Kurt Gödel and the Foundations of Mathematics

Horizons of Truth
Author: Matthias Baaz,Christos H. Papadimitriou,Hilary W. Putnam,Dana S. Scott,Charles L. Harper, Jr
Publisher: Cambridge University Press
ISBN: 1139498436
Category: Mathematics
Page: N.A
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This volume commemorates the life, work and foundational views of Kurt Gödel (1906–78), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency - with the other widely accepted axioms of set theory - of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Gödel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Gödel's fundamental work in mathematics, logic, philosophy and other disciplines for future generations of researchers.

Kurt Gödel

Essays for his Centennial
Author: Solomon Feferman,Charles Parsons,Stephen G. Simpson
Publisher: Cambridge University Press
ISBN: 1139487752
Category: Mathematics
Page: N.A
View: 594
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Kurt Gödel (1906–1978) did groundbreaking work that transformed logic and other important aspects of our understanding of mathematics, especially his proof of the incompleteness of formalized arithmetic. This book on different aspects of his work and on subjects in which his ideas have contemporary resonance includes papers from a May 2006 symposium celebrating Gödel's centennial as well as papers from a 2004 symposium. Proof theory, set theory, philosophy of mathematics, and the editing of Gödel's writings are among the topics covered. Several chapters discuss his intellectual development and his relation to predecessors and contemporaries such as Hilbert, Carnap, and Herbrand. Others consider his views on justification in set theory in light of more recent work and contemporary echoes of his incompleteness theorems and the concept of constructible sets.

After Gödel

Platonism and Rationalism in Mathematics and Logic
Author: Richard Tieszen
Publisher: OUP Oxford
ISBN: 0191619310
Category: Philosophy
Page: 272
View: 862
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Richard Tieszen presents an analysis, development, and defense of a number of central ideas in Kurt Gödel's writings on the philosophy and foundations of mathematics and logic. Tieszen structures the argument around Gödel's three philosophical heroes - Plato, Leibniz, and Husserl - and his engagement with Kant, and supplements close readings of Gödel's texts on foundations with materials from Gödel's Nachlass and from Hao Wang's discussions with Gödel. As well as providing discussions of Gödel's views on the philosophical significance of his technical results on completeness, incompleteness, undecidability, consistency proofs, speed-up theorems, and independence proofs, Tieszen furnishes a detailed analysis of Gödel's critique of Hilbert and Carnap, and of his subsequent turn to Husserl's transcendental philosophy in 1959. On this basis, a new type of platonic rationalism that requires rational intuition, called 'constituted platonism', is developed and defended. Tieszen shows how constituted platonism addresses the problem of the objectivity of mathematics and of the knowledge of abstract mathematical objects. Finally, he considers the implications of this position for the claim that human minds ('monads') are machines, and discusses the issues of pragmatic holism and rationalism.

Correspondence H-Z


Author: Kurt G?del,S. Feferman
Publisher: Oxford University Press
ISBN: 0198500750
Category: Computers
Page: 664
View: 3810
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Kurt Gödel (1906 - 1978) was the most outstanding logician of the twentieth century, famous for his hallmark works on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computability theory, as well as for the strong individuality of his writings on the philosophy of mathematics. He is less well known for his discovery of unusual cosmological models for Einstein's equations, in theory permitting time travel into the past. The Collected Works is a landmark resource that draws together a lifetime of creative thought and accomplishment. The first two volumes were devoted to Gödel's publications in full (both in original and translation), and the third volume featured a wide selection of unpublished articles and lecture texts found in Gödel's Nachlass. These long-awaited final two volumes contain Gödel's correspondence of logical, philosophical, and scientific interest. Volume IV covers A to G, with H to Z in volume V; in addition, Volume V contains a full inventory of Gödel's Nachlass. L All volumes include introductory notes that provide extensive explanatory and historical commentary on each body of work, English translations of material originally written in German (some transcribed from the Gabelsberger shorthand), and a complete bibliography of all works cited. Kurt Gödel: Collected Works is designed to be useful and accessible to as wide an audience as possible without sacrificing scientific or historical accuracy. The only comprehensive edition of Gödel's work available, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science and all others who wish to be acquainted with one of the great minds of the twentieth century.

The Metaphysics of World Order

A Synthesis of Philosophy, Theology, and Politics
Author: Nicolas Laos
Publisher: Wipf and Stock Publishers
ISBN: 1498201024
Category: Philosophy
Page: 244
View: 3047
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In this book, Nicolas Laos studies the meaning of the terms "world" and "order," the moral dimensions of each world order model, and wider issues of meaning and interpretation generated by humanity's attempt to live in a meaningful world and to find the logos of the beings and things in the world. The aim of this book is to propose a unified theory of world order (i.e., a theory that combines philosophy, theology, and political theory). In this context, the author provides a thought-provoking (re)interpretation of classical philosophy (placing particular emphasis on Platonism), an in-depth inquiry into medieval philosophy and spirituality (placing particular emphasis on the cultural differences between the Greek East and the Latino-Frankish West), and an intellectually challenging review and evaluation of modern Western philosophy (including Descartes, Spinoza, Leibniz, Locke, Berkeley, Hume, Rousseau, Kant, Hegel, Kierkegaard, Husserl, and Heidegger) and of Nietzsche's and the postmodernists' revolt against modernity. He then elucidates the philosophical foundations and "pedigree" of each of the three basic political theories of modernity (i.e., Liberalism, Communism, and Fascism), and he studies the basic theoretical debates in International Relations, Geopolitics, and Noopolitics. Finally, Laos proposes a new, "fourth," political theory which he calls "metaphysical republicanism."

Biographisch-literarisches Handworterbuch der exakten Naturwissenschaften, Band VIII, Teil 2, Doppellieferung 1/2


Author: Johann Ch. Poggendorff
Publisher: Wiley-VCH
ISBN: 9783527403028
Category: Science
Page: 160
View: 6940
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Poggendorffs Biographisch-literarisches Handwörterbuch der exakten Naturwissenschaften erscheint seit 1863 und darf als umfassendes Nachschlagewerk in keiner Bibliothek fehlen.

The Encyclopedia of Philosophy

Supplement
Author: Donald M. Borchert
Publisher: Macmillan Reference USA
ISBN: 9780028646299
Category: Philosophy
Page: 775
View: 7762
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The 357 articles address areas of thought that were overlooked by the original set or that have developed since the Encyclopedia's publication, and they do it in a precise manner that will allow all readers to fully understand every side of the topic.

Handbook of the History of Logic: The rise of modern logic : from Leibniz to Frege


Author: Dov M. Gabbay,John Hayden Woods
Publisher: N.A
ISBN: N.A
Category: Induction (Logic)
Page: 770
View: 9201
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With the publication of the present volume, the Handbook of the History of Logic turns its attention to the rise of modern logic. The period covered is 1685-1900, with this volume carving out the territory from Leibniz to Frege. What is striking about this period is the earliness and persistence of what could be called 'the mathematical turn in logic'. Virtually every working logician is aware that, after a centuries-long run, the logic that originated in antiquity came to be displaced by a new approach with a dominantly mathematical character. It is, however, a substantial error to suppose that the mathematization of logic was, in all essentials, Frege's accomplishment or, if not his alone, a development ensuing from the second half of the nineteenth century. The mathematical turn in logic, although given considerable torque by events of the nineteenth century, can with assurance be dated from the final quarter of the seventeenth century in the impressively prescient work of Leibniz. It is true that, in the three hundred year run-up to the Begriffsschrift, one does not see a smoothly continuous evolution of the mathematical turn, but the idea that logic is mathematics, albeit perhaps only the most general part of mathematics, is one that attracted some degree of support throughout the entire period in question. Still, as Alfred North Whitehead once noted, the relationship between mathematics and symbolic logic has been an "uneasy" one, as is the present-day association of mathematics with computing. Some of this unease has a philosophical texture. For example, those who equate mathematics and logic sometimes disagree about the directionality of the purported identity. Frege and Russell made themselves famous by insisting (though for different reasons) that logic was the senior partner. Indeed logicism is the view that mathematics can be re-expressed without relevant loss in a suitably framed symbolic logic. But for a number of thinkers who took an algebraic approach to logic, the dependency relation was reversed, with mathematics in some form emerging as the senior partner. This was the precursor of the modern view that, in its four main precincts (set theory, proof theory, model theory and recursion theory), logic is indeed a branch of pure mathematics. It would be a mistake to leave the impression that the mathematization of logic (or the logicization of mathematics) was the sole concern of the history of logic between 1665 and 1900. There are, in this long interval, aspects of the modern unfolding of logic that bear no stamp of the imperial designs of mathematicians, as the chapters on Kant and Hegcl make clear. Of the two, Hcgel's influence on logic is arguably the greater, serving as a spur to the unfolding of an idealist tradition in logic - a development that will be covered in a further volume, British Logic in the Nineteenth Century.

The Bulletin of Symbolic Logic


Author: N.A
Publisher: N.A
ISBN: N.A
Category: Logic, Symbolic and mathematical
Page: N.A
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Theories of types and proofs


Author: Masako Takahashi,Mitsuhiro Okada,Mariangiola Dezani-Ciancaglini,Nihon Sūgakkai
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: 295
View: 4572
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Choice

Publication of the Association of College and Research Libraries, a Division of the American Library Association
Author: N.A
Publisher: N.A
ISBN: N.A
Category: Academic libraries
Page: N.A
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Set theory

techniques and applications : Curaçao 1995 and Barcelona 1996 conferences
Author: C. A. Di Prisco
Publisher: Kluwer Academic Pub
ISBN: N.A
Category: Mathematics
Page: 226
View: 5503
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During the past 25 years, set theory has developed in several interesting directions. The most outstanding results cover the application of sophisticated techniques to problems in analysis, topology, infinitary combinatorics and other areas of mathematics. This book contains a selection of contributions, some of which are expository in nature, embracing various aspects of the latest developments. Amongst topics treated are forcing axioms and their applications, combinatorial principles used to construct models, and a variety of other set theoretical tools including inner models, partitions and trees. Audience: This book will be of interest to graduate students and researchers in foundational problems of mathematics.

Shadows of Cavernous Shades

Charting the Chiaroscuro of Realistic Computing
Author: Erik Persson
Publisher: N.A
ISBN: N.A
Category: Component software
Page: 727
View: 313
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American Book Publishing Record


Author: N.A
Publisher: N.A
ISBN: N.A
Category: American literature
Page: N.A
View: 3478
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Biographische Enzyklopädie deutschsprachiger Naturwissenschaftler


Author: Dietrich von Engelhardt
Publisher: N.A
ISBN: N.A
Category: Scientists
Page: 1320
View: 5472
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Mathematical Reviews


Author: N.A
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: N.A
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Grundzüge der theoretischen Logik


Author: David Hilbert,Wilhelm Ackermann
Publisher: N.A
ISBN: N.A
Category: Logic, Symbolic and mathematical
Page: 155
View: 6104
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Kurt Gödel: Leben und Werk


Author: John W. Jr. Dawson
Publisher: Springer
ISBN: 9783211831953
Category: Mathematics
Page: 298
View: 888
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Der österreichische Mathematiker Kurt Gödel (1906–1978) ist einer der herausragendsten Logiker des 20. Jahrhunderts. Seine Resultate sind von allerhöchster Bedeutung für die Mathematik, und zunehmend werden auch die Auswirkungen auf unser modernes Weltbild sichtbar. Mit dem Erscheinen von Douglas Hofstadters Buch "Gödel, Escher, Bach" wurde Gödels Werk einem breiten Publikum bekannt gemacht. John W. Dawson jr., Professor für mathematische Logik und Nachlaßverwalter von Gödel, zeichnet in dieser Biographie das Bild eines Mannes, dessen Werk allgemein für abstrus gehalten wurde und dessen Leben Elemente der Rationalität und der Psychopathologie vereinigt. Die lang erwartete Gödel-Biographie ergründet den Mythos des Mathematikers, indem sie gleichermaßen seine Ideen und Arbeit verständlich aufbereitet, als auch die Person zu erfassen vermag.

Kurt Gödel

Wahrheit & Beweisbarkeit
Author: Kurt Gödel,Eckehart Köhler,Bernd Buldt
Publisher: N.A
ISBN: 9783209038340
Category: Logicians
Page: 448
View: 4288
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