Kurt Godel

Text nabízí rozbor části osobní korespondence matematika a filosofa Kurta Gödela (1906–1978), jejímž ústředním tématem je otázka posmrtného života. Primárně vychází ze čtyř dopisů, které Gödel v období mezi červencem a říjnem 1961 adresoval své matce. V příspěvku nabízím rekonstrukci, analýzu a zhodnocení argumentu, který Gödel na podporu posmrtného života matce předkládá. Zkoumané dopisy jsou v textu pojímány jako specifický typ literatury spojený s naplněním konkrétních argumentačních cílů. K analýze argumentace v dopisech využívám nástroje, které do teorie argumentace zavádějí v rámci programu tzv. Nové rétoriky Chaim Perelman a Lucie Olbrechts-Tyteca. Klíčovým přínosem této teorie je zaměření na publikum (univerzální/konkrétní) a s ním spojené zkoumání nástrojů vhodných k jeho přesvědčení. Ukazuji důvody, proč považuji Gödelova čtenáře z hlediska této teorie za tzv. konkrétní publikum, věnuji se charakteristikám, které Gödel tomuto publiku zřejmě přisuzuje, a prostředkům, které považuje za adekvátní vzhledem k jeho úspěšnému přesvědčení. V závěru ukazuji, proč Gödelova argumentační strategie s největší pravděpodobností nebyla jeho čtenářem akceptována.

Kurt Gödel è stato certamente il logico più influente del Novecento. I suoi lavori, visti come delle pietre miliari e come tali richiamati ancora oggi nella letteratura specialistica, investono aspetti fondamentali per la ricerca in campo matematico, quali la relazione tra verità e dimostrabilità o questioni relative alla natura e ai limiti della conoscenza matematica. Questo profilo nasce dal tentativo di correggere l’impressione che la grandezza del personaggio possa essere ricondotta a uno dei suoi molteplici contributi in particolare, come ad esempio il teorema di incompletezza per le teorie assiomatiche al quale egli deve gran parte della sua fama. Al contrario, il volume cerca di offrire una panoramica per quanto possibile esaustiva dell’opera logica e matematica di Gödel, così da valutarne meglio la portata.

Tesis doctoral, defendida en junio de 2019 en la Universidad de Buenos Aires, en la que se analiza el problema de la existencia de los objetos matemáticos. Se estudian las posturas actuales y se propone una solución propia basada en el convencionalismo.

Resumen: Kurt Gödel demostró en 1931, que para todo sistema formal Z recursivo lo suficientemente potente como para derivar los axiomas de Pea-no y que además se suponga como consistente, se tiene que en el sistema hay proposiciones indecidibles, es decir, el sistema no es completo. Por otra parte, Gödel probó que si el sistema Z es consistente entonces no se puede derivar en Z una proposición que afirme la consistencia de Z. Estos resul-tados son los que se conocen como Primer Teorema de Incompletitud Gödel y Segundo Teorema de Incompletitud de Gödel. Dichos resultados tienen un gran impacto sobre la investigación de los fundamentos de la matemática que venía gestándose en los primeros treinta años del siglo pasado, y tiene ade-más consecuencias sobre la filosofía de la matemática de dicha época. Este artículo se encuentra estructurado en tres partes: En una primera parte nos ocupamos de la formulación de los Teoremas de incompletitud y las ideas princi-pales de su demostración en cada caso. Seguidamente mostraremos una apli-cación del Segundo Teorema de Incompletitud en la teoría de conjuntos referente a los cardinales inaccesibles. Por último, desarrollaremos las consecuencias filosóficas que los Teoremas de incompletitud de Gödel tienen sobre el proyecto meta-matemático de David Hilbert.

Es un libro acerca de cómo los logros creativos del lógico Kurt Gödel, el artista M. C. Escher y el compositor Johann Sebastian Bach interactúan. Como el autor indica: "me di cuenta de que Gödel, Escher y Bach eran solamente sombras dirigidas en diversas direcciones de cierta esencia sólida central. Intenté reconstruir el objeto central, y llegué con este libro." El tema central del libro es más abstracto. Hofstadter se pregunta: "¿Siguen las palabras y las ideas reglas formales, o no?" En el prefacio de la edición del vigésimo aniversario, Hofstadter lamenta que su libro haya sido malinterpretado como una mezcolanza de cosas ingeniosas sin un tema central. Indicó: "GEB es una tentativa muy personal de decir cómo es que los seres animados pueden salir de la materia inanimada. ¿Qué es un "uno mismo", y cómo puede un "uno mismo" salir de cosas tan faltas de ser como una piedra o un charco?"

The alleged proof of the non-mechanical, or non-computational, character of the human mind based on Gödel's incompleteness theorem is revisited. Its history is reviewed. The proof, also known as the Lucas argument and the Penrose argument, is refuted. It is claimed, following Gödel himself and other leading logicians, that Anti-Mechanism is not implied by Gödel's theorems alone. The refutation is given the strongest form in the present paper: general theorems implying the inconsistency of Lucas's arithmetic and the semantic inadequacy of Penrose's arithmetic are demonstrated. On the other hand, the limitations regarding our capacity of mechanizing or programming the mind are indicated, as well as two other corollaries of Gödel's theorems: we cannot prove that we are consistent (Gödel's Unknowability Thesis), and we cannot fully describe our notion of a natural number.

THIS PAPER IS AN EXTENSION OF FORMER PAPERS OF THE AUTHOR (AND LARGELY BASED ON HIS BOOK IN POLISH), TO BE PUBLISHED IN A SPECIAL ISSUE OF "Studia Semiotyczne" TOGETHER WITH A NUMBER OF RELATED PAPERS.

PLEASE QUOTE THE FINAL VERSION, Studia semiotyczne  XXXIV, nr 1(2020), 9–56; DOI: 10.26333/sts.xxxiv1.02. [ALSO AVAILABLE HERE AT THIS ACADEMIA.EDU PAGE]

In this article I  focus on some of the paths that help us to understand the philosophical question of time. The interpretative framework is composed of Einstein’s theory of relativity, against which the observations of Gödel and Heidegger on this new worldview emerge as a counterpoint. The thesis I try to support, according to some surprising remarks by Gödel, is that the theory of relativity originates also from a critical analysis of the concept of time in idealist philosophy. Einstein, somehow, follows the path opened up by Kant: time, as reality itself, is nothing. Nevertheless, according to relativity, time still has something objectively real, though its reality does not belong to the subjective world of our perceptions, but to a ‘mathematically real’ world of four dimensions.

lecturer on alchemy and its modern applications. www.DWHauck.com Go directly to the start of the text. Abstract The alchemists believed that no transformation— whether in the laboratory, in the body, or in the soul—could succeed without the presence of a mysterious ingredient known as the Materia Prima (First Matter). Nothing was more important to alchemist's work than this energetic essence, which they believed could be extracted from any substance and actually rendered tangible and visible. This paper examines the nature and properties of the elusive First Matter. We will look at how the alchemists exposed it, accumulated it, and worked with it to accomplish the regeneration of the substance at hand. We will explore how the concept of the First Matter evolved in both the scientific and esoteric traditions to become a fundamental, although often unrecognized, tenet of both modern Hermeticism and modern physics.

Kurt Gödel left with his student Dana Scott two pages of notes in which he sketched a new version of Anselm's ontological proof of God's existence.  In his most recent book, Howard Sobel spends the greater part of a chapter dedicating his considerable talents to an elucidation and critique of Gödel's argument, as well as to an emended version of that argument proposed by Anthony Anderson

"Guillaume Durand et Michel Weber (éditeurs), Les principes de la connaissance naturelle d’Alfred North Whitehead — Alfred North Whitehead’s Principles of Natural Knowledge. Actes des Journées d’étude internationales tenues à l’Université de Nantes, les 3 et 4 octobre 2005 — Proceedings of the Fourth International “Chromatiques whiteheadiennes” Conference. Publiés avec le concours du Département de philosophie de l'Université de Nantes, Frankfurt / Paris / Lancaster, ontos verlag, Chromatiques whiteheadiennes VIII, 2007. (280 p. ; ISBN 978-3-938793-64-0 ; 89 €)
Les quatrièmes journées « Chromatiques » interrogèrent la philosophie des sciences de la nature qui est caractéristique de l’ « époque de Londres », prolongeant ainsi les travaux menés à Liège (2001) puis à Louvain-la-Neuve (2003). L’objet central de cette rencontre était l’œuvre pionnière dans la philosophie whiteheadienne de la nature : An Enquiry Concerning the Principles of Natural Knowledge (1919/1925), dont tous les principaux traits — à commencer par la méthode de l'abstraction extensive — font ici l'objet de nouvelles réflexions.
The fourth international « Chromatiques whiteheadiennes » conference was devoted to the philosophy of the natural sciences that is characteristic of Whitehead’s « London Epoch ». Continuing the efforts of the Liège (2001) and Louvain-la-Neuve (2003) meetings, the conference focused on An Enquiry Concerning the Principles of Natural Knowledge (CUP, 1919/1925) which constitutes Whitehead’s pioneering work in the field. All its main traits — starting with the Method of Extensive Abstraction — are here questioned anew.
"

The main question treated in this book is the relation between statements (sentences, propositions) and reality. The book is the first volume of a larger work about the fundamental (ontological, epistemological, linguistic) character of logic. My viewpoint is a realistic and metaphysical view of logic and truth influenced to a very large extent by works of Plato, Aristotle, Frege, Russell, Gödel, and Hardy, among others. Hence the title "Logical Forms", inspired by Plato. I list below the titles of the chapters in lieu of an abstract.

Introduction. Logic, ontology, and epistemology.
1. Truth, description, and identification.
2. The True and the False.
3. Use, mention, and Russell's theory of descriptions.
4. Arguments for Frege's thesis.
5. Objections to facts.
6. Truth, denotation, and interpretations.
7. Tarski's semantic conception of truth.
8. The True and the False revisited: Frege's logic.
9.Structuring reality: properties, sets, and states of affairs.
10. Identity and extensionality.
11. Senses.
12. Truth and correspondence.

A translation to Spanish--Formas Lógicas I--was published by Eudeba in 2015 and I added some information and a link to Eudeba in TRANSLATIONS.

A translation to Portuguese--Lógica, ontologia e epistemologia --of the Introduction was published in 2006 and I added the pdf in TRANSLATIONS.

This volume continues the discussion initiated in Part 1 of "Logical Forms" and deals with issues of language, logic, proof, and knowledge. I list the chapter titles:
13. Language, meaning, and reference.
14. Syntax and semantics.
15 Grammar and logical form.
16. Propositional logic.
17. Predicate logic.
18. Grammar and logical truth.
19. Proof and logical deduction.
20. Proof and proving.
21. Proof and truth.
22. The tyranny of belief.
23. Ockham's razor.
24. Knowledge and justification.
25. Logic and knowledge.
Epilogue. Plato, Zeno, Parmenides, and Frege.

A translation to Portuguese of chapter 25--Lógica e conhecimento--was published in 2007 and I added the pdf in TRANSLATIONS.

This book discusses the problem of mathematical knowledge, and its broader philosophical ramifications. It argues that the challenge to explain the (defeasible) justification of our mathematical beliefs (‘the justificatory challenge’), arises insofar as disagreement over axioms bottoms out in disagreement over intuitions. And it argues that the challenge to explain their reliability (‘the reliability challenge’), arises to the extent that we could have easily had different beliefs. The book shows that mathematical facts are not, in general, empirically accessible, contra Quine, and that they cannot be dispensed with, contra Field. However, it argues that they might be so plentiful that our knowledge of them is unmysterious. The book concludes with a complementary ‘pluralism’ about modality, logic and normative theory, highlighting its revisionary  implications. Metaphysically, pluralism engenders a kind of perspectivalism and indeterminacy. Methodologically, it vindicates Carnap’s pragmatism, transposed to the key of realism.

Questo lavoro parte dall’idea che la nostra individualità, intesa principalmente come libertà, è facilmente modellabile e assoggettabile da parte di un potere incontrastabile quale è quello che pare oggi imporsi. L’esperienza che sta alla base di questo assunto teoretico è il tragico evento storico costituito dai totalitarismi.  È necessario dunque comprendere siffatti eventi tanto rispetto alla dimensione storica, quanto nel loro aspetto sovra-storico e concettuale, cioè rispetto alla forma di dominio che essi rappresentano.
Kurt Gödel nei primi anni trenta dimostra che la sfera di ciò che è razionalmente dimostrabile non coincide con quella di ciò che è vero. Questo rilievo conduce a individuare nel capolavoro letterario di George Orwell, Nineteen Eighty-Four, un riferimento centrale per la riflessione teorica sul totalitarismo.
La tesi si struttura in tre parti. La prima ricostruisce, brevemente, una genealogia dei totalitarismi principali sia a livello storiografico sia a livello teoretico e inquadra storicamente e biograficamente l’autore e il suo pensiero come espressione di quello che potremmo chiamare, seguendo una certa linea di pensiero, “elenchos sociale”. Nella seconda parte si analizza ciò che è peculiare del totalitarismo perfetto di stampo orwelliano e si indaga come il potere utilizzi la tortura e il dolore per frantumare la soggettività individuale facendo divenire l’uomo, nella sua interezza, parte integrante di un sistema irrazionale. La terza parte, invece, si pone, anche a partire da Kant, il problema di un possibile superamento la logica del negativo che caratterizza siffatto potere.

The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: What does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such as Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics.

The article discusses the hollow myth of modernity and the misguided movement in the present time that puts great faith in a technological future. Critique is directed against subjectivistic philosophy in the guise of Kantianism, Hegelianism, and their continuation in postmodern philosophy. The thought of Neo-Hegelian psychologist/philosopher Wolfgang Giegerich is refuted.

Keywords: Hegel, Romanticism, subjectivism, technology, puer aeternus, individual, collectivism, meaning, World Soul, individuation, Self, Goedel, Kant, Giegerich.

Último sermón de la iglesia del naturalismo fundamentalista por el pastor Hofstadter. Al igual que su mucho más famoso (o infame por sus incesantemente errores filosóficos) trabajo Godel, Escher, Bach, tiene una plausibilidad superficial, pero si se entiende que se trata de un científico rampante que mezcla problemas científicos reales con los filosóficos (es decir, el sólo los problemas reales son los juegos de idiomas que debemos jugar) entonces casi todo su interés desaparece. Proporciono un marco para el análisis basado en la psicología evolutiva y el trabajo de Wittgenstein (ya actualizado en mis escritos más recientes).

Aquellos que deseen un marco completo hasta la fecha para el comportamiento humano de la moderna dos sistemas punto de vista puede consultar mi libros Talking Monkeys 3ª ed (2019), Estructura Logica de Filosofia, Psicología, Mente y Lenguaje en Ludwig Wittgenstein y John Searle 2ª ed (2019), Suicidio pela Democracia 4ª ed (2019), La Estructura Logica del Comportamiento Humano (2019), The Logical Structure de la Conciencia (2019, Comprender las Conexiones entre Ciencia, Filosofía, Psicología, Religión, Política y Economía, Historia y Literatura (2019), Delirios Utópicos Suicidas en el siglo 21  5ª  ed (2019), Observaciones sobre Imposibilidad, Incompletitud, Paraconsistencia, Indecidibilidad, Aleatoriedad, Computabilidad, Paradoja e Incertidumbre en Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych  Berto, Floyd, Moyal-Sharrock y Yanofsky(2019) y otros.

"Podría ser justamente preguntado qué importancia tiene la prueba de Gödel para nuestro trabajo. Para un pedazo de matemáticas no puede resolver problemas del tipo que nos molesten. --La respuesta es que la situación, en la que tal prueba nos trae, es de interés para nosotros. ' ¿Qué vamos a decir ahora? ' --Ese es nuestro tema. Sin embargo, extraña que suene, mi tarea en lo que concierne a la prueba de Gödel parece meramente consistir en aclarar lo que tal proposición como: "Supongamos que esto se puede demostrar" significa en matemáticas. " Wittgenstein "Comentarios sobre los fundamentos de las matemáticas" p337 (1956) (escrito en 1937).

"Mis teoremas sólo muestran que la mecanización de las matemáticas, es decir, la eliminación de la mente y de las entidades abstractas, es imposible, si uno quiere tener una base satisfactoria y un sistema de matemáticas. No he probado que haya preguntas matemáticas que sean indescifrables para la mente humana, pero sólo que no hay máquina (o formalismo ciego) que pueda decidir todas las preguntas de la teoría numérica, (incluso de un tipo muy especial) .... No es la estructura misma de los sistemas deductivos que está siendo amenazado con un descompostura, sino sólo una cierta interpretación de la misma, es decir, su interpretación como un formalismo ciego. " Gödel "Obras Recogidas" VOL 5, p 176-177. (2003)

"Toda inferencia tiene lugar a priori. Los acontecimientos del futuro no pueden deducirse de los del presente. La superstición es la creencia en el nexo causal. La libertad de la voluntad consiste en el hecho de que las acciones futuras no se pueden conocer ahora. Sólo podíamos conocerlas si la causalidad fuera una necesidad interna, como la de la deducción lógica. --La conexión del conocimiento y lo que se conoce es la necesidad lógica. ("A sabe que p es el caso" no tiene sentido si p es una tautología.) Si por el hecho de que una proposición es obvia para nosotros, no sigue que es verdad, entonces la evidencia no es ninguna justificación para creer en su verdad. " TLP 5,133--5,1363

En este artículo se abordan algunos de los caminos que nos ayudan a comprender la cuestión filosófica del tiempo. El marco interpretativo es compuesto por la teoría de la relatividad de Einstein, a la cual hacen de contrapunto las reflexiones de Gödel y de Heidegger sobre esa nueva Gestalt. Una de las tesis que se intenta argumentar –de acuerdo con algunas asombrosas observaciones de Gödel– es que la teoría de la relatividad nace también de una discusión crítica del concepto de tiempo en la filosofía idealista. Einstein, en cierto sentido, sigue el camino trazado por el propio Kant: el tiempo, como realidad en sí, no es nada. Pero a la vez, en la teoría de la relatividad el tiempo sigue manteniendo algo objetivamente real, aunque su realidad no pertenece al mundo subjetivo de nuestras percepciones, sino a un mundo matemáticamente real de cuatro dimensiones.

*Apresentação feita na Jornada de Pós Graduação 2019 da UFSM* O presente trabalho tem como objetivo apresentar de modo simples as bases conceituais e técnicas lógico-matemáticas utilizadas por Kurt Gödel em seu artigo Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I para demonstrar que, em um sistema formal T que comporte o mínimo de Aritmética em seu conjunto de sentenças básicas é incompleto, ou seja, há sentenças enunciadas formalmente em T que são verdadeiras, porém indemonstráveis neste sistema (Primeiro Teorema de Gödel), e que não há como demonstrar a consistência do sistema formal por ele mesmo (Segundo Teorema). Focando na prova do primeiro teorema, faremos uma introdução aos termos gerais dos quais exige-se um entendimento prévio. Em seguida abordaremos um breve contexto histórico, os tipos de paradoxos e quais são relacionados a estratégia e estrutura de prova de Gödel (assim como de outras provas de incompletude), bem como sua técnica de aritmetização de sentenças, o método de prova por diagonalização e o recurso da autorreferência em sentenças de T e a exposição do Teorema do Ponto Fixo, apresentando ao final o esboço da prova do primeiro teorema da incompletude como um caso particular de tal teorema em T.

Continental philosophy of science has developed alongside mainstream analytic philosophy of science. But where continental approaches are inclusive, analytic philosophies of science are not– and they exclude not merely Nietzsche’s philosophy of science but Gödel’s philosophy of
physics. As a radicalization of Kant, Nietzsche’s critical philosophy of science puts science in question and Nietzsche’s critique of the methodological foundations of classical philology
bears on science, particularly evolution as well as style (in art and science). In addition to the critical (in Mach, Nietzsche, Heidegger but also Husserl just to the extent that continental
philosophy of science tends to depart from a reflection on the crisis of foundations), other continental philosophies of science include phenomenology (Husserl, Bachelard, Merleau-Ponty, etc.) and hermeneutics (Heidegger, Gadamer, Heelan, etc.), especially incorporating history of science (Nietzsche, Mach, Duhem, Butterfield, Feyerabend, etc.). Examples are drawn from the philosophy of sciences (chemistry, geology, and biology) other than physics.

In the constructivist movement, which started with Brouwer’s attack on the principles of classical logic and mathematics, the history of mathematics seems to repeat itself in a specific, self-conscious way. The main objective of this paper is to describe this approach by adopting the phenomenological method of Hegel. Its starting point consists in looking at knowledge as a continuous, yet painful, process—the Calvary of the Spirit—with its stations corresponding to the naive, direct concepts of experience as based on sense certainty or belief in the independent realms of objects and their transformation into more sophisticated, socially charged theories. The signs of this advance are the patterns of self-consciousness, such as Cantor’s diagonal results, adopted by constructivism in a different but still powerful way. The key concept against which the progress of this development will be measured within the constructivist movement is the concept of proof, particularly with respect to Gödel theorems and the resulting split of knowledge into proof and truth. I will read this split, with the help of Lorenzen and Brandom, as a relative differentiation between two co-dependent aspects of self-consciousness that are to be prospectively conceived as two idealized dialogue partners.

As Bernays maintained in his famous 1935 article "On Platonism in Mathematics", in mathematical practice objects, functions, relations, properties, structures, etc. are treated as entities that exist independently of our discourse and of our constructions. Bernays maintains that this form of platonism is essentially a manner of speaking, which does not involve a commitment to a strict form of platonism. In this note I defend Bernays' position and propose a more systematic formulation for this form of platonism, combining it with ideas of Frege and of Gödel.

“What do Gödel theorems tell us about Hilbert’s solvability thesis?”, in: M. Peliš (ed.), Logica Yearbook 08, College Publications, London 2009, pp. 83–94

The paper's goal is to present Gödel theorems not as a direct refutation of Hilbert’s solvability axiom (according to which all problems which human mind poses must be solvable) but only as an impulse to phrase it with more caution, in such a way that some famous unsolvable questions such as the Continuum Hypothesis are no longer regarded as real problems.

On pense généralement que l'impossibilité, l'incomplétdulité, la paracohérence, l'indécidabilité, le hasard, la calcul, le paradoxe, l'incertitude et les limites de la raison sont des questions scientifiques physiques ou mathématiques disparates ayant peu ou rien dans terrain d'entente. Je suggère qu'ils sont en grande partie des problèmes philosophiques standard (c.-à-d., jeux de langue) qui ont été la plupart du temps résolus par Wittgenstein plus de 80 ans.
Je fournis un bref résumé de quelques-unes des principales conclusions de deux des plus éminents étudiants du comportement des temps modernes, Ludwig Wittgenstein et John Searle, sur la structure logique de l'intentionnalité (esprit, langue, comportement), en prenant comme point de départ La découverte fondamentale de Wittgenstein, à savoir que tous les problèmes véritablement « philosophiques » sont les mêmes, les confusions sur la façon d'utiliser la langue dans un contexte particulier, et donc toutes les solutions sont les mêmes— en regardant comment la langue peut être utilisée dans le contexte en cause afin que sa vérité (Conditions de satisfaction ou COS) sont claires. Le problème fondamental est que l'on peut dire n'importe quoi, mais on ne peut pas signifier (état clair COS pour) toute déclaration arbitraire et le sens n'est possible que dans un contexte très spécifique.
Je dissé que quelques écrits de quelques-uns des principaux commentateurs sur ces questions d'un point de vue wittgensteinien dans le cadre de la perspective moderne des deux systèmes de pensée (popularisé comme «penser vite, penser lentement»), en utilisant une nouvelle table de intentionnalité et la nomenclature de nouveaux systèmes doubles. Je montre qu'il s'agit d'un puissant heuristique pour décrire la vraie nature de ces questions scientifiques, physiques ou mathématiques putatives qui sont vraiment mieux abordés comme des problèmes philosophiques standard de la façon dont la langue doit être utilisée (jeux de langue dans Wittgenstein terminologie).

Errata in Henkin’s 1950 type-theory completeness paper.
  The first paragraph of Leon Henkin’s influential and widely-read article [1] reads as follows.

The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system.

  As it is written the second sentence says that each formula is a provable logical truth: a patent absurdity. Of course, what was intended was more like following, which moves the clause beginning ‘which becomes’ so it modifies ‘formula of the calculus’.

Roughly speaking, this proof demonstrates that each formula of the calculus which becomes a true sentence under every one of a certain intended class of interpretations of the formal system is a formal theorem.

  Taking more advantage of the disclaimer ‘roughly speaking’, he could say:

Roughly speaking, this proof demonstrates that each formula which is true under every interpretation is a theorem.

  Almost 20 years later, Jaakko Hintikka [2] reprinted [1] along with other important articles. Hintikka noted errata in another Henkin article but not in [1]. There are other mistakes in [1]. Moreover, it contains several passages that, if not actual mistakes, are regrettable. We list and analyse glitches in [1] in order to facilitate study of this landmark article. We also discuss the old question of whether, as logicians including Burton Dreben have charged [personal communication], [1]’s title is misleading in that it seems to contradict an easy consequence of Gödel’s 1931 incompleteness paper. The consequence in question is explicitly mentioned in the second paragraph of [1].
[1] LEON HENKIN, Completeness in the theory of types. Journal of Symbolic Logic, vol. 15 (1950) pp. 81–94.
[2] JAAKKO HINTIKKA, Philosophy of Mathematics, Oxford UP, 1969.
Acknowledgements: Mark Brown, William Demopoulos, Thomas Drucker, Rodolfo Ertola, William Frank, Richard Grandy, Idris Samawi Hamid, Allen Hazen, Leonard Jacuzzo, Miguel León, Kristo Miettinen, Joaquin Miller, Frango Nabrasa, David Plache, Sriram Nambiar, José Miguel Sagüillo, Michael Scanlan, Stewart Shapiro, James Smith, Roberto Torretti, Alasdair Urquhart, Albert Visser, George Weaver, George Williams, Stanley Ziewacz, and others—some of which disagree with some of my points or with my way of making them.

In particular, distinguished and accomplished logicians including Allen Hazen, Christopher Menzel, Alasdair Urquhart, and Albert Visser think that Henkin’s expression of completeness is correct as it is—awkwardness notwithstanding. To be clear, they believe and argue that I am wrong to imply that this is a mistake on Henkin’s part.
WHAT IS YOUR VIEW?

This book focuses on Albert Einstein and his interactions with, and responses to, various scientists, both famous and lesser-known. It takes as its starting point that the discussions between Einstein and other scientists all represented a contribution to the edifice of general relativity and relativistic cosmology. These scientists with whom Einstein implicitly or explicitly interacted form a complicated web of collaboration, which this study explores, focusing on their implicit and explicit responses to Einstein's work.This analysis uncovers latent undercurrents, indiscernible to other approaches to tracking the intellectual pathway of Einstein to his general theory of relativity. The interconnections and interactions presented here reveal the central figures who influenced Einstein during this intellectual period. Despite current approaches to history presupposing that the efforts of scientists such as Max Abraham and Gunnar Nordström, which differed from Einstein’s own views, be relegated to the background, this book shows that they all had an impact on the development of Einstein’s theories, stressing the limits of approaches focusing solely on Einstein. As such, General Relativity Conflict and Rivalries proves that the general theory of relativity was not developed as a single, coherent construction by an isolated, brooding individual, but, rather, that it came to fruition through Einstein's conflicts and interactions with other scientists, and was consolidated by his creative processes during these exchanges.

We calculate the principle of non contradiction (p.d.n.c.) and subsequently we demonstrate it. We map his theory and shift the attention of all the logics to the p.d.n.c. Exactly, with this proof, we want to contain in a single rigor, within the p.d.n.c., all the classic and non classical logics, to conclude that all the logics are given by the p.d.n.c.

An eternally infinite universal Mandelbrot set: All good as usual except the evolutionary anthropological premise Mindly of the unconscious to be unconscious (ref. For the general and intrinsic systematic incompleteness of Gödel) of the Neanderthal inCoScientists themselves which is also phenomenological and cannot be ignored as on the other hand everyone, including Einstein himself, does, for example, relative to entanglement. We are in an eternal and infinite "present" that the particles ignore which is highlighted precisely thanks to the litmus test of the condensed mass and the thing has been consciously perceived by many poets like Leopardi ... BUT relativity is disclosed as relativism and this leads only going around in circles because one cannot shoot all one's points (cf. Putin's impossible multilateralism in an already formed global empire which, like the Roman empire, continues independently of everything and everyone of its own strength, NB that and still there is the 0 1 F 3 0 D

The main purpose of the present paper consists in analyzing the arguments, interpretations and concepts offered by both Gottlob Frege and Kurt Gödel in some of their works in defense of platonic realism in mathematics, where they do not rely on epistemologically complex concepts such as the world of Ideas or an intellectual intuition.

There is an intensive discussion nowadays about the meaning of effective computability, with implications to the status and provability of the Church–Turing Thesis (CTT). I begin by reviewing what has become the dominant account of the way Turing and Church viewed, in 1936, effective computability. According to this account, to which I refer as the Gandy–Sieg account, Turing and Church aimed to characterize the functions that can be computed by a human computer. In addition, Turing provided a highly convincing argument for CTT by analyzing the processes carried out by a human computer. I then contend that if the Gandy–Sieg account is correct, then the notion of effective computability has changed after 1936. Today computer scientists view effective computability in terms of finite machine computation. My contention is supported by the current formulations of CTT, which always refer to machine computation, and by the current argumentation for CTT, which is different from the main arguments advanced by Turing and Church. I finally turn to discuss Robin Gandy's characterization of machine computation. I suggest that there is an ambiguity regarding the types of machines Gandy was postulating. I offer three interpretations, which differ in their scope and limitations, and conclude that none provides the basis for claiming that Gandy characterized finite machine computation.

This book is alleged to be a comprehensive but largely elementary description of mathematical logic including its historical development, its most important achievements and its implications for philosophy. Although the intended audience includes both mathematicians
and philosophers, 'the book presupposes no knowledge of logic and only high school mathematics". The author avows an aim "to stimulate long-lived interest by communicating some of the excitement and beauty of the subject". Details and techniques are eschewed in favor of emphasis on the development of an overview.
The book is also intended by the author to serve as an introduction to logic. His opinion of the shortcomings of most introductions to logic is perceptive and insightful.

We argue that Wittgenstein's philosophical perspective on Gödel's most famous theorem is even more radical than has commonly been assumed. Wittgenstein shows in detail that there is no way that the Gödelian construct of a string of signs could be assigned a useful function within (ordinary) mathematics. — The focus is on Appendix III to Part I of Remarks on the Foundations of Mathematics. The present reading highlights the exceptional importance of this particular set of remarks and, more specifically, emphasises its refined composition and rigorous internal structure.

Translation to Portuguese of chapter 25 of Logical Forms-Part II.

Na parte inicial do texto considero várias perspectivas epistemológicas sobre os problemas de fundamentos levantados pelos paradoxos da lógica e da teoria de conjuntos no início do século vinte. Esta discussão está centrada, principalmente, nos pontos de vista de Russell, Hilbert, Brouwer e Gödel. A parte final do texto consiste de um breve exame de questões filosóficas sobre a noção de prova.

This paper shows what happens when Gödel's theorem is turned on itself, and the logical inconsistency that occurs when selfreferencing statements are converted to Gödel numbers. Gödel's First Incompleteness Theorem (GFIT) is often stated as follows: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F." Here's another way GFIT is stated: "In any reasonable mathematical system there's always true statements that can't be proved."