Automated Deduction – CADE 27, 2019

Full bibliographic details must be given when referring to, or quoting from full items including the author's name, the title of the work, publication details where relevant (place, publisher, date), pagination, and for theses or dissertations the awarding institution, the degree type awarded, and the date of the award.

Revue internationale de philosophie, 2005

Gödel began his 1951 Gibbs Lecture by stating: "Research in the foundations of mathematics during the past few decades has produced some results which seem to me of interest, not only in themselves, but also with regard to their implications for the traditional philosophical problems about the nature of mathematics." (Gödel 1951) Gödel is referring here especially to his own incompleteness theorems (Gödel 1931). Gödel's first incompleteness theorem (as improved by Rosser (1936)) says that for any consistent formalized system F, which contains elementary arithmetic, there exists a sentence G F of the language of the system which is true but unprovable in that system. Gödel's second incompleteness theorem states that no consistent formal system can prove its own consistency. 1 These results are unquestionably among the most philosophically important logicomathematical discoveries ever made. However, there is also ample misunderstanding and confusion surrounding them. The aim of this paper is to review and evaluate various philosophical interpretations of Gödel's theorems and their consequences, as well as to clarify some confusions. I have argued for my own interpretation of Hilbert's program in detail in Raatikainen (2003a).

INTELLETUAL ARCHIVE - CANADA, 2012

Some common fallacies about fundamental themes of Logic are exposed: the First and Second incompleteness Theorem interpretations, Chaitin's various superficialities and the usual classification of the axiomatic Theories in function of its language order.

This article offers a new reflection on the reasoning followed by Gödel in the proof of the incompleteness theorem(s). It takes the form of a short thought experiment using a variant of this theorem, obtained by cloning, and leading to an obviously unacceptable conclusion.

Grail of Science

Plan Statement of the problem Find information to solve the problem Clarification of information to solve the problem Formulation of the lemma to solve the problem Search for a principle to solve the problem Proving the lemma that every mathematical system needs an observer Proving the lemma that every mathematical system needs an observer, whose existence only he can know, because whose existence cannot be proved 8.Mathematical record of problem solving Confirmation of the consistency and completeness of the formal system for one observer. Necessary and sufficient conditions for the formation of a consistent system for society (group of observers) Solving the liar paradox as a byproduct of solving the problem Using observer’s view on The Ship of Theseus The unexpected hanging paradox The sorites paradox The philosophical basis of the theorem proof Some reasonable conclusions from this work that can be applied in other scientific Conclusions of solving the problem My sincere thanks ...

2004

This book is a study of Gödel’s Incompleteness Theorem. The focus here is, first, on the consequences and interpretations of it in the philosophy of mathematics, philosophy of mind, and logic, and second, on a discussion of attempts to apply the theorem in areas of the humanities, such as literary criticism, social studies, and the theory of law. Considerable space is also devoted to the philosophical views and logical achievements of Gödel, widely seen as “the greatest logician since Aristotle”. Chapter I describes the background of Gödel’s work in the study of the foundations of mathematics generally and Hilbert’s program in the early 20th Century. Since this book is not a mathematical textbook, summaries, rather than complete technical presentations, are provided. In addition to standard topics some new developments are mentioned. The reception of Gödel’s work is seen as falling into three periods: from 1931, when his celebrated paper on incompleteness appeared, to the mid-1950s,...

This article raises some important points about logic, e.g., mathematical logic.

2003

La demostración habitual del Segundo Teorema de Incompletud de Gödel a partir de teorías débiles como IΣ 1 es larga y técnicamente intrincada. Raramente se dan todos los detalles y en muchos casos se omiten completamente apelando a la capacidad de lector para completarlos. En la primera parte de este artículo presentamos una guía de los principales puntos técnicos de la demostración habitual del Segundo Teorema de Incompletud de Gödel a partir de teorías débiles. En la segunda parte presentamos una demostración distinta y más simple para la Teoría de Conjuntos de Zermelo-Fraenkel debida a T. Jech ], y observamos que puede ser extendida de forma que englobe teorías débiles, evitando así muchas de las complicaciones técnicas que requieren las demostraciones habituales.

Mathematical Intelligencer, 2006

From the blurb: "In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book - extensively rewritten for its second edition - will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathem...