Godel's Incompleteness Theorem Research Articles

Putnam, Peano, and the Malin Génie: could we possibly bewrong about elementary number-theory?

This article examines Hilary Putnam's work in the philosophy of mathematics and - more specifically - his arguments against mathematical realism or objectivism. These include a wide range of considerations, from Godel's incompleteness-theorem and the limits of axiomatic set-theory as formalised in the Lowenheim-Skolem proof to Wittgenstein's sceptical thoughts about rule-following (along with Saul Kripke's ‘scepticalsolution’), Michael Dummett's anti-realist philosophy of mathematics, and certain problems – as Putnam sees them – with the conceptual foundations of Peano arithmetic. He also adopts a thought-experimental approach – a variant of Descartes' dream scenario – in order to establish the in-principle possibility that we might be deceived by the apparent self-evidence of basic arithmetical truths or that it might be ‘rational’ to doubt them under some conceivable (even if imaginary) set of circumstances. Thus Putnam assumes that mathematical realism involves a self-contradictory ‘Platonist’ idea of our somehow having quasi-perceptual epistemic ‘contact’ with truths that in their very nature transcend the utmost reach of human cognitive grasp. On this account, quite simply, ‘nothing works’ in philosophy of mathematics since wecan either cling to that unworkable notion of objective (recognition-transcendent) truth or abandon mathematical realism in favour of a verificationist approach that restricts the range of admissible statements to those for which we happen to possess some means of proof or ascertainment. My essay puts the case, conversely, that these hyperbolic doubts are not forced upon us but result from a false understanding of mathematical realism – a curious mixture of idealist and empiricist themes – which effectively skews the debate toward a preordained sceptical conclusion. I then go on to mount a defence of mathematical realism with reference to recent work in this field and also to indicate some problems – as I seethem – with Putnam's thought-experimental approach as well ashis use of anti-realist arguments from Dummett, Kripke, Wittgenstein, and others.

Equational fragments of systems for arithmetic

We investigate the equational fragments of formal systems for arithmetic by means of the equational theory of f-rings and of their positive cones, starting from the observation that a model of arithmetic is the positive cone of a discretely ordered ring. A consequence of the discreteness of the order is the presence of a discriminator, which allows us to derive many properties of the models of our equational theories. For example, the spectral topology of discrete f-rings is a Stone topology. We also characterize the equational fragment of Iopen, and we obtain an equational version of Godel's First Incompleteness Theorem. Finally, we prove that the lattice of subvarieties of the variety of discrete f-rings is uncountable, and that the lattice of filters of the countably generated distributive free lattice can be embedded into it.

On Herbrand consistency in weak arithmetic

We prove that the Godel incompleteness theorem holds for a weak arithmetic T = IΔ0 + Ω2 in the form $$$$

Rejoinder to Dejnožka’s Reply

66 Discussion REJOINDER TO DEJNOZKA'S REPLY GARY OSTERTAG Philosophy/ New YorkU. New York,NY 10003, USA G02@NYU.EDU It is common knowledge that Russell does not explicitly endorse modal logic in any of his major logical writings. Nor does my review of BertrandRusseli onModalityand LogicalRelevance' suggest that Jan Dejnozka denies or is somehow unaware of this. On the contrary, I assume it to be obvious that any commitment Russell may have had to modal logic must be implicitin his writings, nor explicit. The real issue is whether there is evidence of any such commitment . I will state, briefly,why I remain sceptical. First, Dejnozka's book lacks any interpretive methodology. When the question arises, In whichcontextsis itpermissibleto translateRusseli's language(formal orinformal)intoa modalsystem?, we get the following: amibute a modal logic to Russell if "it is more reasonable than not to paraphrase Russell's thinking into the modal logic." This condition, we are told, "is met to the extent that acertain modal logic is logicallyimplicit in Russell's thinking." All of which goes without saying, but it simply delays the inevitable question: when is a modal logic logicallyimplicit in a text? Dejnozka's current answer: a modal logic is logically implicit in a text when the text logically implies the paraphrase.2 But the final suggestion is incoherent, since we cannot tell what a text logical1y implies until we have discerned its logicalform-i.e. until we havealready paraphrased it into some formal idiom or other. A related problem is that Dejnozka consistently failsto distinguish two sorts of claims: • Certain passagesin Russell lend themselves to a modal interpretationthey can be captured in a given system of modal logic, e.g. S13. • Such an interpretation reflects Russell'sintentions. That a modal system captures one or another of Russell's accounts of logical truth in itself entails nothing regarding Russell'sintentions. The modal system G described by George Boolos in a munber of publications provides an alterna- ' "Russell's Modal Logic?", Russelin.s. 20 (2000): 165-72. 2 "Reply to Ostertag", Russell, n.s. 21 (2001): 63-5 (at 65). Discussion 67 rive formalization of Peano Arithmetic, enabling the derivation of Godel's incompleteness theorems. Yet the fact that this is possible does not in any way suggest that Godel himself was implicitly committed to G. Unfortunately, Dejnozka often writes as if the mere possibility of a modal interpretation is genuinely revealing as to Russell's intemions. Bm it is just as implausible to maintain this as it is to claim that it was Gi:idel'sintention to present Boolos's G.3 Second, as I maintain in my review, Dejnozka fails to take seriously the many comexts in which Russell is explicitly critical of modality. Dejnozka responds that he has quoted these very passagesand that he "embraces them as half-but only half-of [his] basic message" ("Reply", p. 64). But citing contexts in which Russell appears to find modality congenial does nothing to mitigate the force of the contexts in which Russell explicitly repudiates modality. It is Dejnozka's responsibility to explain how the latter are to cohere with his interpretation. To avoid this issue is to skirt the real philosophical and interpretive challenge that the project entails. Finally, we come to Dejnozka's confusions surrounding MDL-the triad of equivalences that underwrite the interpretation of Principiaas a modal logic. In light of Dejnozka's steadfast refusal to formalize the various modal systems he amibmes to Russell, it hardly makes a differencewhether MDLis a modal logic or merely, to use his terminology, a "modal theory" (assuming that the latter ultimately amounts to something other than a modal logic).4What is importam is whether MDLis trivial. If it is, then it is quite beside the point to pursue the various extensions of MDL.Since I maimain that Russell is not commined, implicitly or otherwise, to a substantive reading of MDL-one in which the modal operators have their conventional meanings-then he is not committed to these extensions, so there is no need for detailed discussion. What remains central is my argtunent for the claim that MDL is trivial. Bm this argmnent (unacknowledged by Dejnozka) is there for...

T-schema deflationism versus Godel's first incompleteness theorem

Journal Article T-schema deflationism versus Gödel’s first incompleteness theorem Get access Christopher Gauker Christopher Gauker University of CincinnatiCincinnati, OH 45221-0374, USAchristopher.gauker@uc.edu Search for other works by this author on: Oxford Academic Google Scholar Analysis, Volume 61, Issue 2, April 2001, Pages 129–136, https://doi.org/10.1093/analys/61.2.129 Published: 01 April 2001

T-schema deflationism versus Godel's first incompleteness theorem

where an instance is the result of writing some sentence of our language in place of 'p' in both places. Call this T-schema deflationism. An immediate objection to T-schema deflationism, as here defined, is that many of the sentences we put in place of 'p' in (T) may be neither true nor false apart from some context of utterance, so that (T) just cannot be the correct scheme for the articulation of the truth conditions of sentences. Still, I think it is worthwhile to consider whether there is any other objection to T-schema deflationism. Let us imagine that our language is one in which every sentence is either true or false or neither absolutely. If Tschema deflationism is objectionable even for such a language, then we may find that any more realistic form of deflationism is objectionable in the same way. T-scheme deflationism, as here defined, is an explicitly vague idea, since it says only that certain instances of (T) comprise a theory of truth for our language and does not say exactly which instances comprise the theory. It is clear that a theory of truth for our language cannot contain all instances of (T), because many instances are liable to be plainly untrue. For instance, suppose our language contains the following sentence:

A strictly finitary non-triviality proof for a paraconsistent system of set theory deductively equivalent to classical ZFC minus foundation

The paraconsistent system CPQ-ZFC/F is defined. It is shown using strong non-finitary methods that the theorems of CPQ-ZFC/F are exactly the theorems of classical ZFC minus foundation. The proof presented in the paper uses the assumption that a strongly inaccessible cardinal exists. It is then shown using strictly finitary methods that CPQ-ZFC/F is non-trivial. CPQ-ZFC/F thus provides a formulation of set theory that has the same deductive power as the corresponding classical system but is more reliable in that non-triviality is provable by strictly finitary methods. This result does not contradict Godel's incompleteness theorem because the proof of the deductive equivalence of the paraconsistent and classical systemss use non-finitary methods.

Between Vienna and Berlin: The Immediate Reception of Godel's Incompleteness Theorems

What were the earliest reactions to Gödel's incompleteness theorems? After a brief summary of previous work in this area I analyse, by means of unpublished archival material, the first reactions in Vienna and Berlin to Gödel's groundbreaking results. In particular, I look at how Carnap, Hempel, von Neumann, Kaufmann, and Chwistek, among others, dealt with the new results.

Artificial intelligence and scientific method

Preface Acknowledgements Chapter 1: The Inductivist Controversy, or Bacon versus Popper 1.1 Bacon's Inductivism 1.2 Popper's Falsificationism 1.3 Kepler's Discovery of the Laws of Planetary Motion 1.4 The Discovery of the Sulphonamide Drugs Chapter 2: Machine Learning in the Turing Tradition 2.1 The Turing Tradition 2.2 The Practical Problem: Expert Systems and Feigenbaum's Bottleneck 2.3 Attribute-based Learning, Decision Trees, and Quinlan's ID3 2.4 GOLEM as an example of Relational Learning 2.5 Bratko's summary of the successes of Machine Learning in the Turing Tradition, 1992 2.6 GOLEM's Discovery of a Law of Nature Chapter 3: How Advances in Machine Learning affect the Inductivist Controversy 3.1 Bacon's Example of Heat 3.2 The Importance of Falsification 3.3 Bacon's Method has only recently come to be used 3.4 The Need for Background Knowledge Chapter 4: Logic and Programming and a New Framework for Logic 4.1 The Development of PROLOG 4.2 PROLOG as a Non-Monotonic Logic 4.3 Two Examples of Translations from One Logical System to Another 4.4 Logic = Inference + Control 4.5 PROLOG introduces Control into Deductive Logic 4.6 PROLOG and Certainty. Is Logic a priori or empirical? Chapter 5: Can there be an Inductive Logic? 5.1 The Divergence between Deductive and Inductive Logic (up to the early 1970s) 5.2 Inductive Logic as Inference + Control 5.3 Confirmation Values as Control in a Deductive Logic 5.4 The Empirical Testing of Rival Logics Chapter 6: Do Godel's Incompleteness Theorems place a Limit on Artificial Intelligence? 6.1 Anxieties caused by Advances in AI 6.2 Informal Exposition of Godel's Incompleteness Theorems 6.3 The Lucas Argument 6.4 Objections to the Lucas Argument: i) Possible Limitations on Self-Knowledge 6.5 Objections to the Lucas Argument: ii) Possible Additions of Learning Systems 6.6 Why Advances in Computing are more likely to Stimulate Human Thinking than to Render it Superfluous Notes References Index

Gödel’s Incompleteness Theorems and Artificial Life

In this paper I discuss whether Godel's incompleteness theorems have any implications for studies in Artificial Life (AL). Since Godel's incompleteness theorems have been used to argue against certain mechanistic theories of the mind, it seems natural to attempt to apply the theorems to certain strong mechanistic arguments postulated by some AL theorists. We find that an argument using the incompleteness theorems can not be constructed that will block the hard AL claim, specifically in the field of robotics. However, we will see that the beginnings of an argument casting doubt on our ability to create living systems entirely resident in a computer environment might be suggested by looking at the incompleteness theorems from the point of view of Godel's belief in mathematical realism.

Self-knowledge and embedded operators

Queen Anne is dead, and it is a fallacy to substitute a definite description for another designator of the same object in stating the content of someone's propositional attitudes. The fallacy can take subtle forms, as when Godel's incompleteness theorems are used to argue against mechanistic views of mind. Some instances of the fallacy exemplify a more general logical phenomenon: the set of principles satisfied by one sentential operator can differ from, and even contradict, the set of principles satisfied by another sentential operator coextensive with the first. These notions will be formally defined. The phenomenon will be explored through a series of example, with particular attention to the misapplication of G6del's results. According to the 'KK' principle, if one knows that something is so, then one knows that one knows that it is so. Although the principle, unqualified, is false, appropriately restricted or idealized versions of it may be true. What modes of presentation of oneself to oneself are relevant to such a qualified KK principle? Use the phrase 'singular term' broadly, for definite descriptions as well as names, demonstratives and pronouns, without prejudice to the question whether definite descriptions are quantifiers. For any singular term 't', 't knows that ...' is therefore a sentence operator. Relative to a fixed context, say that 't' fits the KK principle just in case for all P, if t knows that P, then t knows that t knows that P. Suppose (counterfactually) that in the context of its use by me 'I' fits the KK principle. Suppose also that, although I am in fact the man with a hole in his pocket, I do not know that I am the man with a hole in his pocket. In this context, does the definite description 'the man with a hole in his pocket' fit the KK principle? Suppose that the man with a hole in his pocket knows that it is sunny. Does it follow that the man with a hole in his pocket knows that the man with a hole in his pocket knows that it is sunny? It certainly follows that I know that it is sunny. Since 'I' fits the KK principle in this context, it follows that I know that I know that it is sunny. Thus the man with a hole in his pocket knows that I know that it is sunny. However, it does not follow that the man with a hole in his pocket knows that the man with a hole in his pocket knows that it is sunny, or that I know that the man with

The incompleteness phenomenon: a new course in mathematical logic

The authors have written an introduction to logic taking Godel's incompleteness theorem as a starting point. The book should interest everyone from mathematicians to philosophers and readers who wish to understand the foundations and limitations of rational thinking. It is used as a textbook at major colleges and universities but lends itself to self-study as well.

Vladimir A. Uspensky. Godel's incompleteness theorem. A reprint of LV 889 with minor corrections. Theoretical computer science, vol. 130 (1994), pp. 239–319.

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The Incomplete Universe: Totality, Knowledge, and Truth.

Introduction - some philosphical fragments. Part 1 Apparent lessons of the liar: totality, truth and the liar possible worlds the divine liar possible ways out truth-value gaps, many-valued logics, and strengthened liars the propositional response accepting inconsistency the appeal to hierarchy barwise and etchemendy conclusion. Part 2 Truth, omniscience, and the paradox of the knower: the paradox of the knower the argument the power of the paradox possible ways out propositions and the strengthened knower redundancy theories and operators hierarchy conclusion. Part 3 Epistemic incompleteness: some ideal knowers and the standard Godel result beyond standard systems expressive incompleteness a cantorian generalization of Godel's incompleteness theorem conclusion. Part 4 Classes and quantification - the cantorian argument: there is no set of all truths extensions and applications alternative set theories - a possible way out? the problem without power sets the appeal to quantification a self-reflective problem for the central thesis? conclusion. Part 5 Concluding notes: philosophical fragments within any logic we have ... some twilight speculations.

On Godel's Second Incompleteness Theorem

A very short proof of Godel's second incompleteness theorem (for set theory, second order arithmetic etc.)

Two questions on the geometry of gauge fields

We first show that a theorem by Cartan that generalizes the Frobenius integrability theorem allows us (given certain conditions) to obtain noncurvature solutions for the differential Bianchi conditions and for higher-degree similar relations. We then prove that there is no algorithmic procedure to determine, for a reasonable restricted algebra of functions on spacetime, whether a given connection form satisfies the preceding conditions. A parallel result gives a version of Godel's first incompleteness theorem within an (axiomatized) theory of gauge fields.

The mechanism and freedom of logic

This book uses the friendly format of the computing language Prolog to teach a full formal predicate logic. With Prolog, the scope and limits of both logic and computing can be explored and experimented. Students learning formal logic in a Prolog format can begin using their already developed informal abilities in logic to program in Prolog and conversely learn enough formal logic to examine Prolog and computing in general so major fundamental theorems can be demonstrated. Cases such as Church's Thesis, Church's Theorem, Turing's Halting Problem, and Godel's Incompleteness Theorem provide the author with the means to assess some of the philosophical implications of logic and computing. Henry designed the book for undergraduate students, but it is also useful for philosophers and theologians who wish to see how computer programming serves as a probe into philosophical matters. Contents: The Formalization of Logic; Propositional Logic; Predicate Logic; Prolog: Programming in Logic; Logic Machines; The Scope and Limits of Logic and Logic Machines; Philosophical Reflections; Appendix; Bibliography; Index.

Godel's incompleteness theorems

Godel's incompleteness theorems

Computability theory: concepts and applications

Turing machines turing machines as recognisers universality undecidability alternative models post's correspondence problem recursive function theory formal models of arithmetic Godel's incompleteness theorem computer science and computability theory.

Some properties of the syntactic p-recursion categories generated by consistent, recursively enumerable extensions of Peano arithmetic.

The dominical categories were introduced by Di Paola and Heller, as a first step toward a category-theoretic treatment of the generalized first Godel incompleteness theorem [1]. In his Ph.D. dissertation [7], Rosolini subsequently defined the closely related p-categories, which should prove pertinent to category-theoretic representations of incompleteness for intuitionistic systems. The precise relationship between these two concepts is as follows: every dominical category is a pointed p-category, but there are p-categories, indeed pointed p-isotypes (all pairs of objects being isomorphic) with a Turing morphism that are not dominical. The first of these assertions is an easy consequence of the fact that in a dominical category C by definition the near product functor when restricted to the subcategory Ct, of total morphisms of C (as “total” is defined in [1]) constitutes a bona fide product such that the derived associativity and commutativity isomorphisms are natural on C × C × C and C × C, respectively, as noted in [7]. As to the second, p-recursion categories (that is, pointed p-isotypes having a Turing morphism) that are not dominical were defined and studied by Montagna in [6], the syntactic p-categories ST and S′T associated with consistent, recursively enumerable extensions of Peano arithmetic, PA. These merit detailed investigation on several counts.