Building on previous work by Mints, Buchholz and Schwichtenberg, a simplified version of continuous normalization for the untyped λ-calculus and Gödel’s is presented and analysed in the coalgebraic framework of non-wellfounded terms with so-called repetition constructors.The primitive recursive normalization function is uniformly continuous w.r.t. the natural metric on non-wellfounded terms. Furthermore, the number of necessary repetition constructors is locally related to the number of reduction steps needed to reach the normal form and its size.It is also shown how continuous normal (...) forms relate to derivations of strong normalizability in the typed λ-calculus and how this leads to new bounds for the sum of the height of the reduction tree and the size of the normal form.Finally, the methods are extended to an infinitary λ-calculus with ω-rule and permutative conversions and this is used to derive a strong form of normalization for an iterative version of Gödel’s system , leading to a value table semantics for number-theoretic functions. (shrink)

ZusammenfassungP. Bernays hat darauf hingewiesen, dass man, um die Widerspruchs freiheit der klassischen Zahlentheorie zu beweisen, den Hilbertschen flniter Standpunkt dadurch erweitern muss, dass man neben den auf Symbole sich beziehenden kombinatorischen Begriffen gewisse abstrakte Begriffe zulässt, Die abstrakten Begriffe, die bisher für diesen Zweck verwendet wurden, sinc die der konstruktiven Ordinalzahltheorie und die der intuitionistischer. Logik. Es wird gezeigt, dass man statt deesen den Begriff einer berechenbaren Funktion endlichen einfachen Typs über den natürlichen Zahler benutzen kann, wobei keine anderen (...) Konstruktionsverfahren für solche Funktionen nötig sind, als einfache Rekursion nach einer Zahlvariablen und Einsetzung von Funktionen ineinander .P. Bernays has pointed out that, in order to prove the consistency of classical number theory, it is necessary to extend Hilbert's finitary stand‐point by admitting certain abstract concepts in addition to the combinatorial concepts referring to symbols. The abstract concepts that so far have been used for this purpose are those of the constructive theory of ordinals and those of intuitionistic logic. It is shown that the concept of a computable function of finite simple type over the integers can be used instead, where no other procedures of constructing such functions are necessary except simple recursion by an integral variable and substitution of functions in each other. (shrink)

First English translation of revolutionary paper that established that even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. Introduction by R. B. Braithwaite.

Previously published: Princeton University Press, 1940.

It is known that most existing cosmology models do not include rotation, with few exceptions such as rotating Bianchi and rotating Godel metrics. Therefore in this paper we aim to discuss four possible ways to model rotating universe, including Nurgaliev’s Ermakov-type equation. It is our hope that the new proposed method can be verified with observations, in order to open new possibilities of more realistic nonlinear cosmology models.

Kurt Gödel 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, 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. 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, 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. (shrink)

Kurt Gödel 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, 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, 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. (shrink)

Kurt Gödel was the greatest logician of this century. This third volume of his collected works consists of previously unpublished material, both essays and lectures.

Die Studie zeigt, dass es eine Verkürzung des aufklärerischen Diskurses wäre, betrachtete man das Vorurteil nur als einen der von der Aufklärung anvisierten Angriffspunkte. Der Vorurteilsdiskurs wird vielmehr zum Modus der Selbstaufklärung der Aufklärung. Im literarischen und popularphilosophischen Nachdenken über das Vorurteil erschüttern anthropologiebasierte Argumente die rationale Gewissheit, mit der Vorurteile kritisiert und durch Wahrheit ersetzt werden sollten. Fragen nach der Funktion von Vorurteilen und nach Wegen, mit ihnen umzugehen, ersetzen philosophische Begriffe und Typologien.

Over a period of 22 years (1934-1955), the mathematician Kurt Gödel wrote down a series of philosophical reflections, the so-called Philosophical Remarks (Max Phil). They have been handed down in 15 notebooks written in Gabelsberg shorthand. The first notebook contains general philosophical reflections. Notebooks two and three consist of Gödel's individual ethics. The notebooks that follow clearly show that Gödel had designed a philosophy of science in which he placed his discussions of physics, psychology, biology, mathematics, language, theology, and history (...) within the context of a particular metaphysics. For the first time, the Kurt Gödel Research Center of the Berlin-Brandenburg Academy of Sciences is preparing a complete, historical-critical edition of Gödel's philosophical notebooks. A volume will published each year as a part of this edition. Volume 1, which Gödel entitled "Philosophy I Max 0," covers Gödel´s own philosophical reflections as well as those of other authors that were important to him. These were placed by Gödel at the beginning of his overall philosophical project. The introduction of the editor Eva-Maria Engelen provides an overview of this particular body of work. (shrink)

P. Bernays has pointed out that, in order to prove the consistency of classical number theory, it is necessary to extend Hilbert's finitary standpoint by admitting certain abstract concepts in addition to the combinatorial concepts referring to symbols. The abstract concepts that so far have been used for this purpose are those of the constructive theory of ordinals and those of intuitionistic logic. It is shown that the concept of a computable function of finite simple type over the integers can (...) be used instead, where no other procedures of constructing such functions are necessary except simple recursion by an integral variable and substitution of functions in each other (starting with trivial functions). (shrink)

Badania nad podstawami matematyki przyniosły w ostatnich dziesięcioleciach wyniki, które wydają mi się ciekawe nie tylko dla nich samych, lecz także z uwagi na wnioski, jakie płyną z nich w odniesieniu do tradycyjnych problemów filozoficznych dotyczących natury matematyki. Same wyniki są dość szeroko znane, mimo to jednak sądzę, że warto raz jeszcze przedstawić je w zarysie, zwłaszcza w obliczu faktu, że dzięki pracy szeregu matematyków zyskały one znacznie doskonalszą formę, niż miały pierwotnie. Największy postęp, mający decydujące znaczenie dla tych wyników, (...) stał się możliwy dzięki precyzyjnemu zdefiniowaniu pojęcia skończonej procedury. Definicję taką można uzyskać na różne sposoby, które w efekcie dają dokładnie to samo pojęcie. Najbardziej zadowalający sposób polega moim zdaniem na sprowadzeniu pojęcia skończonej procedury do pojęcia maszyny o skończonej liczbie części, co uczynił brytyjski matematyk Turing. Co do filozoficznych konsekwencji wspomnianych wyników, nie sądzę, by kiedykolwiek należycie je przedyskutowano, czy choćby zwrócono na nie uwagę. (shrink)

Der Mathematiker Kurt Gödel hat über einen Zeitraum von 22 Jahren philosophische Bemerkungen, die so genannten Maximen Philosophie, niedergeschrieben. Sie sind in 15 Notizbüchern in der Kurzschrift Gabelsberger überliefert. Das erste Notizheft enthält allgemeine philosophische Überlegungen, die Hefte zwei und drei bestehen aus Gödels Individualethik. Die dann folgenden Notizbücher machen deutlich, dass Gödel eine Wissenschaftsphilosophie entworfen hat, in der er seine Erörterungen zu Physik, Psychologie, Biologie, Mathematik, Sprache, Theologie und Geschichte in den Kontext einer Metaphysik stellt.Erstmals wird nun an der (...) Kurt-Gödel-Forschungsstelle der Berlin-Brandenburgischen Akademie der Wissenschaften eine vollständige, historisch-kritische Edition von Gödels philosophischen Notizbüchern vorbereitet. Im Rahmen dieser Edition erscheint jährlich ein Band. Band 1, von Gödel mit ›Philosophie I Max 0‹ betitelt, umfasst philosophische Überlegungen Gödels sowie anderer für ihn bedeutender Autoren, die er an den Anfang seines philosophischen Gesamtprojektes gestellt hat. Die Einleitung der Herausgeberin Eva-Maria Engelen bietet einen Überblick über das Korpus. (shrink)

This study explores the substantive relevance of the young Schleiermacher s sermons to his overall theological and philosophical oeuvre. By engaging in a rhetorical analysis of the sermons, it develops insights into the ethical content of the sermons, their underlying theological and philosophical traditions, and into the basic function of mediation in interpreting the world and in homiletics.".

Après avoir recueilli l'enseignement de la sagesse millénaire indienne auprès de Ramana Maharshi et Krishna Menon, Roger Godel a rassemblé dans ses Essais sur l'expérience libératrice (parus en 1952) les conclusions d'une recherche générale sur l'éveil. Son approche de l'éveil se veut d'abord radicalement et volontairement scientifique. L'expérience transcendante est aux yeux de Godel une terra incognita que les scientifiques, et en particulier les psychologues, doivent prendre comme objet d'études faute de quoi ils s'exposent à ne jamais atteindre (...) la vérité ultime. Mais Roger Godel ne fait pas œuvre d'intellectuel en cherchant à décrire l'éveil ; il ne regarde pas l'éveil comme un objet lointain et inconnu ; il écrit, au contraire, en puisant dans sa propre expérience spirituelle. Godel n'est pas seulement un philosophe en quête de sagesse, mais un sage, un homme accompli. Il montre dans ses Essais que les mystiques d'Orient comme d'Occident conduisent toutes à une même expérience spirituelle et qu'il est temps pour l'homme de s'éveiller à sa propre dimension intérieure de divinité. (shrink)