Inhalt: Vorbemerkung Kommentar Titel Vorrede Erster Teil Zweiter Teil Dritter Teil Benutzte Literatur a)Zeitgenossen und Vorläufer Fichtes b)Moderne Interpreten Sachregister zum Fichte-Text Verzeichnis der zitierten Arbeiten Fichtes.

Malgré sa défiance pour la théorie, la recherche scientifique de Goethe n’est pas allée sans un soin tout particulier porté à la méthode qu’elle devait mettre en œuvre. Précisément parce qu’il fallait rendre compte du phénomène dans sa diversité et dans sa totalité, il importait de réfléchir aux moyens qui assureraient la réussite d’une telle démarche. Pour Goethe, il s’agissait de mettre en œuvre un concours harmonieux des différentes facultés de l’esprit humain, seul capable de répondre à la richesse du (...) réel tel qu’il se manifeste à nous, et donc seul capable de fonder adéquatement une démarche expérimentale en science. (shrink)

In this article we show that Morse-Kelley class theory provides us with an adequate framework for class forcing. We give a rigorous definition of class forcing in a model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$$$ \end{document} of MK, the main result being that the Definability Lemma can be proven without restricting the notion of forcing. Furthermore we show under which conditions the axioms are preserved. We conclude by proving that Laver’s Theorem (...) does not hold for class forcings. (shrink)

A discussion of the philosophical prospects for basing a neo-Fregean theory of classes on a principle that attempts to articulate the limitation-of-size conception.

In this article we investigate similarities between the role that ineffability of Absolute Infinity plays in class theory and in theology.

A theory T is tight if different deductively closed extensions of T (in the same language) cannot be bi-interpretable. Many well-studied foundational theories are tight, including $\mathsf {PA}$ [39], $\mathsf {ZF}$, $\mathsf {Z}_2$, and $\mathsf {KM}$ [6]. In this article we extend Enayat’s investigations to subsystems of these latter two theories. We prove that restricting the Comprehension schema of $\mathsf {Z}_2$ and $\mathsf {KM}$ gives non-tight theories. Specifically, we show that $\mathsf {GB}$ and $\mathsf {ACA}_0$ each admit different bi-interpretable (...) extensions, and the same holds for their extensions by adding $\Sigma ^1_k$ -Comprehension, for $k \ge 1$. These results provide evidence that tightness characterizes $\mathsf {Z}_2$ and $\mathsf {KM}$ in a minimal way. (shrink)

Ubiquitous news, global information access, instantaneous reporting, interactivity, multimedia content, extreme customization: Journalism is undergoing the most fundamental transformation since the rise of the penny press in the nineteenth century. Here is a report from the front lines on the impact and implications for journalists and the public alike. John Pavlik, executive director of the Center for New Media at Columbia University's Graduate School of Journalism, argues that the new media can revitalize news gathering and reengage an increasingly distrustful and (...) alienated citizenry. The book is a valuable reference on everything from organizing a new age newsroom to job hunting in the new media. (shrink)

Insatisfait du calcul des classes et des relations de Whitehead et Russell, Leśniewski élabora en 1919-20 une théorie extensionnelle des noms qu’il nomma Ontologie. Sans entrer dans une description technique du formalisme de Leśniewski, nous montrons dans cet article que l’Ontologie permet un traitement général du distributif qui ne s’appuie à aucun moment sur une notion de classe. Nous illustrons enfin cette particularité importante du système de Leśniewski en proposant une définition logiciste de la notion de cardinalité qui répond d’une (...) manière radicale aux impératifs d’une no-class theory. (shrink)

Insatisfait du calcul des classes et des relations de Whitehead et Russell, Leśniewski élabora en 1919-20 une théorie extensionnelle des noms qu’il nomma Ontologie. Sans entrer dans une description technique du formalisme de Leśniewski, nous montrons dans cet article que l’Ontologie permet un traitement général du distributif qui ne s’appuie à aucun moment sur une notion de classe. Nous illustrons enfin cette particularité importante du système de Leśniewski en proposant une définition logiciste de la notion de cardinalité qui répond d’une (...) manière radicale aux impératifs d’une no-class theory. (shrink)

We compare Fregean theorizing about sets with the theorizing of an ontologically non-committal, natural-deduction based, inferentialist. The latter uses free Core logic, and confers meanings on logico-mathematical expressions by means of rules for introducing them in conclusions and eliminating them from major premises. Those expressions (such as the set-abstraction operator) that form singular terms have their rules framed so as to deal with canonical identity statements as their conclusions or major premises. We extend this treatment to pasigraphs as well, in (...) the case of set theory. These are defined expressions (such as ‘subset of’, or ‘power set of’) that are treated as basic in the lingua franca of informal set theory. Employing pasigraphs in accordance with their own natural-deduction rules enables one to ‘atomicize’ rigorous mathematical reasoning. (shrink)

Kripke showed how to restrict Tarski’s schema to grounded sentences. I examine the prospects for an analogous approach to the paradoxes of naive class comprehension. I present new methods to obtain theories of grounded classes and test them against antecedently motivated desiderata. My findings cast doubt on whether a theory of grounded classes can accommodate both the extensionality of classes and allow for class definition in terms of identity.

In this article we introduce and study hyperclass-forcing in the context of an extension of Morse-Kelley class theory, called MK∗∗. We define this forcing by using a symmetry between MK∗∗ models and models of ZFC− plus there exists a strongly inaccessible cardinal. We develop a coding between β-models ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {M}$$ \end{document} of MK∗∗ and transitive models M+ of SetMK∗∗ which will allow us to go from ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...) \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {M}$$ \end{document} to M+ and vice versa. So instead of forcing with a hyperclass in MK∗∗ we can force over the corresponding SetMK∗∗ model with a class of conditions. For class-forcing to work in the context of ZFC− we show that the SetMK∗∗ model M+ can be forced to look like Lκ∗[X]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$L_{\kappa ^*}[X]$$ \end{document}, where κ∗ is the height of M+, κ strongly inaccessible in M+ and X ⊆ κ. Over such a model we can apply definable class forcing and we arrive at an extension of M+ from which we can go back to the corresponding β-model of MK∗∗, which will in turn be an extension of the original ℳ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {M}$$ \end{document}. Our main result combines hyperclass forcing with coding methods of Beller et al. and Friedman to show that every β-model of MK∗∗ can be extended to a minimal such model of MK∗∗ with the same ordinals. A simpler version of the proof also provides a new and analogous minimality result for models of second-order arithmetic. (shrink)

Certain commentators on Russell's “no class” theory, in which apparent reference to classes or sets is eliminated using higher-order quantification, including W. V. Quine and (recently) Scott Soames, have doubted its success, noting the obscurity of Russell’s understanding of so-called “propositional functions”. These critics allege that realist readings of propositional functions fail to avoid commitment to classes or sets (or something equally problematic), and that nominalist readings fail to meet the demands placed on classes by mathematics. I show (...) that Russell did thoroughly explore these issues, and had good reasons for rejecting accounts of propositional functions as extra-linguistic entities. I argue in favor of a reading taking propositional functions to be nothing over and above open formulas which addresses many such worries, and in particular, does not interpret Russell as reducing classes to language. (shrink)

Leon Chwistek's 1924 paper ?The Theory of Constructive Types? is cited in the list of recent ?contributions to mathematical logic? in the second edition of Principia Mathematica, yet its prefatory criticisms of the no-classes theory have been seldom noticed. This paper presents a transcription of the relevant section of Chwistek's paper, comments on the significance of his arguments, and traces the reception of the paper. It is suggested that while Russell was aware of Chwistek's points, they were not (...) important in leading him to the adoption of extensionality that marks the second edition of PM. Rudolf Carnap seems to have independently rediscovered Chwistek's issue about the scope of class expressions in identity contexts in his Meaning and Necessity in 1947. (shrink)

Hamkins and Kikuchi (2016, 2017) show that in both set theory and class theory the definable subset ordering of the universe interprets a complete and decidable theory. This paper identifies the minimum subsystem of,, that ensures that the definable subset ordering of the universe interprets a complete theory, and classifies the structures that can be realised as the subset relation in a model of this set theory. Extending and refining Hamkins and Kikuchi's result for (...) class theory, a complete extension,, of the theory of infinite atomic boolean algebras and a minimum subsystem,, of are identified with the property that if is a model of, then is a model of, where M is the underlying set of, is the unary predicate that distinguishes sets from classes and is the definable subset relation. These results are used to show that that decides every 2‐stratified sentence of set theory and decides every 2‐stratified sentence of class theory. (shrink)

Marxian class theory has been unable to account for the most significant historical developments of the 20th century. The rise of fascism not only belied the hopes put in the revolutionary proletariat, it also brought into the center of the political stage those social strata which the class theory had relegated to, at best, secondary supporting roles. The triumph of the Bolshevik, revolution and the institutionalization and expansion of Soviet socialism has not only failed to issue (...) into the anticipated free, classless society, but has led to the creation of a new system of stratification and domination which is simply unthinkable within the classical Marxian framework. Late or advanced capitalism continues advancing in spite of all its crises and contradictions. (shrink)

This article studies three most basic systems of truth as well as their subsystems over set theory ZF possibly with AC or the axiom of global choice GC, and then correlates them with subsystems of Morse–Kelley class theory MK. The article aims at making an initial step towards the axiomatic study of truth in set theory in connection with class theory. Some new results on the side of class theory, such as conservativity, (...) forcing and some forms of the reflection principle, are also presented. The equivalence results among systems of truth and classes obtained in this article are summarized in Theorem 104 , Theorem 107 and Theorem 108. (shrink)

We study smooth classes without the algebraic closure property. For such smooth classes we investigate the simplicity of the class of generic structures, in the context of Robinson theories.

In Gender, Class, and Freedom in Modern Political Theory, Nancy Hirschmann demonstrates not merely that modern theories of freedom are susceptible to gender and class analysis but that they must be analyzed in terms of gender and class in order to be understood at all. Through rigorous close readings of major and minor works of Hobbes, Locke, Rousseau, Kant, and Mill, Hirschmann establishes and examines the gender and class foundations of the modern understanding of freedom. (...) Building on a social constructivist model of freedom that she developed in her award-winning book The Subject of Liberty: Toward a Feminist Theory of Freedom, she makes in her new book another original and important contribution to political and feminist theory. Despite the prominence of "state of nature" ideas in modern political theory, Hirschmann argues, theories of freedom actually advance a social constructivist understanding of humanity. By rereading "human nature" in light of this insight, Hirschmann uncovers theories of freedom that are both more historically accurate and more relevant to contemporary politics. Pigeonholing canonical theorists as proponents of either "positive" or "negative" liberty is historically inaccurate, she demonstrates, because theorists deploy both conceptions of freedom simultaneously throughout their work. (shrink)

Given a class of linear order types C, we identify and study several different classes of trees, naturally associated with C in terms of how the paths in those trees are related to the order types belonging to C. We investigate and completely determine the set-theoretic relationships between these classes of trees and between their corresponding first-order theories. We then obtain some general results about the axiomatization of the first-order theories of some of these classes of trees in terms (...) of the first-order theory of the generating class C, and indicate the problems obstructing such general results for the other classes. These problems arise from the possible existence of nondefinable paths in trees, that need not satisfy the first-order theory of C, so we have started analysing first order definable and undefinable paths in trees. (shrink)

The article focuses on the temporal and epistemic economy connected to the transatlantic travels of the categorical triad of ‘race-class-gender’. It looks at conditions and forces that have fuelled the dynamics of the discourse on differences and inequality among women and analyses feminist discourse and its aporias as a particular environment for the travels of theories. Furthermore, it follows the changes the triad of ‘race-class-gender’ undergoes on its transatlantic route from the United States to a German-speaking context and (...) it outlines the theoretical challenges connected to an intersectional perspective that aims to overcome a theoretical stagnation that itself finds symptomatic expression in the ritual citing of ‘race-class-gender’. (shrink)

The naive theory of properties states that for every condition there is a property instantiated by exactly the things which satisfy that condition. The naive theory of properties is inconsistent in classical logic, but there are many ways to obtain consistent naive theories of properties in nonclassical logics. The naive theory of classes adds to the naive theory of properties an extensionality rule or axiom, which states roughly that if two classes have exactly the same members, (...) they are identical. In this paper we examine the prospects for obtaining a satisfactory naive theory of classes. We start from a result by Ross Brady, which demonstrates the consistency of something resembling a naive theory of classes. We generalize Brady’s result somewhat and extend it to a recent system developed by Andrew Bacon. All of the theories we prove consistent contain an extensionality rule or axiom. But we argue that given the background logics, the relevant extensionality principles are too weak. For example, in some of these theories, there are universal classes which are not declared coextensive. We elucidate some very modest demands on extensionality, designed to rule out this kind of pathology. But we close by proving that even these modest demands cannot be jointly satisfied. In light of this new impossibility result, the prospects for a naive theory of classes are bleak. (shrink)

This paper constructs models of intuitionistic set theory in suitable categories. First, a Basic Intuitionistic Set Theory (BIST) is stated, and the categorical semantics are given. Second, we give a notion of an ideal over a category, using which one can build a model of BIST in which a given topos occurs as the sets. And third, a sheaf model is given of a Basic Intuitionistic Class Theory conservatively extending BIST. The paper extends the results in (...) [2] by introducing a new and perhaps more natural notion of ideal, and in the class theory of part three. (shrink)

1. The theory of classes presented in this paper is a simplification of that presented by J. von Neumann in his paper Die Axiomatisierung der Mengenlehre. However, this paper is written so that it can be read independently of von Neumann's. The principal modifications of his system are the following.(1) The idea of ordered pair is defined in terms of the other primitive concepts of the system. (See Axiom 4.3 below.)(2) A much simpler proof of the well-ordering theorem, based (...) on von Neumann's equivalence axiom (Axiom 2.2 below) is given. (More exactly, the theory of ordinal numbers, on which the proof of the well-ordering theorem is based, is simplified—see §8.)(3) Functions are not assumed to be defined for all arguments. In place of “ = A” we have “is undefined.” This makes possible a constructive interpretation of the system. (See §2.)2. We wish first to give a rough picture of the system which we are trying to construct. Suppose that we have the idea “class,” but no material from which to construct classes. Nevertheless, we can construct the class 0 having no elements. And then the class 1 having 0 as its only element. And then the classes {1}, and {0, 1}, etc., using always the previously constructed classes as elements. To extend this method to infinite classes, we must give rules telling what elements are to be included. Since the nature of the rules which we can use is not altogether clear, we try to formalize the whole system; that is, to set up a system of axioms which seems to characterize this system of classes. Now in this set of axioms we find it necessary to use the idea of function (in the “Axiom of Replacement”). (shrink)

We define an applicative theoryCL 2 similar to combinatory logic which can be interpreted in classes of functions possessing an enumerating function. In contrast to the models of classical combinatory logic, it is not necessarily assumed that the enumerating function itself belongs to that function class. Thereby we get a variety of possible models including e. g. the classes of primitive recursive, recursive, elementary, polynomial-time comptable ofɛ 0-recursive functions.We show that inCL 2 a major part of the metatheory of (...) enumerated classes of functions can be developed. Namely, a kind of λ-abstraction can be defined and abstract versions of theS n m - and (Primitive) Recursion Theorems are proved. Thereby, a closer analysis of the phenomenon of the different recursion theorems is achieved.A theory closely related toCL 2 can be used to replace the applicative part of Feferman's theories for explicit mathematics. So this work can be seen as an answer to Feferman's question to formulate a theory for explicit mathematics in which operations can be interpreted as primitive recursive or even more feasible ones.Finally it is shown that the proof-theoretical strength of various theoreies for explicit mathematics is preserved when replacing the applicative part of the theories by our theory together with an operation for primitive recursion. (shrink)

For the classA of uncountable Archimedian real closed fields we show that the statement “TheL <ω-theory ofA is complete” is independent of ZFC. In particular we have the following results:Assuming the Continuum-Hypothesis (CH) is incomplete. Conversely it is possible to build a model of set theory in which is complete and decidable. The latter can also be deduced from the Proper Forcing Axiom (PFA). In this case turns out to be equivalent to the elementary theory of the (...) real numbers ℝ (by a quantifier-elimination procedure).Formally: is incomplete. is complete and decidable. (shrink)

A class of metric theories of gravitation on Minkowski spacetime is considered, which is—provided that certain assumptions (staying close to the original ideas of Einstein) are made—the almost most general one that can be considered. In addition to the Minkowskian metric G a dynamical metric H (called the Einstein metric)is defined by means of a second-rank tensor field S (referred to as gravitational potential).The theory is defined by a Lagrangian ℒ, from which the field equations as well as, (...) e.g., the energy-momentum tensor field for the gravitational field follow. The case of weak fields is considered explicitly. The static, spherically and time-inversal symmetric field is calculated, and as a first step to investigate the theory's viability the parameters are fitted to the experimental data of the perihelion advance and the deflection of light at the Sun. Finally the question of gauge freedoms in the gravitational potential is briefly discussed. (shrink)

We recover the essentials of þ-forking, rosiness and super-rosiness for certain amalgamation classes K, and thence of finite-variable theories of finite structures. This provides a foundation for a model-theoretic analysis of a natural extension of the “LkLk-Canonization Problem” – the possibility of efficiently recovering finite models of T given a finite presentation of an LkLk-theory T. Some of this work is accomplished through different sorts of “transfer” theorem to the first-order theory TlimTlim of the direct limit. Our results (...) include, to start with, a recovery of the basic technology of þ-independence [15]) using a rather straightforward transfer. We also recover an analog of the “þ-Independence theorems” of Ealy and Onshuus [7] for amalgamation classes and their limits by showing how to transfer/lift an abstract independence relation on the amalgamation class to the limit theory TlimTlim. We also work out an appropriate notion of Local Character for independence relations over classes finite structures, and we use this to verify that rosiness and super-rosiness-with-finite-UþUþ-ranks coincide in these amalgamation classes and their limit theories. (shrink)