Kurt Gödel and Computability Theory

Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel’s 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church, Kleene, and Rosser. Church and his students studied Gödel 1931, and Gödel taught a seminar at Princeton in 1934. Seen in the historical context, Gödel was an important catalyst for the emergence of computability theory in the mid 1930s

Kurt Gödel: revolución en los fundamentos de las matemáticas

Ofrecemos un repaso a las principales contribuciones de Kurt Gödel en el campo de Lógica y fundamentos de las matemáticas, analizando su impacto, que bien puede llamarse revolucionario. La pretensión es hacer comprensible la tendencia y orientación metodológica de los trabajos de Gödel, y considerar en algún detalle sus repercusiones filosóficas. Así, se ofrece una perspectiva de cómo cambió la filosofía de las matemáticas entre las fechas de nacimiento y muerte del genial lógico matemático.We offer a survey of Kurt Gödel’s main contributions in the field of Logic and foundations of mathematics, and we analyse their impact, which can well be called revolutionary. The aim is to contribute to an understanding of the aims and methodological orientation of Gödel’s work, and to deal in some detail with their philosophical consequences. Thus, a perspective is offered on how the philosophy of mathematics changed between the dates of birth and death of this great mathematical logician

Wittgenstein´s Critique of Gödel´s Incompleteness Results

It is often said that Gödel´s famous theorem of 1931 is\ud equal to the Cretian Liar, who says that everything that he\ud says is a lie. But Gödel´s result is only similar to this\ud sophism and not equivalent to it. When mathematicians\ud deal with Gödel´s theorem, then it is often the case that\ud they become poetical or even emotional: some of them\ud show a high esteem of it and others despise it. Wittgenstein\ud sees the famous Liar as a useless language game\ud which doesn´t excite anybody. Gödel´s first incompleteness\ud theorem shows us that in mathematics there are\ud puzzles which have no solution at all and therefore in\ud mathematics one should be very careful when one\ud chooses a puzzle on which one wants to work. Gödel´s\ud second imcompleteness theorem deals with hidden\ud contradictions – Wittgenstein shows a paradigmatic\ud solution: he simply shrugs his shoulders on this problem\ud and many mathematicians do so today as well. Wittgenstein\ud says than Gödel´s results should not be treated as\ud mathematical theorems, but as elements of the humanistic\ud sciences. Wittgenstein sees them as something which\ud should be worked on in a creative manner

Consequences of a Goedel's misjudgment

The fundamental aim of the paper is to correct an harmful way to interpret a Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the Goedel's fault is rather venial, its misreading has produced and continues to produce dangerous fruits, as to apply the incompleteness Theorems to the full second-order Arithmetic and to deduce the semantic incompleteness of its language by these same Theorems. The first three paragraphs are introductory and serve to define the languages inherently semantic and its properties, to discuss the consequences of the expression order used in a language and some question about the semantic completeness: in particular is highlighted the fact that a non-formal theory may be semantically complete despite using a language semantically incomplete. Finally, an alternative interpretation of the Goedel's unfortunate comment is proposed. KEYWORDS: semantic completeness, syntactic incompleteness, categoricity, arithmetic, second-order languages, paradoxesComment: English version, 19 pages. Fixed and improved terminolog

On a Perceived Expressive Inadequacy of Principia Mathematica

This paper deploys a Cantor-style diagonal argument which indicates that there is more possible mathematical content than there are propositional functions in Russell and Whitehead's Principia Mathematica and similar formal systems. This technical result raises a historical question: "How did Russell, who was himself an expert in diagonal arguments, not see this coming?" It turns out that answering this question requires an appreciation of Russell's understanding of what logic is, and how he construed the relationship between logic and Principia Mathematica

Kriesel and Wittgenstein

Georg Kreisel (15 September 1923 - 1 March 2015) was a formidable mathematical logician during a formative period when the subject was becoming a sophisticated field at the crossing of mathematics and logic. Both with his technical sophistication for his time and his dialectical engagement with mandates, aspirations and goals, he inspired wide-ranging investigation in the metamathematics of constructivity, proof theory and generalized recursion theory. Kreisel's mathematics and interactions with colleagues and students have been memorably described in Kreiseliana ([Odifreddi, 1996]). At a different level of interpersonal conceptual interaction, Kreisel during his life time had extended engagement with two celebrated logicians, the mathematical Kurt Gödel and the philosophical Ludwig Wittgenstein. About Gödel, with modern mathematical logic palpably emanating from his work, Kreisel has reflected and written over a wide mathematical landscape. About Wittgenstein on the other hand, with an early personal connection established Kreisel would return as if with an anxiety of influence to their ways of thinking about logic and mathematics, ever in a sort of dialectic interplay. In what follows we draw this out through his published essays—and one letter—both to elicit aspects of influence in his own terms and to set out a picture of Kreisel's evolving thinking about logic and mathematics in comparative relief.Accepted manuscrip

The epistemological need for qualitative research

La sociedad occidental ha aceptado el concepto matemático de número sin ninguna crítica prácticamente desde la Escuela Pitagórica. Los trabajos de Georg Cantor sobre el concepto de infinito y sobre la teoría de conjuntos y las investigaciones lógicas de Kurt Gödel sobre la incompletitud han zarandeado la claridad de la aritmética elemental. Estas investigaciones junto con las discusiones metodológicas entre frecuencialistas y bayesianos en la estadística epidemiológica y los requerimientos aplicativos de la salud basada en la evidencia hacen necesaria la aportación de la investigación cualitativa para el análisis de la realidad sanitaria. Es deseable asimismo la definición epistemológica dentro de cada proyecto cualitativo.Western society has accepted the mathematical concept of number uncritically almost from the Pythagorean School. Georg Cantor's work on the concept of infinity and on set theory and Kurt Gödel´s logic investigations on incompleteness has shaken clarity of numeracy. These inquiries together with methodological discussions between frecuentists and Bayesian statisticians and applications requirements of evidence-based medicine make necessary the contribution of qualitative research for the analysis of reality of health. It is also desirable epistemological definition within each qualitative project