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Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the tenth publication in the Lecture Notes in Logic series, Per Lindström presents some of the main topics and results in general metamathematics. In addition to standard results of Gödel et al. on incompleteness, (non-)finite axiomatizability, and interpretability, this book contains a thorough treatment of partial conservativity and degrees of interpretability. It comes complete with exercises, and will be useful as a textbook for graduate students with a background in logic, as well as a valuable resource for researchers.
This Element takes a deep dive into Gödel's 1931 paper giving the first presentation of the Incompleteness Theorems, opening up completely passages in it that might possibly puzzle the student, such as the mysterious footnote 48a. It considers the main ingredients of Gödel's proof: arithmetization, strong representability, and the Fixed Point Theorem in a layered fashion, returning to their various aspects: semantic, syntactic, computational, philosophical and mathematical, as the topic arises. It samples some of the most important proofs of the Incompleteness Theorems, e.g. due to Kuratowski, Smullyan and Robinson, as well as newer proofs, also of other independent statements, due to H. Friedman, Weiermann and Paris-Harrington. It examines the question whether the incompleteness of e.g. Peano Arithmetic gives immediately the undecidability of the Entscheidungsproblem, as Kripke has recently argued. It considers set-theoretical incompleteness, and finally considers some of the philosophical consequences considered in the literature.
This book contains the proceedings of the Seventh National Conference of the Italian Systems Society. The title, Systemics of Incompleteness and Quasi-Systems, aims to underline the need for Systemics and Systems Science to deal with the concepts of incompleteness and quasiness. Classical models of Systemics are intended to represent comprehensive aspects of phenomena and processes. They consider the phenomena in their temporal and spatial completeness. In these cases, possible incompleteness in the modelling is assumed to have a provisional or practical nature, which is still under study, and because there is no theoretical reason why the modelling cannot be complete. In principle, this is a matter of non-complex phenomena, to be considered using the concepts of the First Systemics. When dealing with emergence, there are phenomena which must be modelled by systems having multiple models, depending on the aspects being taken into consideration. Here, incompleteness in the modelling is intrinsic, theoretically relating changes in properties, structures, and status of system. Rather than consider the same system parametrically changing over time, we consider sequences of systems coherently. We consider contexts and processes for which modelling is incomplete, being related to only some properties, as well as those for which such modelling is theoretically incomplete—as in the case of processes of emergence and for approaches considered by the Second Systemics. In this regard, we consider here the generic concept of quasi explicating such incompleteness. The concept of quasi is used in various disciplines including quasi-crystals, quasi-particles, quasi-electric fields, and quasi-periodicity. In general, the concept of quasiness for systems concerns their continuous structural changes which are always meta-stable, waiting for events to collapse over other configurations and possible forms of stability; whose equivalence depends on the type of phenomenon under study. Interest in the concept of quasiness is not related to its meaning of rough approximation, but because it indicates an incompleteness which is structurally sufficient to accommodate processes of emergence and sustain coherence or generate new, equivalent or non-equivalent, levels. The conference was devoted to identifying, discussing and understanding possible interrelationships of theoretical disciplinary improvements, recognised as having prospective fundamental roles for a new Quasi-Systemics. The latter should be able to deal with problems related to complexity in more general and realistic ways, when a system is not always a system and not always the same system. In this context, the inter-disciplinarity should consist, for instance, of a constructionist, incomplete, non-ideological, multiple, contradiction-tolerant, Systemics, always in progress, and in its turn, emergent.
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"An introduction to the life and thought of Kurt Gödel, who transformed our conception of math forever"--Provided by publisher.
Age effects have played a particularly prominent role in some theoretical perspectives on second language acquisition. This book takes an entirely new perspective on this issue by re-examining these theories in light of the existence of apparently similar non-native outcomes in adult heritage speakers who, unlike adult second language learners, acquired two or more languages in childhood. Despite having been exposed to their family language early in life, many of these speakers never fully acquire, or later lose, aspects of their first language sometime in childhood. The book examines the structural characteristics of "incomplete" grammatical states and highlights how age of acquisition is related to the type of linguistic knowledge and behavior that emerges in L1 and L2 acquisition under different environmental circumstances. By underscoring age of acquisition as a unifying factor in the study of L2 acquisition and L1 attrition, it is claimed that just as there are age effects in L2 acquisition, there are also age effects, or even perhaps a critical period, in L1 attrition. The book covers adult L2 acquisition, attrition in adults and in children, and includes a comparison of adult heritage language speakers and second language learners.
These lectures on logic, more specifically proof theory, are basically intended for postgraduate students and researchers in logic. The question at stake is the nature of mathematical knowledge and the difference between a question and an answer, i.e., the implicit and the explicit. The problem is delicate mathematically and philosophically as well: the relation between a question and its answer is a sort of equality where one side is ``more equal than the other'': one thus discovers essentialist blind spots. Starting with Godel's paradox (1931)--so to speak, the incompleteness of answers with respect to questions--the book proceeds with paradigms inherited from Gentzen's cut-elimination (1935). Various settings are studied: sequent calculus, natural deduction, lambda calculi, category-theoretic composition, up to geometry of interaction (GoI), all devoted to explicitation, which eventually amounts to inverting an operator in a von Neumann algebra. Mathematical language is usually described as referring to a preexisting reality. Logical operations can be given an alternative procedural meaning: typically, the operators involved in GoI are invertible, not because they are constructed according to the book, but because logical rules are those ensuring invertibility. Similarly, the durability of truth should not be taken for granted: one should distinguish between imperfect (perennial) and perfect modes. The procedural explanation of the infinite thus identifies it with the unfinished, i.e., the perennial. But is perenniality perennial? This questioning yields a possible logical explanation for algorithmic complexity. This highly original course on logic by one of the world's leading proof theorists challenges mathematicians, computer scientists, physicists, and philosophers to rethink their views and concepts on the nature of mathematical knowledge in an exceptionally profound way.
The Software Engineering and Knowledgebase Systems (SOFfEKS) Research Group of the Department of Computer Science, Concordia University, Canada, organized a workshop on Incompleteness and Uncertainty in Information Systems from October 8-9, 1993 in Montreal. A major aim of the workshop was to bring together researchers who share a concern for issues of incompleteness and uncertainty. The workshop attracted people doing fundamental research and industry oriented research in databases, software engineering and AI from North America, Europe and Asia. The workshop program featured six invited talks and twenty other presentations. The invited speakers were: Martin Feather (University of Southern CalifornialInformation Systems Institute) Laks V. S. Lakshmanan (Concordia University) Ewa Orlowska (Polish Academy of Sciences) z. Pawlak (Warsaw Technical University and Academy of Sciences) F. Sadri (Concordia University) A. Skowron (Warsaw University) The papers can be classified into four groups: rough sets and logic, concept analysis, databases and information retrieval, and software engineering. The workshop opened with a warm welcome speech from Dr. Dan Taddeo, Dean, Faculty of Engineering and Computer Science. The first day's presentations were in rough sets, databases and information retrieval. Papers given on the second day centered around software engineering and concept analysis. Sufficient time was given in between presentations to promote active interactions and numerous lively discussions. At the end of two days, the participants expressed their hope that this workshop would be continued.
In the most rigorous articulation of his philosophical system to date, Slavoj Žižek provides nothing short of a new definition of dialectical materialism. In forging this new materialism, Žižek critiques and challenges not only the work of Alain Badiou, Robert Brandom, Joan Copjec, Quentin Meillassoux, and Julia Kristeva (to name but a few), but everything from popular science and quantum mechanics to sexual difference and analytic philosophy. Alongside striking images of the Möbius strip, the cross-cap, and the Klein bottle, Žižek brings alive the Hegelian triad of being-essence-notion. Radical new readings of Hegel, and Kant, sit side by side with characteristically lively commentaries on film, politics, and culture. Here is Žižek at his interrogative best.
The central claim of this powerful philosophical exploration is that within any logic we have, there can be no coherent notion of all truth or of total knowledge. Grim examines a series of logical paradoxes and related formal results to reveal their implications for contemporary epistemology, metaphysics, and the philosophy of religion. He reaches the provocative conclusion that, if the universe is thought of in terms of its truths, it is essentially open and incomplete. The Incomplete Universe includes detailed work on the liar paradox and recent attempts at solution, Kaplan and Montague's paradox of the knower, the Gödel theorems and related incompleteness phenomena, and new forms of Cantorian argument. The emphasis throughout is philosophical rather than formal, with an eye to connection's with possible worlds, the notion of omniscience, and the opening lines of the Tractatus: "The world is all that is the case. "