This book concentrates on research done during the last twenty years on the philosophy of quantum mechanics. In particular, the author focuses on three major issues: whether quantum mechanics is an incomplete theory, whether it is non-local, and whether it can be interpreted realistically. Much of the book is concerned with distinguishing various senses in which these questions can be taken, and assessing the bewildering variety of answers philosophers and physicists have given up to now. The book is self-contained in (...) that it presents the necessary parts of the mathematical formalism of quantum mechanics and also covers other interpretative topics, such as the problem of measurement and the uncertainty relations. A considerable portion of the book is based on original arguments presented by the author in lectures and research papers over the past ten years. However, this material is integrated with a broad coverage of most of the recent research in the field, so as to provide a balanced introduction to the whole subject. (shrink)

The incompleteness problem for virtue ethics is inherited by a virtue-based theory of argumentation as developed by Daniel Cohen. A complete normative theory of argumentation should be able to provide reasons for why argumentative virtues such as open-mindedness are worthwhile, along with being able to resolve conflicts of such virtues. Adumbrating virtue-based argumentation theory with a pragmatic utilitarian approach constitutes a more complete theory that can account for why argumentative virtues are worthwhile.

The purpose of this paper is to discuss two phenomena related to the semantics of definite descriptions: that of incomplete uses of descriptions, and that of the underdetermination of referential uses of descriptions. The Russellian theorist has a way of accounting for incomplete uses of descriptions by appealing to an account of quantifier domain restriction, such as the one proposed in Stanley and Szabó (2000a). But, I argue, the Russellian is not the only one in a position to appeal to (...) such an account of incomplete uses of descriptions. Proponents of other theories, such as the Fregean, which does not treat descriptions as quantifiers, might benefit from this account of domain restriction. In the second part of the paper I discuss referential uses of incomplete definite descriptions. Relative to such uses, Wettstein (1981) and others have argued that the Russellian theory faces a problem of underdetermination of semantic content. Neale (2004) has replied to this objection showing why it does not pose a threat to the Russellian theory. Again, I argue that not only the Russellian, but also the Fregean can subscribe to Neale’s (2004) suggestion. (shrink)

Our purpose is to present some connections between modal incompleteness andmodal logics related to the Gödel-Löb logic GL. One of our goals is to prove that for all m, n, k, l ∈ ℕ the logic K + equation image□i □jp ↔ p) → equation image□ip is incomplete and does not have the fixed point property. As a consequence we shall obtain that the Boolos logic KH does not have the fixed point property.

How is constitutionalism possible, when people disagree on so many questions about what is good and what is right? The answer lies in two kinds of incompletely theorized agreement - both reached amidst the sharpest disagreements about the fundamental issues in social life. The first consist of agreements on abstract formulations ; these agreements are crucial to constitution-making as a social practice. The second consist of agreements on particular doctrines and practices; these agreements are crucial to life and law under (...) existing constitutions. Incompletely theorized agreements help illuminate an enduring constitutional puzzle: how members of diverse societies can work together on terms of mutual respect amidst intense disagreements about both the right and the good. Such agreements help make constitutions and constitutional law possible, even within nations whose citizens cannot concur on the most fundamental matters. (shrink)

The main point of this paper has been to show that the concept of evaluative incompleteness deserves consideration. In addition, I have suggested that it is plausible to accept that certain states of affairs in fact are evaluatively incomplete. But I have not sought to prove that this is so; indeed, I do not know how such proof might be given. Just which states of affairs, if any, are evaluatively incomplete is an extremely vexed question, and it is not (...) one to which I have attempted to supply any systematic answer. My aim has been merely to point out that it is arguable that certain states of affairs are evaluatively incomplete — a fact that ought not to be overlooked due to an unquestioning acceptance of (II) and a fact which, certainly, ought not to be ruled out by fiat due to an adherence to definitions and assumptions which imply that (II) is false. (shrink)

In the early seventies, several logicians developed a semantics for propositional systems of relevance logic. The essential ingredients of this semantics were a privileged point o, an ‘accessibility’ relation R and a special operator * for evaluating negation. Under the truth- conditions of the semantics, each formula A(Pl,…,Pn) could be seen as expressing a first order condition A+(pl,…,pn, o, R,*) on sets p1,…,pn and o, R, *, while each formula-scheme could be regarded as expressing the second-order condition ∀p1,…,∀pn A+(p1,…,pn, o, (...) R, *). It could then be shown that many standard systems of propositional relevance logic were complete in the sense that their theorems were just those formulas true in all models whose components o, R and * conformed to the second-order conditions expressed by the axioms of the system. (shrink)

Gödel's Incompleteness Theorems have the same scientific status as Einstein's principle of relativity, Heisenberg's uncertainty principle, and Watson and Crick's double helix model of DNA. Our aim is to discuss some new faces of the incompleteness phenomenon unveiled by an information-theoretic approach to randomness and recent developments in quantum computing.

This is an exploration of a cluster of related logical results. Taken together these seem to have something philosophically important to teach us: something about knowledge and truth and something about the logical impossibility of totalities of knowledge and truth. The book includes explorations of new forms of the ancient and venerable paradox of the :Liar, applications and extensions of Kaplan and Montague's paradox of the Knower, generalizations of Godel's work on incompleteness, and new uses of Cantorian diagonalization. Throughout, (...) the concern is with metaphysical and epistemological implications of these results: what such results may have to teach us about knowledge, truth, and possibility. (shrink)

Does the quantum state represent reality or our knowledge of reality? In making this distinction precise, we are led to a novel classification of hidden variable models of quantum theory. We show that representatives of each class can be found among existing constructions for two-dimensional Hilbert spaces. Our approach also provides a fruitful new perspective on arguments for the nonlocality and incompleteness of quantum theory. Specifically, we show that for models wherein the quantum state has the status of something (...) real, the failure of locality can be established through an argument considerably more straightforward than Bell’s theorem. The historical significance of this result becomes evident when one recognizes that the same reasoning is present in Einstein’s preferred argument for incompleteness, which dates back to 1935. This fact suggests that Einstein was seeking not just any completion of quantum theory, but one wherein quantum states are solely representative of our knowledge. Our hypothesis is supported by an analysis of Einstein’s attempts to clarify his views on quantum theory and the circumstance of his otherwise puzzling abandonment of an even simpler argument for incompleteness from 1927. (shrink)

Following Post program, we will propose a linguistic and empirical interpretation of Gödel’s incompleteness theorem and related ones on unsolvability by Church and Turing. All these theorems use the diagonal argument by Cantor in order to find limitations in finitary systems, as human language, which can make “infinite use of finite means”. The linguistic version of the incompleteness theorem says that every Turing complete language is Gödel incomplete. We conclude that the incompleteness and unsolvability theorems find limitations (...) in our finitary tool, which is our complete language. (shrink)

Hannah Arendt is commonly criticized for defining action as an end in itself, of horizontal power and of councils as an alternative to representative democracy that is unrealistic and unrealizable. In contrast, I show how much Arendt was concerned about the dangers of these concepts: the replacement of action by fabrication, of power by domination and violence, and the impossibility of a council democracy in the face of a highly administrative and depoliticized society. Arendt shows how these hybrid forms revealed (...) their pure forms as a hidden tradition in revolutionary situations. The supposed incompleteness and failure of a struggle for another society can therefore be understood as a recurring possibility and reality and thus as a specific condition of the political. The critique of the hybrid form also includes the critique of modern scientific thought and political thought. (shrink)

This paper introduces the axiom of Negative Dominance, stating that if a lottery f is strictly preferred to a lottery g, then some outcome in the support of f is strictly preferred to some outcome in the support of g. It is shown that if preferences are incomplete on a sufficiently rich domain, then this plausible axiom, which holds for complete preferences, is incompatible with an array of otherwise plausible axioms for choice under uncertainty. In particular, in this setting, Negative (...) Dominance conflicts with the standard Independence axiom. A novel theory, which includes Negative Dominance, and rejects Independence, is developed and shown to be consistent. (shrink)

According to the displacement model of secularization, religious-theological concepts, themes, and values have been reinterpreted in non-religious contexts without fully dispensing with the religious content. Secularization is thus incomplete. The incomplete secularization argument can be used as a lens through which to read Ethan Kleinberg’s deconstructive approach to the past. In his narrative, as reconstructed here, deconstruction promises to bring us closer to a secular relationship to the past than the ontological realism Kleinberg says still dominates contemporary historical theory. By (...) contrasting Kleinberg’s analysis with Hayden White’s, whose oeuvre can be read as structured by the idea of incomplete secularization and a wish to liberate history from religious themes in order to enable a direct confrontation with meaninglessness, I argue that Kleinberg’s deconstructive approach does not fulfill its promise. Rather, it opens up a post-secular historiography in which religious themes might find a place at the very heart of historical reasoning. (shrink)

This paper addresses the phenomenon of incomplete preferences in disaster risk management. If an agent finds two options to be incomparable and thus has an incomplete preference ordering, i.e., neither prefers one option over the other nor finds them equally as good, it is not possible for the agent to perform a value tradeoff, necessary for an informed decision, between these two options. In this paper we suggest a way to model incomplete preference orderings by means of probabilistic preferences, and (...) how to reveal an agent’s incomplete preference ordering within a behaviorist framework. (shrink)

Incomplete Archaeologies takes a familiar archaeological concept--assemblages--and reconsiders such groupings, collections and sets of things from the perspective of the work required to assemble them. The discussions presented here engage with the practices of collection, construction, performance and creation in the past (and present) which constitute the things and groups of things studied by archaeologists--and examine as well how these things and thing-groups are dismantled, rearranged, and even destroyed, only to be rebuilt and recreated. The ultimate aim is to reassert (...) an awareness of the incompleteness of assemblage, and thus the importance of practices of assembling (whether they seem at first creative or destructive) for understanding social life in the past as well as the present. The individual chapters represent critical engagements with this aim by archaeologists presenting a broad scope of case studies from Eurasia and the Mediterranean. Case studies include discussions of mortuary practice from numerous angles, the sociopolitics of metallurgy, human-animal relationships, landscape and memory, the assembly of political subjectivity and the curation of sovereignty. These studies emphasise the incomplete and ongoing nature of social action in the past, and stress the critical significance of a deeper understanding of formation processes as well as contextual archaeologies to practices of archaeology, museology, art history, and other related disciplines. Contributors challenge archaeologists and others to think past the objects in the assemblage to the practices of assembling, enabling us to consider not only plural modes of interacting with and perceiving things, spaces, human bodies and temporalities in the past, but also to perhaps discover alternate modes of framing these interactions and relationships in our analyses. Ultimately then, Incomplete Archaeologies takes aim at the perceived totality not only of assemblages of artefacts on shelves and desks, but also that of some of archaeology's seeming-seamless epistemological objects"--From publisher's website. (shrink)

I discuss two competing theories of the progressive: the theory proposed in Parsons (1980, 1985, 1989, 1990) and the theory proposed in Landman (1992). These theories differ in more than one way. Landman regards the progressive as an intentional operator, while Parsons doesn't. Moreover, Landman and Parsons disagree on what uninflected predicates denote. For Landman, cross the street has in its denotation complete events of crossing the street; the aspectual contribution of English simple past (perfective aspect) is the identity function. (...) For Parsons, both complete and incomplete events of crossing the street can be in the denotation of the base VP; perfective aspect restricts its denotation to the events that culminate. I present a version of ParsonÕs theory that avoids the problems raised by Landman, in particular the problem posed for Parsons by creation verbs. The repaired version and Landman's theory still differ in the way they analyze uninflected verbs. The repaired version and Landman's theory still differ in the way they analyze uninflected predicates. I present evidence from Slavic languages that both theories are needed. Finally, I discuss some evidence that may favor one or the other approach to the semantics of the English progressive. (shrink)

In the presence of incomplete risk attitudes, choices between noncomparable risky prospects are random. A random choice model advanced by Karni, 2021) includes the hypothesis that choices among noncomparable risky prospects are prompted by signals drawn from personal distributions. This paper introduces a scheme designed to elicit subjects’ assessments of their personal likelihoods of choices among noncomparable risky prospects and describes experiments designed to test the aforementioned hypothesis.

In this paper, an incompleteness theorem for modal extensions of relevant logics is proved. The proof uses elementary methods and builds upon the work of Fuhrmann.

This paper examines the incompleteness of collective preference. We provide a series of Arrovian impossibility theorems without completeness. First, we consider the notion of regularity introduced by Eliaz and Ok (2006, Games and Economic Behavior 56, 61–86); it is an appropriate richness property for strict preference when preference is allowed to be incomplete. We examine the implication of imposing regularity on collective preference. Second, we propose responsiveness, a variation of positive responsiveness. This axiom requires that some changes in individual (...) preferences make an alternative weakly better than another. Third, we consider coherency conditions for collective preferences; this conditionally requires the existence of comparable pairs in a certain manner. We prove an impossibility result for each condition using Arrovian axioms. (shrink)

The dissertation gives a second-order-logic-based explanation of modal incompleteness. The leading concept is that modal incompleteness is to be explained in terms of the incompleteness of standard second-order logic, since modal language is basically a second-order language. The development of Kripke-style semantics for modal logic has been underpinned by the conjecture that all modal systems are characterizable by classes of frames defined by first-order conditions on a binary relation. However, the discovery of certain incomplete modal systems has (...) undermined the all-encompassing feature of Kripke-style approach. There are logics not determined by any class of Kripke frames at all. ;In the dissertation I investigate a normal incomplete sentential modal system due to J. F. A. K. Van Benthem, and I address both the formal and the philosophical facets of modal incompleteness from the vantage point that modal logic is essentially second-order in its nature. Modal systems can be analyzed in terms of structures with a domain of second-order individuals that are assigned under an interpretation to propositional variables within languages of sentential modal logic. Hence, the phenomenon of there being incomplete modal systems can be accounted for through the incompleteness of standard second-order logic. ;Since second-order logic plays a crucial role in the explanation, Quine's animadversions upon second-order logic and in particular his views that second-order logic is 'Set Theory in Sheep's Clothing' are examined. Against Quine's stance I will seek to show that a vindication of second-order logic can be gotten provided a proper due is given to the sharp distinction between the logical notion of set which is the concern of second-order logic and the iterative notion of set which lies within the realm of set theory. (shrink)

When processing of stimuli occurs without attention, phenomenal experience, as well as cognitive access, may be lost. Sensory representations are, however, constructed by neural machinery extending far beyond sensory receptors. In conditions such as cerebral achromatopsia incomplete sensory representations may still elicit phenomenal experience but these representations might be too aberrant to be integrated into the wider cognitive workspace.

Bayesian models typically assume that agents are rational, logically omniscient and opinionated. The last of these has little descriptive or normative appeal, however, and limits our ability to describe how agents make up their minds (as opposed to changing them) or how they can suspend or withdraw their opinions. To address these limitations this paper represents the attitudinal states of non-opinionated agents by sets of (permissible) probability and desirability functions. Several basic ways in which such states of mind can be (...) changed are then characterised and compared with those found in AGM style models of attitude revision. Finally these models are employed to describe how agents make up their mind when deliberating. (shrink)

This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) (...) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimanoff’s paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth. (shrink)

Slavoj Žižek is known for quoting with the same enthusiasm the main names of Western Philosophy and the classics of pop culture, cinema, literature, and music. Therefore, in such rich theoretical framework, it is possible to glimpse a few connections that the philosopher himself has not yet developed in detail. This essay is precisely about of these connections. More specifically, this essay can be seen as an endeavor to think some of Žižek’s writings on music having as a main reference (...) the concept of ontological incompleteness. The philosopher discusses the concept of ontological incompleteness based on several areas, from sciences to movies and literature. Notwithstanding, in this specific case musical examples are left aside. Even though Žižek does not analyse music from the perspective of such concept of ontology, it is possible to argue that Žižek’s writings on music already leave some clues and examples that justify an analysis of that kind. That is the task proposed in this critical essay: to present how an approximation between his thoughts on music and on ontology is pos. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Thomas G. Pavel INCOMPLETE WORLDS, RITUAL EMOTIONS' IN recent years, the notion of "fictional world" has enjoyed a considerable rise in fortune. The expression, however, is not entirely new. To refer to the world of a literary work, of a novel or of a play, has always been a favorite way of speaking for literary critics and aestheticians. In most cases, these were informal worlds. A discussion of the (...) world of Balzac, Dostoyevsky, HenryJames, or Proust often amounts to a description of each author's preferences in die selection and rearrangement of his material. Such discussion primarily fits authors who purposefully attempt to build comprehensive representational artifacts. One speaks quite naturally about l'univers balzacien, or about the universe of Musil. But is one entitled to refer with equal ease to the world ofBenjamin Constant, meagerly perceptible in his Adolphe, a short novel built in the spirit of economy, displaying just a few characters and a minimum of worldly information? Perhaps density and comprehensiveness are not indispensable characteristics of literary worlds. Aestheticians and narratologists using the "fictional worlds" terminology seem to believe mat any kind of artistic text projects a world, be it rich or poor. Such worlds are imagined in the fashion of modern philosophy of logic as states of affairs, more or less comprehensive. Some writers believe that the states of affairs comprised in fictional worlds must be fairly sizable, since diey should include "not only what the author indicates, but also whatever is required (entailed) by that which he indicates." 1 The laws of nature which are not specifically contradicted by the text would belong to its world. Thus, a few notorious cases aside, there is no reason to doubt that every child born in fiction has been engendered by a human fadier as long as the text does not report an exception. And if it does? Then, assuming that the laws of logic still govern if not fiction itself at least our understanding of it, we may appeal to what "possible world" philosophers have called "small miracles." * Editor's note: This article, as well as the contributions by Gerald Prince, Peter van Inwagen, and Kendall Walton which follow it, grew out ofa symposium on "Styles ofFictionality," organized by Professor Pavel and held at Harvard University in 1981. The Fall, 1983 issue of PHILOSOPHY AND LITERATURE will contain additional papers presented at the symposium. 48 Thomas G. Pavel49 Imagine, for instance, the world in which Notre Dame de Paris would be covered with blue paint. While everything else would preserve the properties it possesses in the actual world, with the laws of nature and the caused chains maintained, a small local change, somewhat miraculous, would modify only the color of the venerable building. Similarly, the world of Balzac may be understood as the world of early and mid-nineteenth-century France, slightly modified by the addition of a few hundred imagined characters. That diese conform surprisingly well to the general laws of French society of the time is so much the goal of Balzac's successful project, that for the sociologically-minded reader die' actual nonexistence of Rastignac, Rubempré, Vautrin, or Birotteau may appear a mere accident. These characters, one may surmise, display every trait that actual counterparts of theirs would possess, except existence. Hence, why would Rastignac's or Vautrin's emergence in a world omerwise identical to nineteenth-century France amount to anydiing more than a "small miracle"? On the other hand, Balzac's world being blessed wim hundreds of such miracles, the very magnitude of the project vindicates the attribute of "visionary," ascribed to Balzac by Béguin, or, if metaphorically undercut, mat of"God the Father" given him by Thibaudet. For, would not such magnitude entail more than just local changes in an otherwise faithful representation of the "actual" nineteenthcentury France? Similarly, would we interpret a fictional world in which a child has been engendered not by a human father, but by a divinity, as identical in all respects to the actual world, except for the limited modification attributed to a small miracle? 2 If we grant for a moment the improbable assumption that nineteenth-century France was a given and that Balzac... (shrink)

Early in Principia Mathematica Russell presents an argument that "‘the author of Waverley’ means nothing", an argument that he calls a "definite proof". He generalizes it to claim that definite descriptions are incomplete symbols having meaning only in sentential context. This Principia "proof" went largely unnoticed until Russell reaffirmed a near-identical "proof" in his philosophical autobiography nearly 50 years later. The "proof" is important, not only because it grounds our understanding of incomplete symbols in the Principia programme, but also because (...) failure to understand it fully has been a source of much unjustified criticism of Russell to the effect that he was wedded to a naive theory of meaning and prone to carelessness and confusion in his philosophy of logic and language generally. In my paper, I (1) defend Russell’s "proof" against attacks from several sources over the last half century, (2) examine the implications of the "proof" for understanding Russell’s treatment of class symbols in Principia, and (3) see how the Principia notion of incomplete symbol was carried forward into Russell’s conception of philosophical analysis as it developed in his logical atomist period after 1910. (shrink)

In this paper we formulate a version of Second Incompleteness Theorem. The idea is that a sequential sentence has ‘consistency power’ over a theory if it enables us to construct a bounded interpretation of that theory. An interpretation of V in U is bounded if, for some n , all translations of V -sentences are U -provably equivalent to sentences of complexity less than n . We call a sequential sentence with consistency power over T a pro-consistency statement for (...) T . We study pro-consistency statements. We provide an example of a pro-consistency statement for a sequential sentence A that is weaker than an ordinary consistency statement for A . We show that, if A is $${{\sf S}^{1}_{2}}$$ , this sentence has some further appealing properties, specifically that it is an Orey sentence for EA . The basic ideas of the paper essentially involve sequential theories. We have a brief look at the wider environment of the results, to wit the case of theories with pairing. (shrink)

Cappelen and Lepore (2005) maintain that Incompleteness Arguments for context sensitivity are fallacious. In their view, Incompleteness Arguments are non sequitur fallacies whose conclusions are not logically related to premises. They affirm that the conclusions of Incompleteness Arguments are metaphysical claims about the existence of entities that might be constituents of propositions, while their premises concern psychological data about speakers' dispositions to truth evaluate sentences in contexts of utterance. Cappelen and Lepore reject Incompleteness Arguments because psychological (...) data have no bearing on metaphysical issues. I do not dispute Cappelen and Lepore's claim that psychological data have no relevance to metaphysics. Nonetheless, I argue that Cappelen and Lepore's criticism is vitiated by a misunderstanding of the nature of Incompleteness Arguments that makes them overlook the link between semantics and linguistic competence. I try to shed light on the real nature of Incompleteness Arguments by pointing at the failure of Cappelen and Lepore's criticism. My aim is not to defend Semantic Contextualism, but to bring to attention the link between semantics and linguistic competence that should never be overlooked in debates in the field of semantics. (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.

If market transactions are optimal, why do so many transactions occur within firms themselves? Ronald H. Coase explains this phenomenon by arguing that market transaction costs differ from intra-company transaction costs and that clear intra-intra-firm property rights have the effect of reducing transaction costs. But what exactly are the relevant transaction costs, and what factors determine them? Oliver Hart argues that market contracts are incomplete, and that the key to improving efficiency is putting the power to deal with these unspecified (...) circumstances into the hands of owners within the same entity.In this paper, I argued that, the development of the theory and practice of business ethics as well as China’s innovative practice in recent decades provide a new perspective, one that is especially relevant to the issues raised by Case and Hart and that bear directly on the reform of China’s state-owned enterprises. (shrink)

Why these notes? After all, I’ve written An Introduction to Gödel’s Theorems. Surely that’s more than enough to be going on with? Ah, but there’s the snag. It is more than enough. In the writing, as is the way with these things, the book grew far beyond the scope of the lecture notes from which it started. And while I hope the result is still pretty accessible to someone prepared to put in the time and effort, there’s a lot more (...) in the book than is really needed by philosophers meeting the incompleteness theorems for the first time. After all, you might want to get your heads around only those technical basics which are actually needed for understanding philosophical discussions about incompleteness. So you need a cut-down version of the book – an introduction to the Introduction! Well, isn’t that what lectures are for? Indeed. But there’s another snag. I haven’t got many lectures to play with. So either I crack on at quite a fast pace, cover those basics, but perhaps leave too many people puzzled and alarmed. Or I do relaxed talk’n’chalk, highlighting the really Big Ideas, making sure everyone is grasping them as we go along, but inevitably omit important stuff and leave quite a gap between what happens in the lectures and what happens in the book. What to do? I’m going for plan. But then I still need to do something to fill that gap between lectures and book. Hence these notes. The idea, then, is to give relaxed lectures, highlighting Big Ideas, not worrying too much about depth or fine-detail. Then after the lecture, I’ll write up notes that expand things just enough, and then give pointers to relevant chunks of IGT. The idea, however, is for the notes to be more or less stand-alone, and to tell a brief but coherent story read by themselves. So occasionally I’ll copy a paragraph or two from the book, rather than just refer to them. Warning: just occasionally in these notes, I’ll no doubt apply that good maxim ‘Where it doesn’t itch, don’t scratch’. (shrink)

The truth-table degree of the set of shortest programs remains an outstanding problem in recursion theory. We examine two related sets, the set of shortest descriptions and the set of domain-random strings, and show that the truth-table degrees of these sets depend on the underlying acceptable numbering. We achieve some additional properties for the truth-table incomplete versions of these sets, namely retraceability and approximability. We give priority-free constructions of bounded truth-table chains and bounded truth-table antichains inside the truth-table complete degree (...) by identifying an acceptable set of domain-random strings within each degree. (shrink)

Russell urged that some phrases having no meaning in isolation could nonetheless, Contribute to the meaning of sentences in which they occur. In the case of definite descriptive phrases, A proof is offered. It is argued that russell's proof is valid, Contrary to some commentators. Proper understanding of the notion of "incomplete symbol" plays a key role in the assessment of the argument, As well as in full appreciation of the radical departure of russell's analysis from "surface" grammar.

The implicit content associated with incomplete definite descriptions is contributed in the form of definite descriptions of situations. A definite description of this kind is contributed by a small structure in the syntax, which is interpreted, in general terms, as ‘the situation that bears R to s’.

whole numbers that manages to assert that it itself is unprovable (from a given finite set F of axioms using formal logic). (Gödel's paper is included in the well-known anthology [1].) GF : ``GF cannot be proved from the finite set of axioms F.'' This assertion GF is therefore true if and only if it is unprovable, and the formal axiomatic system F in question either proves falsehoods (because it enables us to prove GF) or fails to prove a true (...) assertion (because it does not enable us to prove GF). If we assume that the former situation is impossible, we conclude that F is necessarily incomplete since it does not permit us to establish the true statement GF. (shrink)

We construct a recursive ultrapower F/U such that F/U is a tame 1-model in the sense of [6, §3] and FU is existentially incomplete in the models of II2 arithmetic. This enables us to answer in the negative a question about closure with respect to recursive fibers of certain special semirings Γ of isols termed tame models by Barback. Erik Ellentuck had conjuctured that all such semirings enjoy the closure property in question. Our result is that while many do, some (...) do not. (shrink)

This paper shows that some classes of multimodal paraconsistent logics endowed with weak forms of negation are incompletable with respect to Kripke semantics. The reach of such incompleteness is discussed, and we argue that this shortcoming, more than just a logical predicament, may be relevant for attempts to characterize quantum logics and to handle quantum information and quantum computation.

In Episode 1, we introduced the very idea of a negation-incomplete formalized theory T . We noted that if we aim to construct a theory of basic arithmetic, we’ll ideally like the theory to be able to prove all the truths expressible in the language of basic arithmetic, and hence to be negation complete. But Gödel’s First Incompleteness Theorem says, very roughly, that a nice theory T containing enough arithmetic will always be negation incomplete. Now, the Theorem comes in (...) two flavours, depending on whether we cash out the idea of being ‘nice enough’ in terms of (i) the semantic idea of T ’s being a sound theory, or (ii) the idea of odel’s own T ’s being a consistent theory which proves enough arithmetic. And we noted that G¨. (shrink)

We present an analogue of Gödel’s second incompleteness theorem for systems of second-order arithmetic. Whereas Gödel showed that sufficiently strong theories that are $\Pi ^0_1$ -sound and $\Sigma ^0_1$ -definable do not prove their own $\Pi ^0_1$ -soundness, we prove that sufficiently strong theories that are $\Pi ^1_1$ -sound and $\Sigma ^1_1$ -definable do not prove their own $\Pi ^1_1$ -soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.

We show how second order logic incompleteness follows from incompleteness of arithmetic, as proved by Gödel.

Full proofs of the Gödel incompleteness theorems are highly intricate affairs. Much of the intricacy lies in the details of setting up and checking the properties of a coding system representing the syntax of an object language (typically, that of arithmetic) within that same language. These details are seldom illuminating and tend to obscure the core of the argument. For this reason a number of efforts have been made to present the essentials of the proofs of Gödel’s theorems without (...) getting mired in syntactic or computational details. One of the most important of these efforts was made by Löb [8] in connection with his analysis of sentences asserting their own provability. Löb formulated three conditions (now known as the Hilbert-Bernays-Löb derivability conditions), on the provability predicate in a formal system which are jointly sufficient to yield the Gödel’s second incompleteness theorem for it. A key role in Löb’s analysis is played by (a special case of) what later became known as the diagonalization or fixed point property of formal systems, a property which had already, in essence, been exploited by Gödel in his original proofs of the incompleteness theorems. The fixed point property plays a central role in Lawvere’s [7] category-theoretic account of incompleteness phenomena (see also [10]). Incompleteness theorems have also been subjected to intensive investigation within the framework of modal logic (see, e.g.[4], [5]). In this formulation the modal operator takes up the role previously played by the provability predicate, and the derivability conditions on the latter are translated into algebraic conditions (the so-called GL, i.e., Gödel–Löb, conditions) on the former. My purpose here is to present a framework for incompleteness phenomena, fully compatible with intuitionistic or constructive principles, in which the idea of a coding system is retained, only in a 2 simple, but very general form, a form wholly free of syntactical notions. As codes we shall take the elements of an arbitrary given nonempty set, possibly, but not necessarily, the set of natural numbers.. (shrink)

We define a property R(A 0 , A 1 ) in the partial order E of computably enumerable sets under inclusion, and prove that R implies that A 0 is noncomputable and incomplete. Moreover, the property is nonvacuous, and the A 0 and A 1 which we build satisfying R form a Friedberg splitting of their union A, with A 1 prompt and A promptly simple. We conclude that A 0 and A 1 lie in distinct orbits under automorphisms of (...) E, yielding a strong answer to a question previously explored by Downey, Stob, and Soare about whether halves of Friedberg splittings must lie in the same orbit. (shrink)