A Mathematical Introduction to Logic, Second Edition, offers increased flexibility with topic coverage, allowing for choice in how to utilize the textbook in a course. The author has made this edition more accessible to better meet the needs of today's undergraduate mathematics and philosophy students. It is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning. Material is presented on computer science issues such as computational complexity and database queries, with additional (...) coverage of introductory material such as sets. Increased flexibility of the text, allowing instructors more choice in how they use the textbook in courses. Reduced mathematical rigour to fit the needs of undergraduate students. (shrink)

We will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and Cut-Free consistencies. If our axiom systems additionally do not recognize Addition as a total function, they will be capable of recognizing the consistency of their Hilbert-style deductive (...) proofs. Our axiom systems will not be strong enough to recognize their Canonical Reflection principle, but they will be capable of recognizing an approximation of it, called the "Tangibility Reflection Principle". We will also prove some new versions of the Second Incompleteness Theorem stating essentially that it is not possible to extend our exceptions to the Incompleteness Theorem much further. (shrink)

§1. Overview. Philosophers and mathematicians have drawn lots of conclusions from Gödel's incompleteness theorems, and related results from mathematical logic. Languages, minds, and machines figure prominently in the discussion. Gödel's theorems surely tell us something about these important matters. But what?A descriptive title for this paper would be “Gödel, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility”. Adding “God and the Devil” would probably be redundant. Despite the breath-taking, whirlwind tour, I have the modest aim of forging (...) connections between different parts of this literature and clearing up some confusions, together with the less modest aim of not introducing any more confusions.I propose to focus on three spheres within the literature on incompleteness. The first, and primary, one concerns arguments that Gödel's theorem refutes the mechanistic thesis that the human mind is, or can be accurately modeled as, a digital computer or a Turing machine. The most famous instance is the much reprinted J. R. Lucas [18]. To summarize, suppose that a mechanist provides plans for a machine,M, and claims that the output ofMconsists of all and only the arithmetic truths that a human, or the totality of human mathematicians, will ever or can ever know. We assume that the output ofMis consistent. (shrink)

Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic systems. Although Gödel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial (...) successes, and generated important advances in logical theory and metatheory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s. (shrink)

Ernest Nagel Lecture, Columbia University, Sept. 27, 2007.

In this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem. We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ and $\textsf (...) {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield’s work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\textsf {G1}$ holds. (shrink)

We give a survey of current research on Gödel’s incompleteness theorems from the following three aspects: classifications of different proofs of Gödel’s incompleteness theorems, the limit of the applicability of Gödel’s first incompleteness theorem, and the limit of the applicability of Gödel’s second incompleteness theorem.

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 antimechanism is not implied by Gödel’s theorems alone. The present paper sets out this refutation in its strongest form, demonstrating general theorems implying the inconsistency of Lucas’s arithmetic and the semantic inadequacy (...) of Penrose’s arithmetic. On the other hand, the limitations to our capacity for mechanizing or programming the mind are also indicated, together with two other corollaries of Gödel’s theorems: that we cannot prove that we are consistent, and that we cannot fully describe our notion of a natural number. (shrink)

The prevalent interpretation of Gödel’s Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of the underlying mathematical theorem depends on several arbitrary formalisation choices. In this paper I examine the theorem’s dependency regarding Gödel numberings. I introducedeviantnumberings, yielding provability predicates satisfying Löb’s conditions, which result in provable consistency sentences. According to the main result of this paper however, these (...) “counterexamples” do not refute the theorem’s prevalent interpretation, since once a natural class ofadmissiblenumberings is singled out, invariance is maintained. (shrink)

In this paper I address the question of whether the incompleteness theorems imply that “the mind cannot be mechanized,” where this is understood in the specific sense that “the mathematical outputs of the idealized human mind do not coincide with the mathematical outputs of any idealized finite machine.” Gödel argued that his incompleteness theorems implied a weaker, disjunctive conclusion to the effect that either “the mind cannot be mechanized” or “mathematical truth outstrips the idealized human mind.” Others, most notably, Lucas (...) and Penrose, have claimed more—they have claimed that the incompleteness theorems actually imply the first disjunct. I will show that by sharpening the fundamental concepts involved and articulating the background assumptions governing them, one can prove Gödel’s disjunction, one can show that the arguments of Lucas and Penrose fail, and one can see what likely led proponents of the first disjunct astray. (shrink)

Reinhardt 's strong mechanistic thesis, a formalization of “I know I am a Turing machine”, is shown to be consistent with Epistemic Arithmetic.

The logician Kurt Godel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is equivalent to a Turing machine (i.e., a computer), or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the (...) mathematical mind is more powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency if rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Godel's disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field. (shrink)

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency. Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does (...) is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency. (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)

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

In his stimulating book SHADOWS OF THE MIND, Roger Penrose presents arguments, based on Gödel's theorem, for the conclusion that human thought is uncomputable. There are actually two separate arguments in Penrose's book. The second has been widely ignored, but seems to me to be much more interesting and novel than the first. I will address both forms of the argument in some detail. Toward the end, I will also comment on Penrose's proposals for a "new science of consciousness".

In his book Shadows of the Mind: A search for the missing science of con- sciousness [SM below], Roger Penrose has turned in another bravura perfor- mance, the kind we have come to expect ever since The Emperor’s New Mind [ENM ] appeared. In the service of advancing his deep convictions and daring conjectures about the nature of human thought and consciousness, Penrose has once more drawn a wide swath through such topics as logic, computa- tion, artificial intelligence, quantum physics (...) and the neuro-physiology of the brain, and has produced along the way many gems of exposition of difficult mathematical and scientific ideas, without condescension, yet which should be broadly appealing. 1 While the aims and a number of the topics in SM are the same as in ENM, the focus now is much more on the two axes that Pen- rose grinds in earnest. Namely, in the first part of SM he argues anew and at great length against computational models of the mind and more specifi- cally against any account of mathematical thought in computational terms. Then in the second part, he argues that there must be a scientific account of consciousness but that will require a (still to be found) non-computational extension or modification of present-day quantum physics. (shrink)

In this article, Lucas maintains the falseness of Mechanism - the attempt to explain minds as machines - by means of Incompleteness Theorem of Gödel. Gödel’s theorem shows that in any system consistent and adequate for simple arithmetic there are formulae which cannot be proved in the system but that human minds can recognize as true; Lucas points out in his turn that Gödel’s theorem applies to machines because a machine is the concrete instantiation of a formal system: therefore, for (...) every machine consistent and able of doing simple arithmetic, there is a formula that it can’t produce as true but that we can see to be true, and so human minds and machines have to be different. Lucas considers as well in this article some possible objections to his argument: for any Gödelian formula we could, for instance, construct a machine able to produce it or we could put the Gödelian formulae that we had proved as axioms of a further machine. However - as Lucas underlines - for every of such machines we could again formulate another Gödelian formula, the Gödelian formula of these machines, that they are not able to proof but that we can recognize as true. More general arguments, such as the possibility to escape Gödelian argument by suggesting that Gödel’s theorem applies to consistent systems while we could be inconsistent ones, are moreover refuted by Lucas by maintaining that our inconsistency corresponds to occasional malfunctioning of a machine and not to his normal inconsistency; indeed, a inconsistent machine is characterized by producing any statement, on the contrary human being are selective and not disposed to assert anything. (shrink)

Gödel appears to have believed strongly that the human mind cannot be explained in terms of any kind of computational physics, but he remained cautious in formulating this belief as a rigorous consequence of his incompleteness theorems. In this chapter, I discuss a modification of standard Gödel-type logical arguments, these appearing to strengthen Gödel’s conclusions, and attempt to provide a persuasive case in support of his standpoint that the actions of the mind must transcend computation. It appears that Gödel did (...) not consider the possibility that the laws of physics might themselves involve noncomputational procedures; accordingly, he found himself driven to the conclusion that mentality must lie beyond the actions of the physical brain. My own arguments, on the other hand, are from the scientific standpoint that the mind is a product of the brain’s physical activity. Accordingly, there must be something in the physical actions of the world that itself transcends computation. We do not appear to find such noncomputational action in the known laws of physics, however, so we must seek it in currently undiscovered laws going beyond presently accepted physical theory. I argue that the only plausibly relevant gap in current understanding lies in a fundamental incompleteness in quantum theory, which reveals itself only with significant mass displacements between quantum states (“Schrödinger’s cats”). I contend that the need for new physics enters when gravitational effects just begin to play a role. In a scheme developed jointly with Stuart Hameroff, this has direct relevance within neuronal microtubules, and I describe this (still speculative) scheme in the following. (shrink)

We give some information about new proofs of the incompleteness theorems, found in 1990s. Some of them do not require the diagonal lemma as a method of construction of an independent statement.

This paper will introduce the notion of a naming convention and use this paradigm to both develop a new version of the Second Incompleteness Theorem and to describe when an axiom system can partially evade the Second Incompleteness Theorem.

In his bestselling work of popular science, Sir Roger Penrose takes us on a fascinating roller-coaster ride through the basic principles of physics, cosmology, mathematics, and philosophy to show that human thinking can never be emulated by a machine.

The paper begins with an examination of Gödel's views on absolute undecidability and related topics in set theory. These views are sharpened and assessed in light of recent developments. It is argued that a convincing case can be made for axioms that settle many of the questions undecided by the standard axioms and that in a precise sense the program for large cardinals is a complete success “below” CH. It is also argued that there are reasonable scenarios for settling CH (...) and that there is not currently a convincing case to the effect that a given statement is absolutely undecidable. (shrink)

Sections 3.16 and 3.23 of Roger Penrose's Shadows of the mind (Oxford, Oxford University Press, 1994) contain a subtle and intriguing new argument against mechanism, the thesis that the human mind can be accurately modeled by a Turing machine. The argument, based on the incompleteness theorem, is designed to meet standard objections to the original Lucas-Penrose formulations. The new argument, however, seems to invoke an unrestricted truth predicate (and an unrestricted knowability predicate). If so, its premises are inconsistent. The usual (...) ways of restricting the predicates either invalidate Penrose's reasoning or require presuppositions that the mechanist can reject. (shrink)

It has been argued, by Penrose and others, that Gödel's proof of his first incompleteness theorem shows that human mathematics cannot be captured by a formal system F: the Gödel sentence G(F) of F can be proved by a (human) mathematician but is not provable in F. To this argment it has been objected that the mathematician can prove G(F) only if (s)he can prove that F is consistent, which is unlikely if F is complicated. Penrose has invented a new (...) argument intended to avoid this objection. In the paper I try to show that Penrose's new argument is inconclusive. (shrink)

It is commonly agreed that the well-known Lucas-Penrose arguments and even Penrose's 'new argument' in [Penrose, R. (1994): Shadows of the Mind, Oxford University Press] are inconclusive. It is, perhaps, less clear exactly why at least the latter is inconclusive. This note continues the discussion in [Lindström, P. (2001): Penrose's new argument, J. Philos. Logic 30, 241-250; Shapiro, S.(2003): Mechanism, truth, and Penrose's new argument, J. Philos. Logic 32, 19-42] and elsewhere of this question.

Kurt Gödel was the greatest logician of this century. This third volume of his collected works consists of previously unpublished material, both essays and lectures.

_'Nagel and Newman accomplish the wondrous task of clarifying the argumentative outline of Kurt Godel's celebrated logic bomb.'_ _– The Guardian_ In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of physicist Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system. The importance of Godel's Proof rests upon its radical implications and has echoed throughout many fields, from maths (...) to science to philosophy, computer design, artificial intelligence, even religion and psychology. While others such as Douglas Hofstadter and Roger Penrose have published bestsellers based on Godel’s theorem, this is the first book to present a readable explanation to both scholars and non-specialists alike. 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. _Kurt Godel_ Born in Brunn, he was a colleague of physicist Albert Einstein and professor at the Institute for Advanced Study in Princeton, N.J. (shrink)

This paper presents a defense of Epistemic Arithmetic as used for a formalization of intuitionistic arithmetic and of certain informal mathematical principles. First, objections by Allen Hazen and Craig Smorynski against Epistemic Arithmetic are discussed and found wanting. Second, positive support is given for the research program by showing that Epistemic Arithmetic can give interesting formulations of Church's Thesis.

In this paper Lucas comes back to Gödelian argument against Mecanism to clarify some points. First of all, he explains his use of Gödel’s theorem instead of Turing’s theorem, showing how Gödel’ theorem, but not Turing’s theorem, raises questions concerning truth and reasoning that bear on the nature of mind and how Turing’s theorem suggests that there is something that cannot be done by any computers but not that it can be done by human minds. He considers moreover how Gödel’s (...) theorem can be interpreted as a sophisticated form of the Cretan paradox, posed by Epimenides, able to escape the viciously self-referential nature of the Cretan paradox, and how it can be used against Mechanism as a schema of disproof. Finally, Lucas suggests some answers to the most recurrent criticisms against his argument: criticisms about the implicit idealisation in the way he set up the context between mind and machine; questions concerning modality and finitude, issues of transfinite arithmetic; questions concerning the need of formalizing rational inference and some questions about consistency. (shrink)

In a recent paper S. McCall adds another link to a chain of attempts to enlist Gödel’s incompleteness result as an argument for the thesis that human reasoning cannot be construed as being carried out by a computer.1 McCall’s paper is undermined by a technical oversight. My concern however is not with the technical point. The argument from Gödel’s result to the no-computer thesis can be made without following McCall’s route; it is then straighter and more forceful. Yet the argument (...) fails in an interesting and revealing way. And it leaves a remainder: if some computer does in fact simulate all our mathematical reasoning, then, in principle, we cannot fully grasp how it works. Gödel’s result also points out a certain essential limitation of self-reflection. The resulting picture parallels, not accidentally, Davidson’s view of psychology, as a science that in principle must remain “imprecise”, not fully spelt out. What is intended here by “fully grasp”, and how all this is related to self-reflection, will become clear at the end of this comment. (shrink)