing notion and An ¦n<ω subsets of P witnessing this. If P, An's and the relation of compatibility are Borel, then P adds a Cohen real. The forcing with a κ-complete ideal over a set X, ¦X¦κ cannot be isomorphic to a Hechler real forcing. This result was claimed in [3], but the proof given there works only for X of cardinality κ. (shrink)