A layman's guide to the mechanics of Gödel's proof together with a lucid discussion of the issues which it raises. Includes an essay discussing the significance of Gödel's work in the light of Wittgenstein's criticisms.

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)

In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system and had radical implications that have echoed throughout many fields. A gripping combination of science and accessibility, Godel’s Proof by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and (...) philosophy the chance to satisfy their intellectual curiosity. (shrink)

This short book has two main purposes. The first is to explain Kurt Gödel's first and second incompleteness theorems in informal terms accessible to a layperson, or at least a non-logician. The author claims that, to follow this part of the book, a reader need only be familiar with the mathematics taught in secondary school. I am not sure if this is sufficient. A grasp of the incompleteness theorems, even at the level of ‘the big picture’, might require some experience (...) with the rigor of mathematical proof. Moreover, since the incompleteness theorems concern formal deductive systems, it would help for a potential reader of this book to have some familiarity with at least elementary logic.There are, of course, a number of informal expositions of the incompleteness theorems. I do not see the need to engage in a comparative study. The second, and more important, goal of the present book is to discuss some alleged consequences of the incompleteness theorems and, in particular, to debunk thoroughly claims in philosophy, religion, and literary criticism that are supposed to be established, or at least bolstered, by incompleteness. The execution of this part of the book is masterly. The arguments are clear and compelling.The book has eight chapters, each between eight and twenty-eight pages. The opening chapter gives a very brief account of the incompleteness theorems and a sketch of Gödel's life and work. Chapter 2 gives the main overview of the incompleteness theorems, plus …. (shrink)

Here’s one version G¨ odel’s 1931 First Incompleteness Theorem: If T is a nice, sound theory of arithmetic, then it is incomplete, i.e. there are arithmetical sentences ϕ such that T proves neither ϕ nor ¬ϕ. There are three things here to explain straight away.

In his paper on the incompleteness theorems, Gödel seemed to say that a direct way of constructing a formula that says of itself that it is unprovable might involve a faulty circularity. In this note, it is proved that ‘direct’ self-reference can actually be used to prove his result.

In “Minds, Machines, and Gödel”, J. R. Lucas claims that Goedel's incompleteness theorem constitutes a proof “that Mechanism is false, that is, that minds cannot be explained as machines”. He claims further that “if the proof of the falsity of mechanism is valid, it is of the greatest consequence for the whole of philosophy”. It seems to me that both of these claims are exaggerated. It is true that no minds can be explained as machines. But it is not (...) true that Goedel's theorem proves this. At most, Goedel's theorem proves that not all minds can be explained as machines. Since this is so, Goedel's theorem cannot be expected to throw much light on why minds are different from machines. Lucas overestimates the importance of Goedel's theorem for the topic of mechanism, I believe, because he presumes falsely that being unable to follow any but mechanical procedures in mathematics makes something a machine. (shrink)

Gödel's Theorem is often used in arguments against machine intelligence, suggesting humans are not bound by the rules of any formal system. However, Gödelian arguments can be used to support AI, provided we extend our notion of computation to include devices incorporating random number generators. A complete description scheme can be given for integer functions, by which nonalgorithmic functions are shown to be partly random. Not being restricted to algorithms can be accounted for by the availability of an arbitrary (...) random function. Humans, then, might not be rule-bound, but Gödelian arguments also suggest how the relevant sort of nonalgorithmicity may be trivially made available to machines. (shrink)

This paper discusses lower bounds for proof length, especially as measured by number of steps (inferences). We give the first publicly known proof of Gödel's claim that there is superrecursive (in fact. unbounded) proof speedup of (i + 1)st-order arithmetic over ith-order arithmetic, where arithmetic is formalized in Hilbert-style calculi with + and · as function symbols or with the language of PRA. The same results are established for any weakly schematic formalization of higher-order logic: this allows all tautologies as (...) axioms and allows all generalizations of axioms as axioms. Our first proof of Gödel's claim is based on self-referential sentences: we give a second proof that avoids the use of self-reference based loosely on a method of Statman. (shrink)

A layman's guide to the mechanics of Gödel's proof together with a lucid discussion of the issues which it raises. Includes an essay discussing the significance of Gödel's work in the light of Wittgenstein's criticisms.

This is a critical analysis of the first part of Go¨del’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Go¨del’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the mathematizing (...) potentialities of human thought, and, if not, there are absolutely unsolvable mathematical problems of diophantine form. (shrink)

G¨odel’s a theorem concerns an arithmetical statement and the truth of this statement does not depend on self-reference; nevertheless its interpretation is of tremendous interest. G¨odel’s theorem allows one to conclude that formal arithmetic is not axiomatizable. But there is another very interesting logico-philosophical result: the possibility of a statement to exist such that it is improvable in the object-theory and at the same time its truth is provable in the metatheory. It seems that in the real history (...) G¨odel’s theorem was absolutely unexpected, and even in retrospect, now it is difficult to point out logico-philosophical reasons for the above possibility. In my opinion though it was still conceivable at that time, but unfortunately the philosophers and logicians missed to notice it. In the terms of recursive arithmetic Hilbert’s programme can be formulated as follows: Build a decidable, complete, consistent logical system such that the class of derivable statements coincides with the class of the mathematical statements which are intuitively true; prove this equivalence by means of an arithmetization of the theory in the framework of the recursive arithmetic! Now, if the system is so strong as to admit the construction of the classical mathematics, then in particular it will contain the classical arithmetic. But Hilbert’s requirements reduce to a proof with the means of the recursive arithmetic, by arithmetization. So, on one hand the system will contain the arithmetic and, on the other hand it will be expressible in it, in order words the system will include its own syntax, as well as its own semantics. Now it is clear that Hilbert’s programme contains a possibility of a situation similar to some well known semantic antinomies, situations of self-reference. (shrink)

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 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. The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic. (shrink)

Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical (...) instrumentalism are defeated by Gödel’s theorem, not all are. By considering inductive reasons in mathematics, we show that some mathematical instrumentalisms survive the theorem. (shrink)

Textbook on Gödel’s incompleteness theorems and computability theory, based on the Open Logic Project. Covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus.

Dialetheism is the view that there exists a true contradiction. This paper ventures to suggest that Priest’s argument for Dialetheism from Gödel’s theorem is unconvincing as the lesson of Gödel’s proof (or Rosser’s proof) is that any sufficiently strong theories of arithmetic cannot be both complete and consistent. In addition, a contradiction is derivable in Priest’s inconsistent and complete arithmetic. An alternative argument for Dialetheism is given by applying Gödel sentence to the inconsistent and complete theory of arithmetic. We (...) argue, however, that the alternative argument raises a circularity problem. In sum, Gödel’s and its related theorem merely show the relation between a complete and a consistent theory. A contradiction derived by the application of Gödel sentence has the value of true sentences, i.e. the both-value, only under the inconsistent models for arithmetic. Without having the assumption of inconsistency or completeness, a true contradiction is not derivable from the application of Gödel sentence. Hence, Gödel’s and its related theorem never can be a ground for Dialetheism. (shrink)

This fourth edition of one of the classic logic textbooks has been thoroughly revised by John Burgess. The aim is to increase the pedagogical value of the book for the core market of students of philosophy and for students of mathematics and computer science as well. This book has become a classic because of its accessibility to students without a mathematical background, and because it covers not simply the staple topics of an intermediate logic course such as Godel's Incompleteness Theorems, (...) but also a large number of optional topics from Turing's theory of computability to Ramsey's theorem. John Burgess has now enhanced the book by adding a selection of problems at the end of each chapter, and by reorganising and rewriting chapters to make them more independent of each other and thus to increase the range of options available to instructors as to what to cover and what to defer. (shrink)

In the paper some applications of Gödel's incompleteness theorems to discussions of problems of computer science are presented. In particular the problem of relations between the mind and machine (arguments by J.J.C. Smart and J.R. Lucas) is discussed. Next Gödel's opinion on this issue is studied. Finally some interpretations of Gödel's incompleteness theorems from the point of view of the information theory are presented.

Gödel's two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues. They concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal (...) system cannot prove that the system itself is consistent (assuming it is indeed consistent). These results have had a great impact on the philosophy of mathematics and logic. There have been attempts to apply the results also in other areas of philosophy such as the philosophy of mind, but these attempted applications are more controversial. The present entry surveys the two incompleteness theorems and various issues surrounding them. (shrink)

In his recent article Christopher Gauker (2001) has presented a thoughtprovoking argument against deflationist theories of truth. More exactly, he attacks what he calls ‘T-schema deflationism’, that is, the claim that a theory of truth can simply take the form of certain instances of the T-schema.

In 1929 Carnap gave a paper in Prague on Investigations in General Axiomatics; a briefsummary was published soon after. Its subject lookssomething like early model theory, and the mainresult, called the Gabelbarkeitssatz, appears toclaim that a consistent set of axioms is complete justif it is categorical. This of course casts doubt onthe entire project. Though there is no furthermention of this theorem in Carnap''s publishedwritings, his Nachlass includes a largetypescript on the subject, Investigations inGeneral Axiomatics. We examine this work (...) here,showing that it provides important insights intoCarnap''s development during this critical period, thetransition from Aufbau to Syntax,especially regarding the nature and motivation ofCarnap''s logicism. Moreover, we show how theAxiomatics influenced Carnap''s student Gödel inreaching the fundamental logical results that soonafterwards undermined Carnap''s project. (shrink)

In this paper, we attempt to show that a weak version of Hilbert's metamathematics is compatible with Gödel's Incompleteness Theorems by employing only what are clearly natural prov‐ ability predicates. Defining first “T proves the consistency of a theory S indirectly in one step”, we subsequently prove “PA proves its own consistency indirectly in one step” and sketch the proof for “If S is a recursively enumerable extension of , S proves its own consistency indirectly in one step”. The formalizations (...) of the metatheoretical consistency assertions that occur in these theorems are clearly the natural ones. We conclude the paper with reflections on indirect consistency proofs and soundness proofs. (shrink)