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The Schrodinger’s cat paradox has troubled physicists for many years and many ways to solve it has been provided so far. In this paper I will try to provide a different method of solving the Paradox with the help of logical machine and it will look like a hybrid of Wigner’s no-go theorem and the machine of Godel in his Incompleteness Theorems. And due to the machine the paradox will not require any human intervention to measure the state of the cat, whether it is alive or dead, while the measurement will be made by humans with regard to the statement given out by the machine.

We give a survey of current research on G\"{o}del's incompleteness theorems from the following three aspects: classifications of different proofs of G\"{o}del's incompleteness theorems, the limit of the applicability of G\"{o}del's first incompleteness theorem, and the limit of the applicability of G\"{o}del's second incompleteness theorem., Comment: 54 pages, final accepted version, to appear in The Bulletin of Symbolic Logic

We investigate relationships between versions of derivability conditions for provability predicates. We show several implications and non-implications between the conditions, and we discuss unprovability of consistency statements induced by derivability conditions. First, we classify already known versions of the second incompleteness theorem, and exhibit some new sets of conditions which are sufficient for unprovability of Hilbert--Bernays' consistency statement. Secondly, we improve Buchholz's schematic proof of provable $\Sigma_1$-completeness. Then among other things, we show that Hilbert--Bernays' conditions and L\"ob's conditions are mutually incomparable. We also show that neither Hilbert--Bernays' conditions nor L\"ob's conditions accomplish G\"odel's original statement of the second incompleteness theorem., Comment: 36 pages

The proofs of Gödel (1931), Rosser (1936), Kleene (first 1936 and second 1950), Chaitin (1970), and Boolos (1989) for the first incompleteness theorem are compared with each other, especially from the viewpoint of the second incompleteness theorem. It is shown that Gödel’s (first incompleteness theorem) and Kleene’s first theorems are equivalent with the second incompleteness theorem, Rosser’s and Kleene’s second theorems do deliver the second incompleteness theorem, and Boolos’ theorem is derived from the second incompleteness theorem in the standard way. It is also shown that none of Rosser’s, Kleene’s second, or Boolos’ theorems is equivalent with the second incompleteness theorem, and Chaitin’s incompleteness theorem neither delivers nor is derived from the second incompleteness theorem. We compare (the strength of) these six proofs with one another.

In this article, one presents how some of the most important mathematicians of the early twentieth century – those responsible for organizing mathematics into algebraic, topological and order structures, which one studied in undergraduate courses in mathematics – received the results established by Godel´s Incompleteness Theorem. One understands that the stance taken by Bourbaki group and revealed in his works thus claims how the mathematicians of that time welcomed the incompleteness theorem and handled with its consequences. The attitude assumed by the group was to continue doing Mathematics with the same ideal of complete formalization, although they faced with the proof of the existence of a non-empty set of true and undemonstrable propositions – via mathematical tools for producing this science – and of the incompleteness of any theory that contains Peano axioms. One also presents Wittgenstein's perspective on Godel's Incompleteness Theorem. Finally, one apprehends that this theorem is understood by the mathematical community as a messenger of incompleteness as a characteristic inherent to axiomatization, not as an impediment to the continuation of the activity with formal systems, but as an invigorating result for Mathematics.

A cornerstone of modern mathematical logic is the diagonal lemma of Gödel and Carnap. It is used in, for example, the classical proofs of the theorems of Gödel, Rosser, and Tarski. From its first explication in 1934, just essentially one proof has appeared for the diagonal lemma in the literature; a proof that is so tricky and hard to relate that many authors have tried to avoid the lemma altogether. As a result, some so-called diagonal-free proofs have been given for the above-mentioned fundamental theorems of logic. In this paper, we provide new proofs for the semantic formulation of the diagonal lemma, and for a weak version of the syntactic formulation of it.

Informally speaking, the categoricity of an axiom system means that its non-logical symbols have only one possible interpretation that renders the axioms true. Although non-categoricity has become ubiquitous in the second half of the twentieth century whether one looks at number theory, geometry or analysis, the first axiomatizations of such mathematical theories by Dedekind, Hilbert, Huntington, Peano and Veblen were indeed categorical. A common resolution of the difference between the earlier categorical axiomatizations and the more modern non-categorical axiomatizations is that the latter derive their non-categoricity from Skolem's Paradox and Godel's Incompleteness Theorems, while the former, being second order, suffer from a heavy reliance on meta-theory, where the Skolem-Godel phenomenon re-emerges. Using second-order meta-theory to avoid non-categoricity of the meta-theory would only seem to lead to an infinite regress. In this article we maintain that internal categoricity breaks with this traditional picture. It applies to both first- and second-order axiomatizations, although in the first-order case we have so far only examples. It does not depend on the meta-theory in a way that would lead to an infinite regress. It also covers the classical categoricity results of early researchers. In the first-order case it is weaker than categoricity itself, and in the second-order case stronger. We provide arguments to suggest that internal categoricity is the "right" concept of categoricity.

It is quite well-known from Kurt Godel's (1931) ground-breaking result on the Incompleteness Theorem that rudimentary relations (i.e., those definable by bounded formulae) are primitive recursive, and that primitive recursive functions are representable in sufficiently strong arithmetical theories. It is also known, though perhaps not as well-known as the former one, that some primitive recursive relations (and functions) are not rudimentary. We present a simple and elementary proof of this fact in the first part of this paper. In the second part, we review some possible notions of representability of functions studied in the literature, and give a new proof of the equivalence of the weak representability with the (strong) representability of functions in sufficiently strong arithmetical theories. Our results shed some new light on the notions of rudimentary, primitive recursive, and representable functions and relations, and clarify, hopefully, some misunderstandings and confusing errors in the literature.

While Garfinkel’s early work, captured in Studies in Ethnomethodology (Garfinkel 1967), has received a lot of attention and discussion, this has not been the case for his later work since the 1970s (e.g., Garfinkel 1986, 2002). In this paper, we critically examine the aims of Garfinkel’s later ethnomethodological studies of work programme and evaluate key ideas such as the ‘missing what’ in the sociology of work, ‘the unique adequacy requirements of methods’, and the notion of ‘hybrid studies’. We do so through a detailed engagement with a study that has frequently been singled out as exemplary by Garfinkel for his studies of work programme, namely Livingston’s (1986) The Ethnomethodological Foundations of Mathematics. We show how Livingston uses the proof of Godel’s Incompleteness Theorem as a way to exhibit the work involved in understanding mathematical proofs. We then discuss how Livingston uses this example to introduce a distinction between the written ‘proof account’ (or ‘proof description’) and the associated ‘lived work’ of working through that proof, which we argue allows Livingston to provide a powerful critique of a formalist understanding of the objectivity of mathematics. We then discuss three aspects that we find problematic in Livingston’s and Garfinkel’s claims about Livingston’s study. Firstly, we question whether written proofs are best conceived of as descriptions or accounts. Secondly, we interrogate whether an ethnomethodological study could teach mathematicians how to make discoveries. Thirdly, we throw doubt on Garfinkel’s claim that Livingston’s results are results in mathematics. We conclude this paper with a discussion of how Livingston’s study Godel’s proof highlights key tensions in Garfinkel’s later work. Firstly, we argue that there exists an ambiguity in Garfinkel’s treatment of texts as ‘incompetent’. Secondly, we show that Garfinkel’s attempt to extend his idea of classical studies from sociology to other professions and disciplines is problematic. Finally, we question Garfinkel’s proposals to reorient ethnomethodological studies away from sociological audiences and ask whether ethnomethodological studies promise the delivery of a ‘large prize’ or, rather, provide something like ‘helpful therapy’.

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.

The article attempts to provide a rather concise overview of the life and work of Jean van Heijenoort (1912–1986) in the field of the history of logic. Information of a biographical nature is given, in which the scientist’s attitude towards Marxism is noted and his close cooperation with L.D. Trotsky in a certain period; disappointment in the key principles of the Marxist doctrine and especially its practical implementation, but the preservation of the general interest in this political trend. Based on some of the unpublished materials obtained from the American Mathematics Archive in Austin, Texas, van Heijenoort’s works in the field of the history of logic described. It is noted that he received a mathematical education as a geometer and a topologist, but his interests were transferred to logic, which, by the nature of its search for proof and the canons of reasoning, were close to geometry. Particular emphasis placed in the article on the circumstances of the work on the fundamental anthology of the development of logical thought "From Frege to Godel", compiled and commented by van Heijenoort and a number of his colleagues. Attention paid to the fact that the algebraic direction of the development of mathematical logic remains outside the anthology. We describe the works of van Heijenoort relating to the incompleteness theorem, and the first two volumes of the collected works of K. Godel in the early 1980s, interrupted by the death of van Heijenoort. The assessments of his colleagues on van Heijenoort’s legacyin the history of logic provided.

Kurt Gödel’s incompleteness theorems and the limits of knowledge In this paper a presentation is given of Kurt Gödel’s pathbreaking results on the incompleteness of formal arithmetic. Some biographical details are provided but the main focus is on the analysis of the theorems themselves. An intermediate level between informal and formal has been sought that allows the reader to get a sufficient taste of the technicalities involved and not lose sight of the philosophical importance of the results. Connections are established with the work of Alan Turing and Hao Wang to show the present-day relevance of Gödel’s research and how it relates to the limitations of human knowledge, mathematical knowledge in particular.

It is a widespread belief that results like Godel’s incompleteness theorems or the intrinsic randomness of quantum mechanics represent fundamental limitations to humanity’s strive for scientific knowledge. As the argument goes, there are truths that we can never uncover with our scientific methods, hence we should be humble and acknowledge a reality beyond our scientific grasp. Here, I argue that this view is wrong. It originates in a naive form of metaphysics that sees the physical and Platonic worlds as a collection of things with definite properties such that all answers to all possible questions exist ontologically somehow, but are epistemically inaccessible. This view is not only a priori philosophically questionable, but also at odds with modern physics. Hence, I argue to replace this perspective by a worldview in which a structural notion of ‘real patterns’, not ‘things’ are regarded as fundamental. Instead of a limitation of what we can know, undecidability and unpredictability then become mere statements of undifferentiation of structure. This gives us a notion of realism that is better informed by modern physics, and an optimistic outlook on what we can achieve: we can know what there is to know, despite the apparent barriers of undecidability results.

Although there does not seem to be a direct influence of utilitarianism on David Hilbert and his School, there are some pragmatic aspects in the foundational investigations of the father of modern axiomatics and of some of his scholars that can be read in utilitarian terms. In this article, I detail these aspects, focusing on the foundational research of Hilbert throughout his career and the later methodological reflections of John von Neumann on the mathematical sciences. As for research on Hilbert, much attention is paid to the role played by the notion of success as an alternative epistemological criterion for the introduction of powerful conceptual tools within axiomatic theories, as well as for the justification of these theories in the case where consistency proofs were not available. Ironically, after the discovery of Gödel’s incompleteness theorems, success became a plain epistemological criterion for the justification of axiomatized sciences, together with all the conceptual tools contained therein. In his later foundational reflections, von Neumann, in particular, a former mathematician of Hilbert’s School, subscribed to an opportunistic methodology of science, i.e. a methodology that tried to coherently develop Hilbert’s thought on success, as well as his axiomatic reform of mathematical sciences. There seems to be, therefore, a clear evolutionary line that can be traced from Hilbert to von Neumann, which also reveals utilitarian aspects (broadly conceived) not only in science, but in axiomatics as well. Bien qu’il ne semble pas y avoir d’influence directe de l’utilitarisme sur Hilbert et son École, dans la recherche fondamentale du père de l’axiomatique moderne, comme dans celles de certains de ses élèves, il y a des aspects pragmatiques qui peuvent être lus en termes utilitaires. Dans cet article, je considère ces aspects en m’attardant sur toute la recherche fondamentale de Hilbert et sur les réflexions méthodologiques ultérieures de von Neumann sur les sciences mathématiques. Quant à Hilbert, je me concentre, en particulier, sur le rôle que la notion de succès a joué dans sa pensée comme critère épistémologique alternatif pour l’introduction d’outils conceptuels très puissants au sein de théories scientifiques construites axiomatiquement et pour la justification de ces mêmes théories dans le cas où des preuves de cohérence appropriées ne seraient pas disponibles. Étonnamment, après la découverte par Gödel des théorèmes d’incomplétude, le succès est devenu en fait un critère épistémologique pour la justification des sciences construites de manière axiomatique et des outils conceptuels qu’elles contiennent. Dans ses dernières considérations fondamentales, von Neumann, l’un des mathématiciens ayant appartenu à l’École de Hilbert dans sa jeunesse, a adopté une méthodologie opportuniste explicite de la science: une méthodologie de la science, c’est-à-dire qui cherchait à développer de manière cohérente les considérations de Hilbert sur le succès et en même temps proposer une réforme axiomatique des sciences mathématiques. Il semble donc y avoir une ligne évolutive claire qui peut être tracée de Hilbert à von Neumann et qui révèle des aspects utilitaires (au sens large) présents non seulement dans la science, mais aussi dans l’axiomatique lui-même. Sebbene non sembri esserci un’influenza diretta dell’utilitarismo su Hilbert e sulla sua Scuola, nelle ricerche fondazionali del padre dell’assiomatica moderna, come pure in quelle di alcuni suoi allievi, sono presenti aspetti pragmatici che è possibile leggere in termini utilitaristici. Nell’articolo considero questi aspetti soffermandomi sull’intera indagine fondazionale di Hilbert e sulle riflessioni metodologiche più tarde di von Neumann relativamente alle scienze matematiche. Quanto ad Hilbert, mi concentro, in particolare, sul ruolo che nel suo pensiero la nozione di successo svolgeva in quanto criterio epistemologico alternativo per l’introduzione di strumenti concettuali molto potenti all’interno di teorie scientifiche costruite in modo assiomatico e per la giustificazione di queste stesse teorie nel caso in cui non fossero state disponibili opportune dimostrazioni di coerenza. Sorprendentemente, dopo la scoperta dei teoremi di incompletezza da parte di Gödel, il successo divenne a tutti gli effetti un criterio epistemologico per la giustificazione delle scienze costruite in modo assiomatico e degli strumenti concettuali contenuti al loro interno. Nelle sue tarde considerazioni fondazionali, von Neumann, uno dei matematici che in gioventù fece parte della scuola di Hilbert, sposò un’esplicita metodologia opportunistica della scienza: una metodologia della scienza, cioè, che cercava di sviluppare in modo coerente le considerazioni di Hilbert sul successo e che proponeva nel contempo una riforma assiomatica delle scienze matematiche. Sembra così esserci una chiara linea evolutiva che può essere tracciata da Hilbert a von Neumann e che rivela aspetti utilitaristici (intesi in senso ampio) presenti non solo nella scienza, ma anche nella stessa assiomatica.

Publisher Copyright: © Sandu G. This paper contains some philosophical reflections on Gödelian (undecidable) sentences and the recognition of their truth using semantic arguments. These reflections are not new, similar matters have been extensively addressed in the philosophical literature. The matter is rather one of emphasis.

It is known from history of mathematics, that Gödel submitted his two incompleteness theorems, which can be considered as one of hallmarks of modern mathematics in 20th century. Here we argue that Gödel incompleteness theorem and its self-referential paradox have not only put Hilbert’s axiomatic program into question, but he also opened up the problem deep inside the then popular Aristotelian Logic. Although there were some attempts to go beyond Aristotelian binary logic, including by Lukasiewicz’s three-valued logic, here we argue that the problem of self-referential paradox can be seen as reconcilable and solvable from Neutrosophic Logic perspective. Motivation of this paper: These authors are motivated to re-describe the self-referential paradox inherent in Godel incompleteness theorem. Contribution: This paper will show how Neutrosophic Logic offers a unique perspective and solution to Godel incompleteness theorem.

Self-reference has played a prominent role in the development of metamathematics in the past century, starting with Gödel’s first incompleteness theorem. Given the nature of this and other results in the area, the informal understanding of self-reference in arithmetic has sufficed so far. Recently, however, it has been argued that for other related issues in metamathematics and philosophical logic a precise notion of self-reference and, more generally, reference is actually required. These notions have been so far elusive and are surrounded by an aura of scepticism that has kept most philosophers away. In this paper I suggest we shouldn’t give up all hope. First, I introduce the reader to these issues. Second, I discuss the conditions a good notion of reference in arithmetic must satisfy. Accordingly, I then introduce adequate notions of reference for the language of first-order arithmetic, which I show to be fruitful for addressing the aforementioned issues in metamathematics.

In this article I am going to argue for the possibility of a transcendental source of logic based on a phenomenologically motivated approach. My aim will be essentially carried out in two succeeding steps of reduction: the first one will be the indication of existence of an inherent temporal factor conditioning formal predicative discourse and the second one, based on a supplementary reduction of objective temporality, will be a recourse to a time-constituting origin which has to be assumed as a non-temporal, transcendental subjectivity and for that reason as possibly the ultimate transcendental root of pure logic. In the development of the argumentation and taking into account W.V. Quine’s views in his well-known Word and Object, a special emphasis will be given to the fundamentally temporal character of universal and existential predicative forms, to their status in logical theories in general, and to their underlying role in generating an inherently non-finitistic character reflected, for instance, in the undecidability of certain infinity statements in formal mathematical theories. This is shown also to concern metatheorems of such vital importance as Godel’s incompleteness theorems in mathematical foundations. Moreover in the course of the discussion the quest for the ultimate limits of predication will lead to the notions of separation and intentional correlation between an ‘observing’ subject and the object of ‘observation’ as well as to the notion of syntactical individuals taken as the irreducible non-analytic nuclei-forms within analytical discourse.

Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyondNP≠coNP. These conjectures formally connect computational complexity with the difficulty of proving some sentences, which means that high computational complexity of a problem associated with a sentence implies that the sentence is not provable in a weak theory, or requires a long proof. Another reason for putting forward these conjectures is that some results in proof complexity seem to be special cases of such general statements and we want to formalize and fully understand these statements. Roughly speaking, we are trying to connect syntactic complexity, by which we mean the complexity of sentences and strengths of the theories in which they are provable, with the semantic concept of complexity of the computational problems represented by these sentences.We have introduced the most fundamental conjectures in our earlier works [27, 33–35]. Our aim in this article is to present them in a more systematic way, along with several new conjectures, and prove new connections between them and some other statements studied before.

The paper examines Husserl’s interactions with logicians in the 1930s in order to assess Husserl’s awareness of Gödel’s incompleteness theorems. While there is no mention about the results in Husserl’s known exchanges with Hilbert, Weyl, or Zermelo, the most likely source about them for Husserl is Felix Kaufmann (1895–1949). Husserl’s interactions with Kaufmann show that Husserl may have learned about the results from him, but not necessarily so. Ultimately Husserl’s reading marks on Friedrich Waismann’s Einführung in das mathematische Denken: die Begriffsbildung der modernen Mathematik, 1936, show that he knew about them before his death in 1938.

Gödel introduced the original provability predicate in the proofs of Gödel’s incompleteness theorems, and Rosser defined a new one. They are equivalent in the standard model ${\mathbb N}$ of arithmetic or any nonstandard model of ${\rm PA} + {\rm Con_{PA}} $, but the behavior of Rosser’s provability predicate is different from the original one in nonstandard models of ${\rm PA} + \neg {\rm Con_{PA}} $. In this paper, we investigate several properties of the derivability conditions for Rosser provability predicates, and prove the existence of a Rosser provability predicate with which we can define any consistent complete extension of ${\rm PA}$ in some nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $. We call it a universal Rosser predicate. It follows from the theorem that the true arithmetic ${\rm TA}$ can be defined as the set of theorems of ${\rm PA}$ in terms of a universal Rosser predicate in some nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $. By using this theorem, we also give a new proof of a theorem that there is a nonstandard model M of ${\rm PA} + \neg {\rm Con_{PA}} $ such that if N is an initial segment of M which is a model of ${\rm PA} + {\rm Con_{PA}} $ then every theorem of ${\rm PA}$ in N is a theorem of $\rm PA$ in ${\mathbb N}$. In addition, we prove that there is a Rosser provability predicate such that the set of theorems of $\rm PA$ in terms of the Rosser provability predicate is inconsistent in any nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $.

We present a new manifestation of Gödel’s second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert’s program. Specifically, we consider a proper extension of Peano arithmetic ( $\mathbf {PA}$ ) by a mathematically meaningful axiom scheme that consists of $\Sigma ^0_2$ -sentences. These sentences assert that each computably enumerable ( $\Sigma ^0_1$ -definable without parameters) property of finite binary trees has a finite basis. Since this fact entails the existence of polynomial time algorithms, it is relevant for computer science. On a technical level, our axiom scheme is a variant of an independence result due to Harvey Friedman. At the same time, the meta-mathematical properties of our axiom scheme distinguish it from most known independence results: Due to its logical complexity, our axiom scheme does not add computational strength. The only known method to establish its independence relies on Gödel’s second incompleteness theorem. In contrast, Gödel’s theorem is not needed for typical examples of $\Pi ^0_2$ -independence (such as the Paris–Harrington principle), since computational strength provides an extensional invariant on the level of $\Pi ^0_2$ -sentences.

This paper surveys the interactions between Russell and Gödel, both personal and intellectual. After a description of Russell’s influence on Gödel, it concludes with a discussion of Russell’s reaction to the incompleteness theorems.

On Three Appearances of Difference in Mathematics. This article proposes to examine three types of relations between man and equality, as they are embodied in the relation to the minimal condition of the mathematical: the sentence of identity: I=I. Starting our examination from the current common conception of science and mechanism, we aim to reveal that behind the dominating logic of identity there are two other systems of logic, which have arisen during the history of humanity and that of mathematics. Following Serres, the first one might be revealed during the passage from the Babylonian mathematics to the interaction between Thales and the Greek gnomon. The Babylonian epoch marks an era, in which the logic of sameness operates; logic which offers a place for the mathematician. This era is followed by a science, working under the second logic: the logic of identity, which consecrates an axiomatic definition-based approach, at the expense of the human being. The third logic looms after the discovery of Godel's incompleteness theorems: that an undecidable statement can be derived within every arithmetical system. Under this logic, the roles of man and the allegedly mathematical automatic machine become even more intertwined: the newly found logic promises instability not only for man, but also for the machinic-mathematics.

In a note appended to the translation of “On consistency and completeness” (1967), Godel reexamined the problem of the unprovability of consistency. Godel here focuses on an alternative means of expressing the consistency of a formal system, in terms of what would now be called a ‘reflection principle’, roughly, the assertion that a formula of a certain class is provable in the system only if it is true. Godel suggests that it is this alternative means of expressing consistency that we should be interested in from a foundational point of view, and he gives a result that shows certain reflection principles to be underivable in extensions of elementary number theory under conditions significantly weaker than the Hilbert-Bernays derivability conditions. In this paper I shall discuss the background to Godel's result and the foundational significance he claims for it. Along the way, I shall present a new proof of the result which places it in an even more general context than the one considered by Godel in the 1960s.

In the 1951 Gibbs lecture, Godel asserted his famous dichotomy, where the notion of informal proof is at work. G. Priest developed an argument, grounded on the notion of naive proof, to the effect that Godel’s first incompleteness theorem suggests the presence of dialetheias. In this paper, we adopt a plausible ideal notion of naive proof, in agreement with Godel’s conception, superseding the criticisms against the usual notion of naive proof used by real working mathematicians. We explore the connection between Godel’s theorem and naive proof so understood, both from a classical and a dialetheic perspective.

In this article we study proofs of some general forms of the Second Incompleteness Theorem. These forms conform to the Feferman format, where the proof predicate is fixed and the representation of the set of axioms varies. We extend the Feferman framework in one important point: we allow the interpretation of number theory to vary.

A notion of delegated causality is introduced. This subtle kind of causality is dual to interventional causality. Delegated causality elucidates the causal role of dynamical systems at the "edge of chaos", explicates evident cases of downward causation, and relates emergent phenomena to Godel's incompleteness theorem. Apparently rich implications are noticed in biology and Chinese philosophy., Causal duality is formulated; 18 pages

Graham Priest (In contradiction) introduces an informal but presumably rigorous and sharp ‘provability predicate’. He argues that this predicate yields inconsistencies, along the lines of the paradox of the Knower. One long-standing claim of Priest’s is that a dialetheist can have a complete, decidable, and yet sufficiently rich mathematical theory. After all, the incompleteness theorem is, in effect, that for any recursive theory A, if A is consistent, then A is incomplete. If the antecedent fails, as it might for a dialetheist, then the consequent may also fail to hold. One somewhat friendly purpose of my ‘Incompleteness and inconsistency’ was to improve the technical situation for the dialetheist, eschewing reliance on an informal provability predicate. Another, less friendly purpose was to bring out what I took to be some untoward consequences of the situation. It seems that Priest accepted at least some of the improvements that I attempted. In the second edition of In contradiction, he responded to the alleged untoward consequences. One purpose of this note is to revisit the technical and philosophical situation. There were some errors in my original presentation, brought out by discussion with Priest and by Hartry Field’s analysis of the second incompleteness theorem in such contexts. A second task here is to present a sort of Curry version of the Godel incompleteness situation. I tentatively conclude that even for a dialetheist, an interesting and complete theory is not as easy to come by as it may look—at least not for theories of arithmetic that are plausible for a dialetheist.

In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem (1979), “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class ofcompletely additivemodal algebras, or as we call them,${\cal V}$-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to${\cal V}$-baos, namely the provability logic$GLB$(Japaridze, 1988; Boolos, 1993). We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is${\cal V}$-complete. After these results, we generalize the Blok Dichotomy (Blok, 1978) to degrees of${\cal V}$-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness.

This piece continues the tradition of arguments by John Lucas, Roger Penrose and others to the effect that the human mind is not a machine. Kurt Gödel thought that the intensional paradoxes stand in the way of proving that the mind is not a machine. According to Gödel, a successful proof that the mind is not a machine would require a solution to the intensional paradoxes. We provide what might seem to be a partial vindication of Gödel and show that if a particular solution to the intensional paradoxes is adopted, one can indeed give an argument to the effect that the mind is not a machine.

Evolutionary Debunking Arguments (EDAs) are defined as arguments that appeal to the evolutionary genealogy of our beliefs to undermine their justification. Recently, Helen De Cruz and her co-authors supported the view that EDAs are self-defeating: if EDAs claim that human arguments are not justified, because the evolutionary origin of the beliefs which figure in such arguments undermines those beliefs, and EDAs themselves are human arguments, then EDAs are not justified, and we should not accept their conclusions about the fact that human arguments are unjustified. De Cruz's objection to EDAs is similar to the objection raised by Reuben Hersh against the claim that, since by Godel's second incompleteness theorem the purpose of mathematical logic to give a secure foundation for mathematics cannot be achieved, mathematics cannot be said to be absolutely certain. The response given by Carlo Cellucci to Hersh's objection shows that the claim that by Godel's results mathematics cannot be said to be absolutely certain is not self-defeating, and can be adopted to show that EDAs are not self-defeating as well in a twofold sense: an argument analogous to Cellucci's one may be developed to face De Cruz's objection, and such argument may be further refined incorporating Cellucci's response itself in it, to make it stronger. This paper aims at showing that the accusation of being self-defeating moved against EDAs is inadequate by elaborating an argument which can be considered an EDA and which can also be shown not to be self-defeating.

EnglishIn The Oxford Handbook of Wittgenstein, Michael Potter’s essay ‘Wittgenstein on mathematics’ offers not only an analysis on why Wittgenstein’s philosophy of mathematics is so controversial among philosophers and mathematicians but also justifies such controversy. For that purpose, Potter emphasizes that Wittgenstein’s discussions on math are unfinished. This article sheds light on some inconsistencies and contradictions pertaining Wittgenstein’s mathematic thoughts which, although reinforcing the decade-long criticism on the author, did not interest scholars as much as Wittgenstein’s comments on Kurt Godel’s incompleteness theorems and his dialogues with Alan Turing on the foundations of mathematics. portuguesNo ensaio “Wittgenstein on mathematics”, publicado no Oxford Handbook of Wittgenstein, Michael Potter procura nao apenas analisar por que a filosofia da matematica de Wittgenstein e tao controvertida entre filosofos e matematicos como justificar essa situacao. Com esse intuito, Potter enfatiza o carater inacabado das reflexoes de Wittgenstein sobre a matematica. Neste artigo, tem-se por objetivo explicitar algumas inconsistencias e contradicoes no pensamento matematico de Wittgenstein que ratificam as criticas que esse autor vem recebendo ha decadas, mas que nao tiveram a mesma atencao dos especialistas que os comentarios de Wittgenstein sobre os teoremas da incompletude de Kurt Godel e suas discussoes com Alan Turing sobre os fundamentos da matematica.

Rene Descartes’ enduring contribution to philosophy, natural science and mathematics includes the unresolved residue of Cartesian dualism, as well as a singular ‘bottom-up’ interpretation of reductive logic sustained within the modern structure of the reductive natural science paradigm. Application of strong reductive logic leads to the perplexing reductive epiphenomenalism proposition.

This paper provides examples in arithmetic of the account of rational intuition and evidence developed in my book After Godel: Platonism and Rationalism in Mathematics and Logic (Oxford: Oxford University Press, 2011). The paper supplements the book but can be read independently of it. It starts with some simple examples of problem-solving in arithmetic practice and proceeds to general phenomenological conditions that make such problem-solving possible. In proceeding from elementary ‘authentic’ parts of arithmetic to axiomatic formal arithmetic, the paper exhibits some elements of the genetic analysis of arithmetic knowledge that is called for in Husserl’s philosophy. This issues in an elaboration on a number of Godel’s remarks about the meaning of his incompleteness theorems for the notion of evidence in mathematics.

Solving numeric, logic and language puzzles and paradoxes is common within a wide community of high school and university students, fact witnessed by the increasing number of books published by mathematicians such as Martin Gardner (popular books as old as Gardner in Aha! insight. W. H. Freeman & Co., London, 1978, Wheels, life and other mathematical amusements. W H Freeman & Co., London, 1985), Douglas Hofstadter [in one of the best popular science books on paradoxes (Hofstadter in Godel, escher, bach: an eternal golden braid, Penguin, London, 2000)], inspired by Godel’s incompleteness theorems), Patrick Hughes and George Brecht (see Hughes and Brecht in Vicious circles and infinity, an anthology of paradoxes. Penguin Books, London, 1993) and Raymond M. Smullyan (the most well known being Smullyan in Forever undecided, puzzle guide to godel. Oxford Paperbacks, Oxford 1988, To Mock a Mockingbird and other logic puzzles. Oxford Paperbacks, Oxford 2000, The lady or the tiger? And other logic puzzles. Dover Publications Inc., Mineola 2009), inter alia. Books by Smullyan (such as Smullyan 1988, 2000) are, however, much more involved, since they introduce learning trajectories and strategies across several subjects of mathematical logic, as difficult as combinatorial logic (see, e.g., Smullyan 2000), computability theory (see Smullyan 1988), and proof theory (see Smullyan 1988, 2009). These books provide solutions to their suggested exercises. Both statements and their solutions are written in the natural language, introducing some informal algorithms. As an exercise in Mathematics we wonder if an easy proof system could be devised to solve the amusing equations proposed by Smullyan in his books. Moreover, university students of logic could well train themselves in constructing deductive systems to solve puzzles instead of a non-uniform treatment one by one. In this paper, addressing students, we introduce one such formal systems, a tableaux approach able to provide the solutions to the puzzles involving either propositional logic, first order logic, or aspect logic. Let the reader amuse herself or himself!

The essence of number was regarded by the ancient Greeks as the root cause of the existence of the universe, but it was only towards the end of the 19th century that mathematicians initiated an in-depth study of the nature of numbers. The resulting unavoidable actuality of infinities in the number system led mathematicians to rigorously investigate the foundations of mathematics. The formalist approach to establish mathematical proof was found to be inconclusive: Gödel showed that there existed true propositions that could not be proved to be true within the natural number universe. This result weighed heavily on proposals in the mid-20th century for digital models of the universe, inspired by the emergence of the programmable digital computer, giving rise to the branch of philosophy recognised as digital philosophy. In this article, the models of the universe presented by physicists, mathematicians and theoretical computer scientists are reviewed and their relation to the natural numbers is investigated. A quantum theory view that at the deepest level time and space may be discrete suggests a profound relation between natural numbers and reality of the cosmos. The conclusion is that our perception of reality may ultimately be traced to the ontology and epistemology of the natural numbers.

Conventionalism about mathematics claims that mathematical truths are true by linguistic convention. This is often spelled out by appealing to facts concerning rules of inference and formal systems, but this leads to a problem: since the incompleteness theorems we’ve known that syntactic notions can be expressed using arithmetical sentences. There is serious prima facie tension here: how can mathematics be a matter of convention and syntax a matter of fact given the arithmetization of syntax? This challenge has been pressed in the literature by Hilary Putnam and Peter Koellner. In this paper I (i) sketch a conventionalist theory of mathematics, (ii) show that this conventionalist theory can meet the challenge just raised (this is done by considering how arithmetical coding works in non-standard models of arithmetic), and (iii) clarify the type of mathematical pluralism endorsed by the conventionalist by introducing the notion of a semantic counterpart. The paper’s aim is an improved understanding of conventionalism, pluralism, and the relationship between them.

We shall use Godel’s Second Incompleteness Theorem to show that consistency is not possibility, and then argue that the argument does serious damage to some theories of modality where consistency plays a major but not exclusive role.

An ultimate Universal theory – a complete theory that accounts, via few and simple first principles, for all the phenomena already observed and that will ever be observed – has been, and still is, the aspiration of most physicists and scientists. Yet, a basic principle that is embodied in the results of Gödel’s incompleteness theorems is that self-referencing leads to logical conflict or failure, as in the liar paradox or Russell’s paradox. In physical theories self-referencing necessarily occurs when it is realized that the observer is also a participant in the experienced phenomena – we, humans, are part of the Universe while observing it. Therefore self-referencing, and consequently logical conflicts, are unavoidable, and any theory pretending to be Universal is bound to be incomplete.

The so-called “Slingshot” argument purports to show that an ontology of facts is untenable. In this paper, we address a minimal slingshot restricted to an ontology of physical facts as truth-makers for empirical physical statements. Accepting that logical matters have no bearing on the physical facts that are truth-makers for empirical physical statements and that objects are themselves constituents of such facts, our minimal slingshot argument purportedly shows that any two physical statements with empirical content are made true by one and the same fact. It is well-known that Russell’s theory of descriptions may be employed to reveal a scope fallacy in the slingshot argument. This paper reveals that there is a quite independent Russellian criticism of the slingshot argument based on the thesis that facts are structured entities.

It is widely considered that Godel’s and Rosser’s proofs of the incompleteness theorems are related to the Liar Paradox. Yablo’s paradox, a Liar-like paradox without self-reference, can also be used to prove Godel’s first and second incompleteness theorems. We show that the situation with the formalization of Yablo’s paradox using Rosser’s provability predicate is different from that of Rosser’s proof. Namely, by using the technique of Guaspari and Solovay, we prove that the undecidability of each instance of Rosser-type formalizations of Yablo’s paradox for each consistent but not Σ1-sound theory is dependent on the choice of a standard proof predicate.

I argue that three main interpretations of the aim of Russell’s early logicism in The Principles of Mathematics (1903) are mistaken, and propose a new interpretation. According to this new interpretation, the aim of Russell’s logicism is to show, in opposition to Kant, that mathematical propositions have a certain sort of complete generality which entails that their truth is independent of space and time. I argue that on this interpretation two often-heard objections to Russell’s logicism, deriving from Godel’s incompleteness theorem and from the non-logical character of some of the axioms of Principia Mathematica respectively, can be seen to be inconclusive. I then proceed to identify two challenges that Russell’s logicism, as presently construed, faces, but argue that these challenges do not appear unanswerable.

We argue that the set of humanly known mathematical truths (at any given moment in human history) is finite and so recursive. But if so, then given various fundamental results in mathematical logic and the theory of computation (such as Craig’s in J Symb Log 18(1): 30–32(1953) theorem), the set of humanly known mathematical truths is axiomatizable. Furthermore, given Godel’s (Monash Math Phys 38: 173–198, 1931) First Incompleteness Theorem, then (at any given moment in human history) humanly known mathematics must be either inconsistent or incomplete. Moreover, since humanly known mathematics is axiomatizable, it can be the output of a Turing machine. We then argue that any given mathematical claim that we could possibly know could be the output of a Turing machine, at least in principle. So the Lucas-Penrose (Lucas in Philosophy 36:112–127, 1961; Penrose, in The Emperor’s new mind. Oxford University Press, Oxford (1994)) argument cannot be sound.

Kurt Godel (1906–1978) began to study the philosophy of Edmund Husserl (1859–1938) in 1959. In this paper I present an overview of central themes in Godel’s study of Husserl’s phenomenology. Since many of Godel’s ideas concerning Husserl were never put into a systematic form by Godel himself, I quote fairly extensively in the paper from several sources in order to inform the reader of the nature of Godel’s interest in Husserl. Godel prepared one manuscript specifically on Husserl, as we will see below, and many of Godel’s comments on Husserl are included in the books of Hao Wang. I will also quote some relevant texts from the Godel Nachlass. In accordance with these various sources, I provide a brief overview in a later section of the paper of Godel’s interest in eidetic transcendental phenomenology as a new type of monadology. The relationship of Godel’s incompleteness theorems to Husserl’s notion of ‘definite’ axiom systems is also discussed briefly.

Husserl’s first work formulated what proved to be an algorithmically complete arithmetic, lending mathematical clarity to Kronecker’s reduction of analysis to finite calculations with integers. Husserl’s critique of his nominalism led him to seek a philosophical justification of successful applications of symbolic arithmetic to nature, providing insight into the “wonderful affinity” between our mathematical thoughts and things without invoking a pre-established harmony. For this, Husserl develops a purely descriptive phenomenology for which he found inspiration in Mach’s proposal of a “universal physical phenomenology.” To account for applications to any domain, Husserl envisages a theory of all possible deductive systems, which he develops extensively in his Gottingen lectures wherein he engages with Hilbert’s work on deductive systems for geometry, real arithmetic, and physics. This leads Husserl to formulate claims of decidability and proofs of completeness for various arithmetics that result from his analysis of Kronecker’s general arithmetic. Careful attention to these proofs seem to show that Husserl was not oblivious to problems that underlie our incompleteness theorems, namely that of showing that some inversions of his algorithmic arithmetic are undefined. His growing preoccupation with the issue of a pre-established harmony between mathematical thought and reality motivate his pursuit of a “supramathematics” of all possible complete theory forms to demystify such harmony, by having such a form on hand for describing any empirical domain. He soon decides that a transcendental idealism of nature will reveal the wonderful affinity of thoughts and things comprising such harmony, to be a wonderful “parallelism of objective unities and constituted manifolds of consciousness.” But the paradoxes of logic and set theory cloud the clarity of mathematics, which Weyl would restore with Brouwer’s intuitionism and Hilbert with his metamathematics. Husserl informed Weyl that his student Becker had formulated a phenomenological foundation not only for Weyl’s generalization of relativity theory but also for the Brouwer-Weyl continuum. But Weyl eventually rejected much of Becker’s work, especially when it became clear that his phenomenological intuitionism could not account for the success of Hilbert’s transfinite mathematics in quantum physics. Becker responded to this “crisis of phenomenological method” with his mantic phenomenology celebrating the magic of mathematical mysticism, which Husserl finally rejects in favor of a pluralistic phenomenology of mathematics and nature.