Five recursively axiomatizable theories extending Kleene's intuitionistic theory FIM of numbers and numbertheoretic sequences are introduced and shown to be consistent, by a modified relative realizability interpretation which verifies that every sequence classically defined by a Π11 formula is unavoidable and that no sequence can fail to be classically Δ11. The analytical form of Markov's Principle fails under the interpretation. The notion of strongly inadmissible rule of inference is introduced, with examples.

This paper introduces an extension A of Kleene's axiomatization of Brouwer's intuitionistic analysis, in which the classical arithmetical and analytical hierarchies are faithfully represented as hierarchies of the domains of continuity. A domain of continuity is a relation R(α) on Baire space with the property that every constructive partial functional defined on {α : R(α)} is continuous there. The domains of continuity for A coincide with the stable relations (those equivalent in A to their double negations), while every relation R(α) (...) is equivalent in A to ∃αA(α, β) for some stable A(α, β) (which belongs to the classical analytical hierarchy). The logic of A is intuitionistic. The axioms of A include countable comprehension, bar induction, Troelstra's generalized continuous choice, primitive recursive Markov's Principle and a classical axiom of dependent choices proposed by Krauss. Constructive dependent choices, and constructive and classical countable choice, are theorems, A is maximal with respect to classical Kleene function realizability, which establishes its consistency. The usual disjunction and (recursive) existence properties ensure that A preserves the constructive sense of "or" and "there exists.". (shrink)

In the author's Relative lawlessness in intuitionistic analysis [this JOURNAL. vol. 52 (1987). pp. 68-88] and An intuitionistic theory of lawlike, choice and lawless sequences [Logic Colloquium '90. Springer-Verlag. Berlin. 1993. pp. 191-209] a notion of lawless ness relative to a countable information base was developed for classical and intuitionistic analysis. Here we simplify the predictability property characterizing relatively lawless sequences and derive it from the new axiom of closed data (classically equivalent to open data) together with a natural principle (...) of invariance under finite translation. We characterize relative lawlessness in terms of a notion of forcing. Finally, we study relative lawlessness on an arbitrary fan and show that the collection of lawless binary sequences (which is comeager in the sense of Baire) has probability measure zero. The reasoning is predominantly constructive. (shrink)

Kripke recently suggested viewing the intuitionistic continuum as an expansion in time of a definite classical continuum. We prove the classical consistency of a three-sorted intuitionistic formal system IC, simultaneously extending Kleene’s intuitionistic analysis I and a negative copy C° of the classically correct part of I, with an “end of time” axiom ET asserting that no choice sequence can be guaranteed not to be pointwise equal to a definite sequence. “Not every sequence is pointwise equal to a definite sequence” (...) is independent of IC. The proofs are by Crealizability interpretations based on classical ω-models ${\cal M}$ = $\left$ of C°. (shrink)

This paper introduces, as an alternative to the (absolutely) lawless sequences of Kreisel and Troelstra, a notion of choice sequence lawless with respect to a given class D of lawlike sequences. For countable D, the class of D-lawless sequences is comeager in the sense of Baire. If a particular well-ordered class F of sequences, generated by iterating definability over the continuum, is countable then the F-lawless, sequences satisfy the axiom of open data and the continuity principle for functions from lawless (...) to lawlike sequences, but fail to satisfy Troelstra's extension principle. Classical reasoning is used. (shrink)

This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and number-theoretic sequences, we propose some modified choice principles and other function existence axioms which may be of use in reverse constructive analysis. Specifically, we consider the function comprehension principles assumed by the two minimal theories EL and M, introduce an axiom schema CFd asserting that every decidable property (...) of numbers has a characteristic function, and use it to describe a precise relationship between the minimal theories. We show that the axiom schema AC00 of countable choice can be decomposed into a monotone choice schema AC00m (which guarantees that every Cauchy sequence has a modulus) and a bounded choice schema BC00. We relate various (classically correct) axiom schemas of continuous choice to versions of the bar and fan theorems, suggest a constructive choice schema AC1/2,0 (which incidentally guarantees that every continuous function has a modulus of continuity), and observe a constructive equivalence between restricted versions of the fan theorem and correspondingly restricted bounding axioms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${AB_{1/2,0}^{2^{\mathbb{n}}}}$$\end{document}. We also introduce a version WKL!! of Weak König’s Lemma with uniqueness which is intermediate in strength between WKL and the decidable fan theorem FTd. (shrink)

Realizabilities are powerful tools for establishing consistency and independence results for theories based on intuitionistic logic. Troelstra discovered principles ECT 0 and GC 1 which precisely characterize formal number and function realizability for intuitionistic arithmetic and analysis, respectively. Building on Troelstra's results and using his methods, we introduce the notions of Church domain and domain of continuity in order to demonstrate the optimality of “almost negativity” in ECT 0 and GC 1 ; strengthen “double negation shift” DNS 0 to DNS (...) 1 and use it with GC 1 and Markov's Principle MP to characterize the classically provably realizable formulas of analysis; and show that intuitionistic analysis with GC 1 , MP and DNS 1 is consistent and satisfies Troelstra's and Church's Rules. We end with a discussion of semi-intuitionistic theories and a conjecture. (shrink)

Metaphysics, morals and science -- The classics : value -- The neo-classics : utility -- The Keynesian revolution -- Development and under-development -- What are the rules of the game?

In spite of feminist recognition that hierarchical organizations are an important location of male dominance, most feminists writing about organizations assume that organizational structure is gender neutral. This article argues that organizational structure is not gender neutral; on the contrary, assumptions about gender underlie the documents and contracts used to construct organizations and to provide the commonsense ground for theorizing about them. Their gendered nature is partly masked through obscuring the embodied nature of work.jobs and hierarchies, common concepts in organizational (...) thinking, assume a disembodies and universal worker. This worker is actually a man; men's bodies, sexuality, and relationships to procreation and paid work are subsumed in the image of the worker. Images of men's bodies and masculinity pervade organizational processes, marginalizing women and contributing to the maintenance of gender segregation in organizations. The positing of gender-neutral and disembodied organizational structures and work relations is part of the larger strategy of control in industrial capitalist societies, which, at least partly, are built upon a deeply embedded substructure of gender difference. (shrink)

In this article, the author addresses two feminist issues: first, how to conceptualize intersectionality, the mutual reproduction of class, gender, and racial relations of inequality, and second, how to identify barriers to creating equality in work organizations. She develops one answer to both issues, suggesting the idea of “inequality regimes” as an analytic approach to understanding the creation of inequalities in work organizations. Inequality regimes are the interlocked practices and processes that result in continuing inequalities in all work organizations. Work (...) organizations are critical locations for the investigation of the continuous creation of complex inequalities because much societal inequality originates in such organizations. Work organizations are also the target for many attempts to alter patterns of inequality: The study of change efforts and the oppositions they engender are often opportunities to observe frequently invisible aspects of the reproduction of inequalities. The concept of inequality regimes may be useful in analyzing organizational change projects to better understand why these projects so often fail and why they succeed when this occurs. (shrink)

Not only can the influence of Gottlob Frege be found in contemporary work in logic, the philosophy of mathematics, and the philosophy of language, but his projects—and the very terminology he employed in pursuing those projects—are still current in contemporary philosophy. This is undoubtedly why it seems so reasonable to assume that we can read Frege' s writings as if he were one of us, speaking to our philosophical concerns in our language. In Joan Weiner's view, however, Frege's words (...) can be accurately interpreted only if we set that assumption aside. Weiner here offers a challenging new approach to the philosophy of this central figure in analytic philosophy. Weiner finds in Frege's corpus, from Begriffsschrift on, a unified project of remarkable ambition to which each of the writings in that corpus makes a distinct contribution—a project whose motivation she brings to life through a careful reading of his Foundations of Arithmetic. The Frege that Weiner brings into clear view is very different from the familiar figure. Far from having originated one of the standard positions on the nature of reference, Frege turns out not to have had positive doctrines on anything like what contemporary philosophers mean by "reference." Far from having served as a standard-bearer for those who take the realists' side of contemporary disputes with anti-realists, Frege turns out to have had no stake in either side of the controversy. Through Weiner's lens, Frege emerges as a thinker who has principled reasons for challenging the very assumptions and motivations that animate philosophers to dispute these doctrines. This lucidly written and accessible book will generate controversy among all readers with an interest in epistemology, philosophy of language, history of philosophy, and the philosophy of mathematics. (shrink)

Hailed by the Bulletin of the American Mathematical Society as "easy to use and a pleasure to read," this research monograph is recommended for students and professionals interested in model theory and definability theory. The sole prerequisite is a familiarity with the basics of logic, model theory, and set theory. 1974 edition.

Gender, ‘race’, poverty, health and discourses of health reform in the context of globalization: a postcolonial feminist perspective in policy researchIn this paper, I draw on extant literature and my empirical work to discuss the impact of globalization and healthcare reform on the lives of women — those from countries of the South as well as of the North. First, I review briefly the economic hardships identified in different sectors of the population that have been attributed to how globalization is (...) now working. Second, I examine what these global processes mean for health, with particular focus on poverty, gender, racialization and health. Third, I reflect on how nurse scientists might develop research agendas in the 21st century that would foster social transformation and social justice for all people. The position taken here is not an indictment of globalization. Rather, I argue that globalization is a fact in all of our lives. There are positive aspects of globalization. There are also negative aspects which we must collectively address, given that the issues identified can have deleterious consequences for the world’s poor, women in particular. I suggest that, to construct knowledge for practice and praxis, research agendas of the future should be inclusive of subaltern voices. I argue that drawing on a postcolonial feminist epistemology might help us to define such agendas, and express the multilayered sociopolitical contexts of health and illness in advocacy with policy‐makers. (shrink)

Here is the final collection of articles and speeches by the bestselling and world-renowned novelist, essayist, and philosopher.

There is a section in Samuel Delany’s magnificent autobiographical meditation, The Motion of Light in Water, that dramatically raises the problem of writing the history of difference, the history, that is, of the designation of “other,” of the attribution of characteristics that distinguish categories of people from some presumed norm.1 Delany recounts his reaction to his first visit to the St. Marks bathhouse in 1963. He remembers standing on the threshold of a “gym-sized room” dimly lit by blue bulbs. The (...) room was full of people, some standing, the rest an undulating mass of naked, male bodies, spread wall to wall. My first response was a kind of heart-thudding astonishment very close to fear. I have written of a space at certain libidinal saturation before. That was not what frightened me. It was rather that the saturation was not only kinesthetic but visible.2Watching the scene establishes for Delany a “fact that flew in the face” of the prevailing representation of homosexuals in the 1950s as “isolated perverts,” as subjects “gone awry.” The “apprehension of massed bodies” gave him a “sense of political power”:what this experience said was that there was a population—not of individual homosexuals … not of hundreds, not of thousands, but rather of millions of gay men, and that history had, actively and already, created for us whole galleries of institutions, good and bad, to accommodate our sex. [M, p. 174] 2. Samuel R. Delany, The Motion of Light in Water: Sex and Science Fiction Writing in the East Village, 1957-1965 , p. 173; hereafter abbreviated M. Joan W. Scott is professor of social science at the Institute for Advanced Study in Princeton, New Jersey. She is the author, most recently, of Gender and the Politics of History and is currently at work on a history of feminist claims for political rights in France during the period 1789-1945 as a way of exploring arguments about equality and difference. (shrink)

The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V or HOD is “far” from V. The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the (...) third author has provided evidence that there is an ultimate inner model—Ultimate-L—and he has isolated a natural conjecture associated with the model—the Ultimate-L Conjecture. This conjecture implies that that the first alternative holds—HOD is “close” to V. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where chaos prevails. (shrink)