A motivation and a presentation of the semantics of a formal system which proceeds from von neumann's "eine axiomatisierung der mengenlehre" by taking as fundamental notions the notion of partial functions, his principles of reification on the grounds of a "method of simulation" of meaningless terms and formulas by "ideal" terms and "virtual" formulas. theory of sets appears as the end of axiomatic extensions of the theory of objects (the intermediary extensions being the theories of classes, (...) of universes, of aggregates). (shrink)

Theories of the present have converged on changes in spatialization or the spatial order of societies. This article discusses the focus on borders and boundaries in programmatic statements on reflexive modernity or remodernization (RM) by Latour and Beck. It is insufficient to say that boundary-marking and border-making become simply more fraught or obvious. There is an historicity and dynamic quality which are central to these analyses which are best understood in terms of the intangible aspects, or virtuality, of borders and (...) boundaries. A dynamic, four-part ontology is advanced to elucidate the functioning of borders as interfaces and liminal zones with their own internal semiotics and emergent meanings and effects. This is the basis of a critique of RM. Further, a more diverse set of methods than might be supposed is required to research boundary phenomena, which social sciences are ill-equipped to undertake. (shrink)

This paper shows that several live philosophical and scientific hypotheses – including the holographic principle and multiverse theory in quantum physics, and eternalism and mind-body dualism in philosophy – jointly imply an audacious new theory of free will. This new theory, "Libertarian Compatibilism", holds that the physical world is an eternally existing array of two-dimensional information – a vast number of possible pasts, presents, and futures – and the mind a nonphysical entity or set of properties that (...) "read" that physical information off to subjective conscious awareness (in much the same way that a song written on an ordinary compact-disc is only played when read by an outside medium, i.e. a CD-player). According to this theory, every possible physical “timeline” in the multiverse may be fully physically deterministic or physically-causally closed but each person’s consciousness still entirely free to choose, ex nihilo, outside of the physical order, which physically-closed timeline is experienced by conscious observers. Although Libertarian Compatibilism is admittedly fantastic, I show that it not only follows from several live scientific and philosophical hypotheses, I also show that it (A) is a far more explanatorily powerful model of quantum mechanics than more traditional interpretations (e.g. the Copenhagen, Everett, and Bohmian interpretations), (B) makes determinate, testable empirical predictions in quantum theory, and finally, (C) predicts and explains the very existence of a number of philosophical debates and positions in the philosophy of mind, time, personal identity, and free will. First, I show that whereas traditional interpretations of quantum mechanics are all philosophically problematic and roughly as ontologically “extravagant” as Libertarian Compatibilism – in that they all posit “unseen” processes – Libertarian Compatibilism is nearly identical in structure to the only working simulation that human beings have ever constructed capable of reproducing (and so explaining) every general feature of quantum mechanics we perceive: namely, massive-multiplayer-online-roleplaying videogames (or MMORPGs). Although I am not the first to suggest that our world is akin to a computer simulation, I show that existing MMORPGs (online simulations we have already created) actually reproduce every general feature of quantum mechanics within their simulated-world reference-frames. Second, I show that existing MMORPGs also replicate (and so explain) many philosophical problems we face in the philosophy of mind, time, personal identity, and free will – all while conforming to the Libertarian Compatibilist model of reality. -/- I conclude, as such, that as fantastic and metaphysically extravagant as Libertarian Compatibilism may initially seem, it may well be true. It explains a number of features of our reality that no other physical or metaphysical theory does. (shrink)

The focus in this essay will be upon the paradoxes, and foremostly in set theory. A central result is that the librationist set theory £ extension \Pfund $\mathscr{HR}(\mathbf{D})$ of \pounds \ accounts for \textbf{Neumann-Bernays-Gödel} set theory with the \textbf{Axiom of Choice} and \textbf{Tarski's Axiom}. Moreover, \Pfund \ succeeds with defining an impredicative manifestation set $\mathbf{W}$, \emph{die Welt}, so that \Pfund$\mathscr{H}(\mathbf{W})$ %is a model accounts for Quine's \textbf{New Foundations}. Nevertheless, the points of view developed support the view that (...) the truth-paradoxes and the set-paradoxes have common origins, so that the librationist resolutions of the set theoretic paradoxes are at the same time resolutions of the truth theoretic paradoxes. Both the librationist resolutions of the set theoretic paradoxes and the truth theoretic paradoxes have non-trivial philosophical implications: librationist set theories have the consequence that there are no absolutely uncountable sets, and librationist truth theories allow the use of syntactical modalities in ways which circumvent limitations as those of \parencite{Montague1963}, and a truth predicate which is useful for more precise philosophical discourse. (shrink)

This work applies the competitive exclusion principle and the concept of potential competitors as simple axiomatic tools to generalized situations in ecology. These tools enable apparent competition and its dual counterpart to be explicitly evaluated in poorly understood ecological systems. Within this set-theory framework we explore theoretical symmetries and invariances, De Morgan’s laws, frozen evolutionary diversity and virtual processes. In particular, we find that the exclusion principle compromises the geometrical growth of the number of species. By theoretical extending (...) this principle, we can describe interspecific depredation in the dual case. This study also briefly considers the debated situation of intraspecific competition. The ecological consequences of our findings are discussed; particularly, the use of our framework to reinterpret coupled mathematical differential equations describing certain ecological processes. (shrink)

We approach the virtual reality phenomenon by studying its relationship to set theory. This approach offers a characterization of virtual reality in set theoretic terms, and we investigate the case where this is done using the wellfoundedness property. Our hypothesis is that non-wellfounded sets (so-called hypersets) give rise to a different quality of virtual reality than do familiar wellfounded sets. To elaborate this hypothesis, we describe virtual reality through Sommerhoff’s categories of first- and (...) second-order self-awareness; introduced as necessary conditions for consciousness in terms of higher cognitive functions. We then propose a representation of first- and second-order self-awareness through sets, and assume that these sets, which we call events, originally form a collection of wellfounded sets. Strong virtual reality characterizes virtual reality environments which have the limited capacity to create only events associated with wellfounded sets. In contrast, the logically weaker and more general concept of weak virtual reality characterizes collections of virtual reality mediated events altogether forming an entirety larger than any collection of wellfounded sets. By giving reference to Aczel’s hyperset theory we indicate that this definition is not empty because hypersets encompass wellfounded sets already. Moreover, we argue that weak virtual reality could be realized in human history through continued progress in computer technology. Finally, within a more general framework, we use Baltag’s structural theory of sets (STS) to show that within this hyperset theory Sommerhoff’s first- and second-order self-awareness as well as both concepts of virtual reality admit a consistent mathematical representation. To illustrate our ideas, several examples and heuristic arguments are discussed. (shrink)

Introduction -- Prelude: from gesture to virtual agency -- Foundations for a theory of agency -- Virtual environmental forces and gestural energies: actants as agential -- Virtual embodiment: from actants to virtual human agents -- Virtual identity and actorial continuity -- Interlude I: from embodiment to subjectivity -- Staging virtual subjectivity -- Virtual subjectivity and aesthetically warranted emotions -- Staging virtual narrative agency -- Performing agency -- An integrative agential interpretation of (...) Chopin's Ballade in F minor, op. 52 -- Interlude II: hearing agency: a complex cognitive task -- Other perspectives on virtual agency -- Postlude. (shrink)

In this paper we introduce a theory of finite sets FST with a strong negation of the axiom of infinity asserting that every set is provably bijective with a natural number. We study in detail the role of the axioms of Power Set, Choice, Regularity in FST, pointing out the relative dependences or independences among them. FST is shown to be provably equivalent to a fragment of Alternative Set Theory. Furthermore, the introduction of FST is motivated in (...) view of a non-standard development. MSC: 03E30, 03E35. (shrink)

Although Descartes has often been portrayed as the father of the modern concept of mind, his approach to consciousness is notoriously problematic. What makes it particularly hard to assess his role in the development of the theories of consciousness is the difficulty of clarifying the kind of consciousness he might have in mind when using the associated Latin terms (conscius, cogitatio, conscium esse, etc.). In this article, I analyse Antoine Arnauld’s early interpretation of the passages in Descartes that refer to (...) the issue of consciousness. I argue for two separate but interconnected claims. Firstly, I show that when Arnauld sets out to make a case for Descartes’ concept of cogitatio, he reads the central passages in light of some scholastic theories of cognition, in particular, the concept of ‘reflexio virtualis’ which, far from being a Cartesian invention, comes from the late scholastic discourse. Secondly, I argue that by talking about virtual reflection Arnauld provides an interpretation of Descartes’ views in terms of the intrinsic structure of the first-order thought – a reading which is still plausible, even by our contemporary standards. (shrink)

This revision of an important and lucid account of the various systems of axiomatic set theory preserves the basic format and essential ingredients of its highly regarded original. Quine's innovative exploitation of the virtual theory of classes in order to develop a considerable portion of set theory without ontological commitment to the existence of classes remains unchanged. So, too, does the list of topics treated--the theory of sets up to transfinite ordinal and cardinal numbers, (...) the axiom of choice and its equivalents, and a beautiful comparison of the various major formulations of set theory. There have, however, been a number of important changes, generally in the interest of greater elegance and clarity. One of the most notable of these is in the treatment of ordinal numbers. In the fourth section of that chapter an axiom schema is added to the effect that if α is a function and x an ordinal number, then there exists a class identical with the image of x under α. Although Quine, in general prefers to limit his existence assumptions as severely as possible, this added item possesses two attractive and closely related advantages: it yields the existence of all classes up to the size of any ordinal, and it greatly simplifies the subsequent treatment of ordinals and of transfinite recursion. The treatment of this latter subject, incidentally, has also been extensively modified to square with some difficulties first pointed out by Charles Parsons in The Journal of Symbolic Logic in 1964. Another important change consists in a considerable reworking of the section on infinite cardinals. The result is a more simplified and natural treatment. Finally, some new material on the theory of types and consequences of the axiom schema of comprehension has been added. Particularly with regard to the theory of types, this new material--due largely to the work of David Kaplan--has enabled Quine to drop, at one point, the assumption that there exists an object of lowest type. Altogether, this is a generally improved version of an originally masterful and brilliant work.--H. P. K. (shrink)

This is a text for a one or two semester course on axiomatic set theory; the goal is to introduce and develop one system of set theory in a complete and thorough way, presupposing only the elusive "mathematical maturity" of the reader. There are nine chapters which begin with a development of propositional and predicate logic oriented toward set theory and develop the Zermelo-Fraenkel system in exceptional detail. The book starts slowly, the first 120 pages being devoted (...) to logical preliminaries and the introduction of axioms; it gathers speed, however, and the remaining chapters include treatment of the algebra of classes, relations and functions, order of various sorts and Zorn's lemma, real numbers and equivalence classes, the equipollence of sets, similarity as a relation of sets, the ordinal numbers and their arithmetic, the cardinal numbers and their arithmetic. These later chapters are extremely detailed and comprehensive, presenting a wealth of material. There are numerous exercises, many quite difficult, at the end of each chapter, along with a summary of that chapter's content. Because of the single-mindedness of the book, there is little reference to other systems of set theory, but this is not a drawback at all. The presentation is orthodox, but not dull; careful, but not generally pedantic or repetitive. This makes it a good text for self-study.—P. J. M. (shrink)

In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, there (...) was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

We explore a non-classical, universal set theory, based on a purely 'structural' conception of sets. A set is a transfinite process of unfolding of an arbitrary binary structure, with identity of sets given by the observational equivalence between such processes. We formalize these notions using infinitary modal logic, which provides partial descriptions for set structures up to observational equivalence. We describe the comprehension and topological properties of the resulting set-theory, and we use it to give non-classical (...) solutions to classical paradoxes, to prove fixed-point theorems that relate recursion and corecursion, to formalize 'super-large', reflexive categories and 'super-large' circular modes, and to provide 'natural' solutions for domain equations. (shrink)

During the 20th century two major ventures were launched to advance Darwinian evolution theory. Both involved historic visions and were vital steps for science and society, but then something happened on the way to the millennium. By mid-century the first venture had become a virtual scientific monopoly governed by the biology of the neoDarwinian paradigm. The second venture then set out in the 1980s to remedy the inadequacies of the neoDarwinian paradigm by widening the prospects for evolution (...) class='Hi'>theory. But overwhelmed by the underlying mismatch between scientific abstraction and evolutionary reality the first venture established, it soon settled into a fierce attempt to further expand the territory for the neoDarwinian monopoly into what became a militant ideology for sociobiology and evolutionary psychology. This special issue of World Futures contains the papers of a small "task force" of the General Evolution Research Group that set out in the summer of 2000 to try to put behind us what increasingly looms as the "old" paradigm, as well as the "old" story, of evolution. (shrink)

Georg Cantor a cherché à assurer les fondements de sa théorie des ensembles. Cet article présente les differentiations cantoriennes concernant la notion d’infinité et une perspective historique de l’émergence de sa notion d’ensemble.

Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a (...) discussion of the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians. (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)

In the present paper we use the theory of exact completions to study categorical properties of small setoids in Martin-Löf type theory and, more generally, of models of the Constructive Elementary Theory of the Category of Sets, in terms of properties of their subcategories of choice objects. Because of these intended applications, we deal with categories that lack equalisers and just have weak ones, but whose objects can be regarded as collections of global elements. In this (...) context, we study the internal logic of the categories involved, and employ this analysis to give a sufficient condition for the local cartesian closure of an exact completion. Finally, we apply this result to show when an exact completion produces a model of CETCS. (shrink)

We propose a theory for modeling concepts that uses the state-context-property theory (SCOP), a generalization of the quantum formalism, whose basic notions are states, contexts and properties. This theory enables us to incorporate context into the mathematical structure used to describe a concept, and thereby model how context influences the typicality of a single exemplar and the applicability of a single property of a concept. We introduce the notion `state of a concept' to account for this contextual (...) influence, and show that the structure of the set of contexts and of the set of properties of a concept is a complete orthocomplemented lattice. The structural study in this article is a preparation for a numerical mathematical theory of concepts in the Hilbert space of quantum mechanics that allows the description of the combination of concepts. (shrink)

This book will appeal to mathematicians and philosophers interested in the foundations of mathematics.

My research into Virtual Reality technology and its central property of immersion has indicated that immersion in Virtual Reality (VR) electronic systems is a significant key to the understanding of contemporary culture as well as considerable aspects of previous culture as detected in the histories of philosophy and the visual arts. The fundamental change in aesthetic perception engendered by immersion, a perception which is connected to the ideal of total-immersion in virtual space, identifies certain shifts in ontology (...) which are relevant to a better understanding of the human being. This understanding was achieved through a broad inquiry into the histories of Virtual Reality, philosophy, and the visual arts and has lead to the formulation of an aesthetic theory of immersive consciousness indicative of immersive culture. The primary subject of this discourse is immersion then: an experience which will be identified within the dissertation as the indispensable characteristic of Virtual Reality. The understanding of immersion arrived at here will be used to fashion a synchronous theory of art particularly informed by encounters and concepts of immersion into virtuality. To sufficiently address this subject in a scholarly fashion, I have researched, found and accumulated aesthetic and philosophic examples of immersive tendencies, as found within the histories of art and philosophy, which subsequently contributed towards the articulation of what I have come to call immersive consciousness. As a result of formulating such an immersive consciousness, a good deal of the basis for the questioning of the Western ontological tradition has been found in the Western tradition itself when we look with new eyes and ask new uncertain questions. Moreover, this immersive consciousness will be used to propose some abstract questions encircling today's electronic-based culture. Through the structuring of the argument within the thesis - and overtly within the conclusion - I have articulated a non-teleological creative strategy which provides the basis for an unconstraining integration of noologies (ways of semblancing the thinking process). This strategy provides a means of exemplifying - and for honoring - various methods of thinking. This structuring strategy is consistent with the 'hacker ethic' as defined by Steven Levy, as a demand that access to computers - and anything which might teach us something about the way the world works - should be unlimited and total. To follow this strategy, this dissertation has set out to understand how topical conceptions of virtual immersion connect to pre-existing systems of thought as revealed in art as they have extended out of antecedent ontological self-understandings, historical human self-understandings which have evidenced themselves in the elaboration of technological objectives. To do this I have forged a certain rhizomatic paternity/maternity for Virtual Reality within this dissertation by joining choice immersive examples of simulacra technology into mental connections with the relevant examples culled from the histories of art, architecture, information-technology, sex, myth, space, consciousness and philosophy. (shrink)

I provide an examination and comparison of modal theories for underwriting different non-modal theories of sets. I argue that there is a respect in which the `standard' modal theory for set construction---on which sets are formed via the successive individuation of powersets---raises a significant challenge for some recently proposed `countabilist' modal theories (i.e. ones that imply that every set is countable). I examine how the countabilist can respond to this issue via the use of regularity axioms and (...) raise some questions about this approach. I argue that by comparing them with the `standard' uncountabilist theory, a new approach that brings in arbitrariness rather than the strict controls of forcing is desirable. (shrink)

A DO model (here also referred to a Paris model) is a model of set theory all of whose ordinals are first order definable in . Jeffrey Paris (1973) initiated the study of DO models and showed that (1) every consistent extension T of ZF has a DO model, and (2) for complete extensions T, T has a unique DO model up to isomorphism iff T proves V=OD. Here we provide a comprehensive treatment of Paris models. Our results include (...) the following:1. If T is a consistent completion of ZF+V≠OD, then T has continuum-many countable nonisomorphic Paris models.2. Every countable model of ZFC has a Paris generic extension.3. If there is an uncountable well-founded model of ZFC, then for every infinite cardinal κ there is a Paris model of ZF of cardinality κ which has a nontrivial automorphism.4. For a model ZF, is a prime model ⇒ is a Paris model and satisfies AC⇒ is a minimal model. Moreover, Neither implication reverses assuming Con(ZF). (shrink)

To unify rival theories is to combine their key insights into a single coherent framework. It is often achieved by integrating the theories and forging new connections between their explanatory fac...

Proceedings of the Steklov Institute of Mathematics is a cover-to-cover translation of the Trudy Matematicheskogo Instituta imeni V.A. Steklova of the Russian Academy of Sciences. Each issue ordinarily contains either one book-length article or a collection of articles pertaining to the same topic.

We develop a new version of the causal theory of spacetime. Whereas traditional versions of the theory seek to identify spatiotemporal relations with causal relations, the version we develop takes causal relations to be the grounds for spatiotemporal relations. Causation is thus distinct from, and more basic than, spacetime. We argue that this non-identity theory, suitably developed, avoids the challenges facing the traditional identity theory.

Volume II of Classical Recursion Theory describes the universe from a local (bottom-up or synthetical) point of view, and covers the whole spectrum, from the recursive to the arithmetical sets. The first half of the book provides a detailed picture of the computable sets from the perspective of Theoretical Computer Science. Besides giving a detailed description of the theories of abstract Complexity Theory and of Inductive Inference, it contributes a uniform picture of the most basic complexity (...) classes, ranging from small time and space bounds to the elementary functions, with a particular attention to polynomial time and space computability. It also deals with primitive recursive functions and larger classes, which are of interest to the proof theorist. The second half of the book starts with the classical theory of recursively enumerable sets and degrees, which constitutes the core of Recursion or Computability Theory. Unlike other texts, usually confined to the Turing degrees, the book covers a variety of other strong reducibilities, studying both their individual structures and their mutual relationships. The last chapters extend the theory to limit sets and arithmetical sets. The volume ends with the first textbook treatment of the enumeration degrees, which admit a number of applications from algebra to the Lambda Calculus. The book is a valuable source of information for anyone interested in Complexity and Computability Theory. The student will appreciate the detailed but informal account of a wide variety of basic topics, while the specialist will find a wealth of material sketched in exercises and asides. A massive bibliography of more than a thousand titles completes the treatment on the historical side. (shrink)

We discuss several axiomatizations of set theory in first order predicate calculus with epsilon and a constant symbol W, starting with the simple system K(W) which has a strong equivalence with ZF without Foundation. The other systems correspond to various extensions of ZF by certain large cardinal hypotheses. These axiomatizations are unusually simple and uncluttered, and are highly suggestive of underlying philosophical principles that generate higher set theory.