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)

Presents a new framework for the complexity of algorithms, for all readers interested in the theory of computation.

On the Contemplative Life is known for its depiction of a philosophical group of Jewish men and women known as the 'Therapeutae'. Yet the reasons for their depiction have been little understood. In the first commentary on the treatise in English for over 100 years, the social, cultural and political background of the times in which Philo lived are shown to be crucial in understanding Philo's purposes. As Alexandrian Jews were vilified and attacked, Philo went to Rome to present the (...) case for his community, faced with intense opposition. Side-stepping direct confrontation, Philo here cleverly presents the Therapeutae as the pinnacle of excellence, most especially in their communal meal, while ridiculing his accusers in a stinging parody of a festive banquet. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in (...) the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

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)

Women and religion -- Reproductive futurism -- Political emancipation -- From the Cold War to the clash of civilizations -- Sexual emancipation.

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)

Este artículo analiza el prólogo de 1812 a la Ciencia de la lógica (WdL) y responde a los planteos: ¿de qué trata esta obra?, ¿qué hay de lógico en la expresión “lógica”?, ¿es ”lógica” un nombre sustituto para “metafísica” o, más bien, la WdL es la inauguración de una disciplina completamente nueva y que dista de la metafísica en el sentido tradicional? Concluyo que la lógica es aquí el desarrollo del concepto expresada como libertad social.