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.

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)

Previously published: Princeton University Press, 1940.

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)

Der Mathematiker Kurt Gödel hat über einen Zeitraum von 22 Jahren philosophische Bemerkungen niedergeschrieben. Sie sind in 15 Notizbüchern in der Kurzschrift Gabelsberger überliefert. Das erste Heft enthält allgemeine philosophische Überlegungen, die Hefte zwei und drei bestehen aus Gödels Individualethik. Die dann folgenden zeigen, 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. Mit Notizbuch ›Maximen III‹ beginnt ein neuer Abschnitt in Gödels Philosophischen Notizbüchern. Das Verhältnis verschiedener Disziplinen zueinander wird ebenso beleuchtet wie ihre jeweilige Bedeutung sowohl für den denkenden und arbeitenden Wissenschaftler als auch für das Wissenssystem. Zu nennen sind hier insbesondere Psychologie, Mathematik und Theologie. Daneben finden sich Überlegungen zur Heuristik und zu wissenschaftlichen Methoden. Auch die Rolle grundlegender oder einfacher Begriffe für das Denken wird erkennbar. (shrink)

This book is a record of a seminar on mathematical logic Kurt Gödel and Hans Hahn held in Vienna in 1931-32. The seminar proceedings, given in English translation, are a unique witness of the state of research in logic and foundations of mathematics right after Gödel had published his celebrated incompleteness theorems. The seminars explain Gödel's results in logic in detail, in contrast to his publications of the time that often were quite laconic and extremely short. This book also contains (...) Gödel's trial lecture on intuitionistic logic held in Vienna in 1933. The manuscript, recently found among the Gödel papers kept in Princeton, is preserved in Gödel's forgotten German shorthand and published here in an English translation. The book also adds an important aspect to the intellectual history of Vienna, as both Gödel and Hahn were members of the Vienna Circle group of philosophers and scientists. (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.

The initial volume of a comprehensive edition of Gödel's works, this book makes available for the first time in a single source all his publications from 1929 to 1936. The volume begins with an informative overview of Gödel's life and work and features facing English translations for all German originals, extensive explanatory and historical notes, and a complete biography. Volume 2 will contain the remainder of Gödel's published work, and subsequent volumes will include unpublished manuscripts, lectures, correspondence and extracts from (...) the notebooks. (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)

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)

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)

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.".

Brouwer has, as is well known, put up the claim that the law of excluded middle is no reliable means of proof in mathematics. His justification is that for many a mathematical proposition, it is today completely out of the question to decide whether it is true or false – and such a decision is perhaps impossible in general. The claim that each proposition should be either true or false, completely disregarding whether one can determine the one or the other, (...) that Brouwer holds to be senseless. Brouwer has later drawn various conclusions from his claim, in a more or less systematic way. It is only with his student Heyting that this question has been engaged with axiomatically. Namely, he has put up an axiom system for logic, more precisely the narrowest part of logic, the propositional calculus, from which the law of excluded middle cannot be derived. (shrink)

Lucky Gödel who, after his pièce célèbre on incompleteness of 1931, faced with enthusiasm the nascent science of mathematical logic. That very same year, he was invited to write a summary of the state of research in logic and foundations, under the title Mathematische Grundlagenforschung. His mathematics professor Hans Hahn made him also responsible for a Seminar on Mathematical Logic in Vienna in 1931/32, to which he contributed profusely through his results on completeness and incompleteness, and by many further results (...) about the nature of intuitionistic logic. (shrink)

The Russell-Whitehead calculus. – Truth value tables – Basic concepts – Basic propositions – Rules of inference – Theorems about negation and disjunction as propositional functions – Conjunction – Equivalence.

The philosophical remarks of Kurt Gödel have been handed down in the form of 15 notebooks. Gödel's overall philosophical project encompasses both his individual ethics and a philosophy of science within the context of a metaphysics. The present vo.

Preliminary Material /Hans Adler and Rainer Godel -- Formen des Nichtwissens im Zeitalter des Fragens /Hans Adler and Rainer Godel -- Das gewisse Etwas der Aufklärung /Hans Adler -- Zur Prekarität der Aufklärung. Vernunftkritik und das Paradigma der Anthropologie (Taine, Horkheimer / Adorno, Foucault, Lyotard) /Heinz Thoma -- Von den berechenbaren Grenzen des Nichtwissens zur Zeit der Aufklärung /Eberhard Knobloch -- L'effi cace de la raison /Bertrand Binoche -- Aufgeklärtes Nicht-Wissen /Rainer Enskat -- 'Fabelhaft' und 'wunderbar' in Aufklärungsdiskursen. Zur Genese (...) des modernen Mythosbegriffs /Stefan Matuschek -- Un-Möglichkeit der Ausnahme: Die Wunderkritik der Aufklärung /Daniel Weidner -- La sagacité de l'esprit. La conjecture au XVIIIe siècle /Denis Thouard -- Séjourner dans la contradiction. L'idée de «nouvelles Lumières» et les contradictions de l'universalism /Etienne Balibar -- Doing Without Knowing in Kant and Diderot: Experiments in Enlightenment /Claudia Brodsky -- Ways of Being Ignorant: A Field Guide /Arnd Bohm -- Wissenwollen trotz Nichtwissenkönnens. Von den Erkenntnisgrenzen der psychologischen und pädagogischen Wissenschaften des späten 18. Jahrhunderts /Gunhild Berg -- "Ein Abgrund innerer dunkeln Kräfte". Nichtwissen in den Bildungsdiskursen der Aufklärung /Stefan Metzger -- "There is nothing so intrinsically rational as religion is." Zum Versuch einer vernünftigen Entschlüsselung des Nicht-mehr-Wissbaren in der Frühaufklärung /Insa Kringler -- Ein falsches Bild. Oder: Was passiert, wenn Katharsis Form wird? Zu Lessings Emilia Galotti /Ralf Simon -- Die Schattenseiten der Seele: Zu Johann Georg Sulzers Theorie der dunklen Vorstellungen /Élisabeth Décultot -- Arten der Unwissenheit im anthropologischen Diskurs der Aufklärung /Angela Holzer -- Die Zone der Unbestimmtheit - Melancholiediskurs zwischen Wissen und Nichtwissen /Cornelia Bogen -- Die Produktion des Nicht-Wissens vom Selbst: Autobiographie bei Goethe und Hegel /Fritz Breithaupt -- "Charakter" - Die Konkurrenz einer metaphorischen und einer begriffl ichen Erkenntnisform des Nicht-/Wissens vom Menschen /Monika Sproll -- Unzuverlässige Leser - unzuverlässige Erzähler; oder: Literarische Wege aus dem Nicht-Wissen /Rainer Godel -- Labyrinthe der Aufklärung. Goethes Roman Wilhelm Meisters Lehrjahre /Harald Tausch -- Nichtwissen in Johann Gottfried Herders Geschichtsphilosophie /Luca Crescenzi -- Ins Narrenschiff mit Sokrates! Formen des Abderitismus bei Christoph Martin Wieland /Geneviève Espagne -- Zur Interdependenz von Wissen und Nichtwissen. Anthropologie als Aufklärungskritik bei Christoph Martin Wieland /Antonio Roselli -- Nichtwissen und Glaube. Zur Erkenntnistheorie und Ethik Friedrich Heinrich Jacobis /Mark-Georg Dehrmann -- Darstellung und Gegendarstellung von Aufklärung. Handlung, Erzählung und Schein in Schillers Geisterseher /Dirk Oschmann -- Wissen und Nicht-Wissen von anderen Menschen. Das Problem der Gemütererkenntnis von Gracián bis Schiller /Daniel Fulda -- Historische Vaterschaftsprozesse: Nichtwissen, verborgenes und verkehrtes Wissen bei Goethe, Hogarth und Kleist /Alexander Košenina -- Translating Knowledge and Prescribing Taste: English Enlightenment Critics /Sabine Volk-Birke -- Autorenverzeichnis /Hans Adler and Rainer Godel -- Namenregister /Hans Adler and Rainer Godel. (shrink)

Kurt Godel was the most outstanding logician of the 20th century. This second volume of Godel's works collects the remainder of his published work, covering the period 1938-1974. Each article or closely related group of articles is preceded by an introductory note that elucidates it and places it in its historical context.

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. (shrink)

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. 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. (shrink)

Nous publions sous ce titre la traduction française de l'essai philosophique de Kurt Gödel intitulé: «Is Mathematics Syntax of Language?» Inédit jusqu'à présent, l'original paraîtra dans le 3e volume des Collected Works de Gödel, dont la publication est imminente. Nous savons par Hao Wang que, le 15 mai 1953, Paul Arthur Schilpp avait invité Gödel à apporter sa contribution au volume consacré à Carnap dans The Library of Living Philosophers. Le manuscrit de Gödel «Carnap and the Ontology of Mathematics», devait (...) être remis au plus tard le 2 avril 1954. Dés le 2 juillet 1953, Gödel donnait son accord à Schilpp pour la rédaction d'un article intitulé «Some Observations on the Nominalistic View of the Nature of Mathematics.» Cependant, le volume consacré à Carnap paraîtra en 1963 sans qu'y figure la contribution de Gödel. En effet — toujours selon Wang — Gödel avait informé Schilpp, le 2 février 1959, qu'il renonçait à l'envoi de l'article, dont il avait composé, surtout en 1954–1955, plusieur s versions sous le titre «Is Mathematics Syntax of Language?» Pour expliquer cet abandon, Wang avance les raisons suivantes: En premier lieu, Gödel était bien convaincu d'avoir réfuté la thése syntaxique, mais il n'était pas satisfait de son argumentation en faveur du platonisme. Deuxiémement, les critiques de Gödel s'adressent aux théses soutenues par Carnap en 1934 plutôt qu'à celles que ce dernier défendra vingt ans aprés, et qui paraissent moins hostiles au platonisme de son interlocuteur. II pouvait paraître douteux, en troisieme lieu, que Carnap ait eu le temps de rédiger, en 1959, la réponse qu'il aurait du faire, conformément aux régies de la collection, à Gödel. Or, en 1943 déjà, Gödel avait voulu retirer sa contribution au volume consacré à Russell dans la même serie, parce que ce dernier n'avait pu répondre à l'article de Gödel dont la rédaction lui avait été communiquée trop tardivement. (shrink)

Der Mathematiker Kurt Gödel hat über einen Zeitraum von 22 Jahren (1934-1955) philosophische Bemerkungen, die so genannten Maximen Philosophie (Max Phil), niedergeschrieben. Sie sind in 15 Notizbüchern in der Kurzschrift Gabelsberger überliefert. Das erste Heft enthält allgemeine philosophische Überlegungen, die Hefte zwei und drei bestehen aus Gödels Individualethik. Die dann folgenden zeigen, 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. In Band 4 (...) ist fast die Hälfte der Bemerkungen Grundlagenfragen der Mathematik und Logik sowie Fragen der Philosophie der Mathematik gewidmet. 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. In Band 4 setzt Gödel sich mit Grundlagenfragen der Mathematik und Logik sowie der Philosophie der Mathematik auseinander. Zudem stehen das Verhältnis verschiedener wissenschaftlicher Disziplinen zueinander und deren spezifische Fragestellungen im Zentrum seiner Überlegungen. Zu nennen sind hier insbesondere Philosophie, Psychologie, aber auch Theologie. (shrink)

Over a period of 22 years (1934-1955), the mathematician Kurt Gödel wrote down philosophical remarks, the so-called Maximen Philosophie (Max Phil). They are preserved in 15 notebooks in Gabelsberger shorthand. The first booklet contains general philosophical considerations, booklets two and three consist of Gödel's individual ethics. The following books show that Gödel developed a philosophy of science in which he places his discussions on physics, psychology, biology, mathematics, language, theology and history in the context of a metaphysics. A complete, historical-critical (...) edition of Gödel's Philosophical Notebooks is now being prepared for the first time at the Kurt Gödel Research Centre of the Berlin-Brandenburg Academy of Sciences and Humanities. One volume will be published each year as part of this edition. Volume 5 is particularly interesting because Gödel reveals his satisfaction with it at the beginning of 'Maximen V'. He states that he has arrived at independent thought here, which has not primarily come about through discussion with other authors. Furthermore, in addition to numerous remarkable philosophical considerations, there is a previously unknown interpretation of his (second) incompleteness theorem that goes beyond the mathematical one. (shrink)

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)

Volume 2 contains both notebooks of "Time Management (Max) I and II" and thereby Gödel’s applied individual ethics, which he received among others through his teacher Heinrich Gomperz. Gödel thus incorporates the ethical ideal of self-perfection into his opus. The volume is prefaced by an introduction to relevant considerations from the ethics of the Stoics as well as ancient dietetics, which provide the philosophical background to understand Gödel’s approach. In addition, editor Eva-Maria Engelen presents how this fits into the context (...) of Gödel’s Philosophical Notebooks. (shrink)

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)