John W. Dawson

The Reception of Gödel's Incompleteness Theorems

According to several commentators, Kurt Gödels incompleteness discoveries were assimilated promptly and almost without objection by his contemporaries - - a circumstance remarkable enough to call for explanation. Careful examination reveals, however, that there were doubters and critics, as well as defenders and rival claimants to priority. In particular, the reactions of Carnap, Bernays, Zermelo, Post, Finsler, and Russell, among others, are considered in detail. Documentary sources include unpublished correspondence from Gödels Nachlass.

Kurt Gödel in sharper focus

The lives of great thinkers are sometimes overshadowed by their achievements—a phenomenon perhaps no better exemplified than by the life and work of Kurt Godel, a reclusive genius whose incompleteness theorems and set-theoretic consistency proofs are among the most celebrated results of twentieth-century mathematics, yet whose life history has until recently remained almost unknown.The lives of great thinkers are sometimes overshadowed by their achievements—a phenomenon perhaps no better exemplified than by the life and work of Kurt Godel, a reclusive genius whose incompleteness theorems and set-theoretic consistency proofs are among the most celebrated results of twentieth-century mathematics, yet whose life history has until recently remained almost unknown.

Boolean-valued set theory and forcing

This article is meant to serve as a brief introduction to Boolean-valued set theory. Our acquaintance with this subject began in January 1966 in a seminar conducted by Professor Dana Scott. This seminar included several lectures by Scott himself and by Kenneth Kunen. We have since benefited greatly from papers by Solovay, Solovay and Tennenbaum, and Shoenfield, as well as various classes, seminars, and private discussions. Our purpose here is to summarize the main theoretical aspects of the subject as we have learned it and synthesized it from these sources. The authors regard this paper as a substitute for the famous unwritten paper [7] of Scott and Solovay. We have set ourselves the goal of writing that paper as we would like to have seen it. In order to do this, we have introduced a major technical change in their approach by substituting Shoenfields definition of forcing [8] for the original Scott-Solovay definition. This has made the proof of Lemma 1.9 a pedagogical possibility; previously it has usually been left to the reader. We have also gone beyond the Scott-Solovay treatment by including a section on the independence of Martins axiom. We feel it is important to present this application within our framework, in contrast to the treatment of Solovay and Tennenbaum [11] using the system from [8]. It is our intention that this paper be regarded as sequel to Krivines book Introduction to Axiomatic Set Theory [4], and we assume that the reader is familiar with that text.

Other Case Studies

I hope that the case studies in the preceding chapters have convinced the reader that the comparative study of alternative proofs is a worthwhile endeavor and that the informal criteria for distinguishing proofs described in Chapter 1 serve that purpose well. I hope too that some of the proofs discussed in those chapters will have been new to most readers, who will have found them to possess both intrinsic interest and pedagogical value.This final chapter highlights some other theorems whose proofs have been the subject of comparative studies by other scholars. In addition, some theorems whose alternative proofs appear to be worthy subjects for further investigation are indicated.

Jean van Heijenoort and the Gödel Editorial Project

A colleague’s personal recollections of Jean van Heijenoort’s contributions to the editing of volumes I–III of Gödel’s Collected Works and of his interactions with the other editors.

Classical Logic's Coming of Age

Publisher Summary This chapter discusses the evolution of classical logic. The displacement of the concepts and methods of Aristotelian logic by modern symbolic and mathematical ones began in the nineteenth century with the works of George Boole, Charles Sanders Peirce, Ernst Schroder, Georg Cantor, Giuseppe Peano, and Gottlob Frege. During the period from 1847 to 1947, three formative periods in the development of logic may be distinguished: its youth, from 1847 until around 1900; its adolescence, from then until 1928; and its coming of age, during the following two decades. By 1950, the major branches of modern logic, with the exception of model theory, had largely assumed their modern form. First-order logic had become the focus of study. The questions that Hilbert had posed to logicians in his 1900 address and his 1928 text with Ackermann had for the most part been answered (though not, in many cases, in the way he had expected). The future direction of proof-theoretic studies had been established through the work of Godel and Gentzen. Recursion theory had become an important and highly technical specialty, poised to play a central role in the nascent field of computer science. The foundational controversies had died down. And set theory had become widely regarded as the basis for all of mathematics.

Gödel and the origins of computer science

The centenary of Kurt Godel (1906–78) is an appropriate occasion on which to assess his profound, yet indirect, influence on the development of computer science. His contributions to and attitudes toward that field are discussed, and are compared with those of other pioneer figures such as Alonzo Church, Emil Post, Alan Turing, and John von Neumann, in order better to understand why Godels role was no greater than it was.

Logical contributions to the Menger colloquium

Apart from articles on set-theoretical topics, the eight volumes of Ergebnisse eines mathematischen Kolloquiums contain reports of sixteen contributions to the field of mathematical logic. Three other presentations on logical topics, entitled “Uber einige fundamentale Begriffe der Metamathematik”, “Uber die Vollstandigkeit der Axiome des logischen Funktionenkalkuls”, and “Bemerkungen von Freges und Russells Definition der Zahl”, are noted as having been delivered to the colloquium as well, during its eleventh, fifteenth and ninetieth sessions (20 February and 14 May, 1930, and 5 June 1935, respectively), but their contents were not published in the colloquium proceedings. In the first two cases, the speakers involved (Alfred Tarski and Kurt Godel) presumably preferred that the results they discussed on those occasions appear in a journai having wider circulation1. Whether the third contribution (by Friedrich Waismann) ever appeared elsewhere is unknown to this commentator. The question is worth exploring, especially in view of the tantalizing comment (volume 7, page 15 of these Ergebnisse) that the following discussion included remarks by Godel, Menger and Tarski.

Facets of Incompleteness

Great intellectual discoveries, like fine gems held up before the eyes, continue to fascinate and to reveal new facets as they are beheld in the eye of the mind. As an example of this phenomenon, we may consider the Gődel incompleteness theorems, which, during the 55 years since their discovery, have repeatedly yielded new insights in the course of ongoing reinterpretation.

The Irreducibility of the Cyclotomic Polynomials

The irreducibility of the cyclotomic polynomials is a fundamental result in algebraic number theory that has been proved many times, by many different authors, in varying degrees of generality and using a variety of approaches and methods of proof. We examine these in the spirit of our inquiry here.