Classical interpretations of Goedels formal reasoning, and of his conclusions, implicitly imply that mathematical languages are essentially incomplete, in the sense that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is, both, non-algorithmic, and essentially unverifiable. However, a language of general, scientific, discourse, which intends to mathematically express, and unambiguously communicate, intuitive concepts that correspond to scientific investigations, cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth (...) verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Goedel's reasoning, under which any formal system of Peano Arithmetic---classically accepted as the foundation of all our mathematical Languages---is verifiably complete in the above sense. We show how some paradoxical concepts of Quantum mechanics can, then, be expressed, and interpreted, naturally under a constructive definition of mathematical truth. (shrink)

The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological, and logical character. This chapter focuses on two arguments from logic. First, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to which every proposition is either true or false, no matter whether the (...) proposition is about the past, present or future. In particular, the argument goes, whatever one does or does not do in the future is determined in the present by the truth or falsity of the corresponding proposition. The second argument coming from logic is much more modern and appeals to Gödel's incompleteness theorems to make the case against determinism and in favour of free will, insofar as that applies to the mathematical potentialities of human beings. The claim more precisely is that as a consequence of the incompleteness theorems, those potentialities cannot be exactly circumscribed by the output of any computing machine even allowing unlimited time and space for its work. The chapter concludes with some new considerations that may be in favour of a partial mechanist account of the mathematical mind. (shrink)

Although written in Japanese, this article handles historical and technical survey of Gödel's incompleteness theorem thoroughly.

The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin's famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure (...) of the strength of the theory. I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental. (shrink)

The foundations of mathematics and physics no longer start with fundamental entities and their properties like spatial extension, points, lines or the billiard ball like particles of Newtonian physics. Mathematics has abolished these from its foundations in set theory by making all assumptions explicit and structural. Particle physics has become completely mathematical, connecting to physical reality only through experimental technique. Applying the principles guiding the foundations of mathematics and physics to philosophical analysis underscores that only conscious experience has an intrinsic (...) nature. This leads to a version of realistic monism in which the essence and totality of the existence of physical structure is immediate experience in some form. Identifying physical structure with conscious experience allows the application of mathematics to the evolution of consciousness. Some of the implications from Goedel’s Incompleteness Theorem are connected to creativity and ethics. (shrink)

A survey of more philosophical applications of Gödel's incompleteness results.

In spite of the many efforts made to clarify von Neumann’s methodology of science, one crucial point seems to have been disregarded in recent literature: his closeness to Hilbert’s spirit. In this paper I shall claim that the scientific methodology adopted by von Neumann in his later foundational reflections originates in the attempt to revaluate Hilbert’s axiomatics in the light of Gödel’s incompleteness theorems. Indeed, axiomatics continues to be pursued by the Hungarian mathematician in the spirit of Hilbert’s (...) school. I shall argue this point by examining four basic ideas embraced by von Neumann in his foundational considerations: a) the conservative attitude to assume in mathematics; b) the role that mathematics and the axiomatic approach have to play in all that is science; c) the notion of success as an alternative methodological criterion to follow in scientific research; d) the empirical and, at the same time, abstract nature of mathematical thought. Once these four basic ideas have been accepted, Hilbert’s spirit in von Neumann’s methodology of science will become clear. (shrink)

A resolution to the Russell Paradox is presented that is similar to Russell's “theory of types” method but is instead based on the definition of why a thing exists as described in previous work by this author. In that work, it was proposed that a thing exists if it is a grouping tying "stuff" together into a new unit whole. In tying stuff together, this grouping defines what is contained within the new existent entity. A corollary is that a thing, (...) such as a set, does not exist until after the stuff is tied together, or said another way, until what is contained within is completely defined. A second corollary is that after a grouping defining what is contained within is present and the thing exists, if one then alters what is tied together (e.g., alters what is contained within), the first existent entity is destroyed and a different existent entity is created. A third corollary is that a thing exists only where and when its grouping exists. Based on this, the Russell Paradox's set R of all sets that aren't members of themselves does not even exist until after the list of the elements it contains (e.g. the list of all sets that aren't members of themselves) is defined. Once this list of elements is completely defined, R then springs into existence. Therefore, because it doesn't exist until after its list of elements is defined, R obviously can't be in this list of elements and, thus, cannot be a member of itself; so, the paradox is resolved. This same type of reasoning is then applied to Godel's first Incompleteness Theorem. Briefly, while writing a Godel Sentence, one makes reference to a future, not yet completed and not yet existent sentence, G, that claims its unprovability. However, only once the sentence is finished does it become a new unit whole and existent entity called sentence G. If one then goes back in and replaces the reference to the future sentence with the future sentence itself, a totally different sentence, G1, is created. This new sentence G1 does not assert its unprovability. An objection might be that all the possibly infinite number of possible G-type sentences or their corresponding Godel numbers already exist somehow, so one doesn't have to worry about references to future sentences and springing into existence. But, if so, where do they exist? If they exist in a Platonic realm, where is this realm? If they exist pre-formed in the mind, this would seem to require a possibly infinite-sized brain to hold all these sentences. This is not the case. What does exist in the mind is the system for creating G-type sentences and their corresponding numbers. This mental system for making a G-type sentence is not the same as the G-type sentence itself just as an assembly line is not the same as a finished car. In conclusion, a new resolution of the Russell Paradox and some issues with proofs of Godel's First Incompleteness Theorem are described. (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.

This chapter describes Kurt Gödel's paper on the incompleteness theorems. Gödel's incompleteness results are two of the most fundamental and important contributions to logic and the foundations of mathematics. It had been assumed that first-order number theory is complete in the sense that any sentence in the language of number theory would be either provable from the axioms or refutable. Gödel's first incompleteness theorem showed that this assumption was false: it states that there (...) are sentences of number theory that are neither provable nor refutable. The first theorem is general in the sense that it applies to any axiomatic theory, which is ω-consistent, has an effective proof procedure, and is strong enough to represent basic arithmetic. Their importance lies in their generality: although proved specifically for extensions of system, the method Gödel used is applicable in a wide variety of circumstances. Gödel's results had a profound influence on the further development of the foundations of mathematics. It pointed the way to a reconceptualization of the view of axiomatic foundations. (shrink)

The previous two parts of the paper demonstrate that the interpretation of Fermat’s last theorem (FLT) in Hilbert arithmetic meant both in a narrow sense and in a wide sense can suggest a proof by induction in Part I and by means of the Kochen - Specker theorem in Part II. The same interpretation can serve also for a proof FLT based on Gleason’s theorem and partly similar to that in Part II. The concept of (probabilistic) measure of a subspace (...) of Hilbert space and especially its uniqueness can be unambiguously linked to that of partial algebra or incommensurability, or interpreted as a relation of the two dual branches of Hilbert arithmetic in a wide sense. The investigation of the last relation allows for FLT and Gleason’s theorem to be equated in a sense, as two dual counterparts, and the former to be inferred from the latter, as well as vice versa under an additional condition relevant to the Gödel incompleteness of arithmetic to set theory. The qubit Hilbert space itself in turn can be interpreted by the unity of FLT and Gleason’s theorem. The proof of such a fundamental result in number theory as FLT by means of Hilbert arithmetic in a wide sense can be generalized to an idea about “quantum number theory”. It is able to research mathematically the origin of Peano arithmetic from Hilbert arithmetic by mediation of the “nonstandard bijection” and its two dual branches inherently linking it to information theory. Then, infinitesimal analysis and its revolutionary application to physics can be also re-realized in that wider context, for example, as an exploration of the way for physical quantity of time (respectively, for time derivative in any temporal process considered in physics) to appear at all. Finally, the result admits a philosophical reflection of how any hierarchy arises or changes itself only thanks to its dual and idempotent counterpart. (shrink)

The aim of this paper is to argue that the (alleged) indeterminism of quantum mechanics, claimed by adherents of the Copenhagen interpretation since Born (1926), can be proved from Chaitin's follow-up to Goedel's (first) incompleteness theorem. In comparison, Bell's (1964) theorem as well as the so-called free will theorem-originally due to Heywood and Redhead (1983)-left two loopholes for deterministic hidden variable theories, namely giving up either locality (more precisely: local contextuality, as in Bohmian mechanics) or free choice (i.e. uncorrelated (...) measurement settings, as in 't Hooft's cellular automaton interpretation of quantum mechanics). The main point is that Bell and others did not exploit the full empirical content of quantum mechanics, which consists of long series of outcomes of repeated measurements (idealized as infinite binary sequences): their arguments only used the long-run relative frequencies derived from such series, and hence merely asked hidden variable theories to reproduce single-case Born probabilities defined by certain entangled bipartite states. If we idealize binary outcome strings of a fair quantum coin flip as infinite sequences, quantum mechanics predicts that these typically (i.e. almost surely) have a property called 1-randomness in logic, which is much stronger than uncomputability. This is the key to my claim, which is admittedly based on a stronger (yet compelling) notion of determinism than what is common in the literature on hidden variable theories. (shrink)

On the heels of Franzén's fine technical exposition of Gödel's incompleteness theorems and related topics (Franzén 2004) comes this survey of the incompleteness theorems aimed at a general audience. Gödel's Theorem: An Incomplete Guide to its Use and Abuse is an extended and self-contained exposition of the incompleteness theorems and a discussion of what informal consequences can, and in particular cannot, be drawn from them.

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)

This article is an update of our previous article in this SGJ journal, titled: On Gödel's Incompleteness Theorem, Artificial Intelligence & Human Mind. We provide some commentary on the latest developments around AI, humanoid robotics, and future scenario. Basically, we argue that a more thoughtful approach to the future is "techno-realism.".

In the present paper, these authors argue on actual reasons why Hilbert’s axiomatic program to unify gravitation theory and electromagnetism failed completely. An outline of plausible resolution of this problem is given here, based on: a) Gödel’s incompleteness theorem, b) Newton’s aether stream model. And in another paper we will present our calculation of receding Moon from Earth based on such a matter creation hypothesis. More experiments and observations are called to verify this new hypothesis, albeit it is inspired (...) from Newton’s theory himself. (shrink)

An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics (...) match with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question. (shrink)

Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation. Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity. That is nothing else than a new reading at issue and comparative interpretation of Gödel’s papers (1930; 1931) meant here. Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of infinity. The most (...) utilized example of those generalizations is the complex Hilbert space. Any generalization of Peano arithmetic consistent to infinity, e.g. the complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself. (shrink)

Hofstadter [1979, 2007] offered a novel Gödelian proposal which purported to reconcile the apparently contradictory theses that (1) we can talk, in a non-trivial way, of mental causation being a real phenomenon and that (2) mental activity is ultimately grounded in low-level rule-governed neural processes. In this paper, we critically investigate Hofstadter’s analogical appeals to Gödel’s [1931] First Incompleteness Theorem, whose “diagonal” proof supposedly contains the key ideas required for understanding both consciousness and mental causation. We maintain that bringing (...) sophisticated results from Mathematical Logic into play cannot furnish insights which would otherwise be unavailable. Lastly, we conclude that there are simply too many weighty details left unfilled in Hofstadter’s proposal. These really need to be fleshed out before we can even hope to say that our understanding of classical mind-body problems has been advanced through metamathematical parallels with Gödel’s work. (shrink)

I have read many recent discussions of the limits of computation and the universe as computer, hoping to find some comments on the amazing work of polymath physicist and decision theorist David Wolpert but have not found a single citation and so I present this very brief summary. Wolpert proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv.org) on the limits to inference (computation) that are so general they are independent of the device doing the computation, (...) and even independent of the laws of physics, so they apply across computers, physics, and human behavior. They make use of Cantor's diagonalization, the liar paradox and worldlines to provide what may be the ultimate theorem in Turing Machine Theory, and seemingly provide insights into impossibility,incompleteness, the limits of computation,and the universe as computer, in all possible universes and all beings or mechanisms, generating, among other things,a non-quantum mechanical uncertainty principle and a proof of monotheism. (shrink)

Whether it was John Searle’s Chinese Room argument (Searle, 1980) or Roger Penrose’s argument of the non-computable nature of a mathematician’s insight – an argument that was based on Gödel’s Incompleteness theorem (Penrose, 1989), we have always had skeptics that questioned the possibility of realizing strong Artificial Intelligence (AI), or what has become known by Artificial General Intelligence (AGI). But this new book by Landgrebe and Smith (henceforth, L&S) is perhaps the strongest argument ever made against strong AI. It (...) is a very extensive review of what building a mind essentially amounts to drawing on insights and results from biology, physics, linguistics, computability, philosophy, and mathematics. (shrink)

REVIEW OF: Automated Development of Fundamental Mathematical Theories by Art Quaife. (1992: Kluwer Academic Publishers) 271pp. Using the theorem prover OTTER Art Quaife has proved four hundred theorems of von Neumann-Bernays-Gödel set theory; twelve hundred theorems and definitions of elementary number theory; dozens of Euclidean geometry theorems; and Gödel's incompleteness theorems. It is an impressive achievement. To gauge its significance and to see what prospects it offers this review looks closely at the book and (...) the proofs it presents. (shrink)

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

Introduction to mathematical logic. Part 2.Textbook for students in mathematical logic and foundations of mathematics. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem.

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

I have read many recent discussions of the limits of computation and the universe as computer, hoping to find some comments on the amazing work of polymath physicist and decision theorist David Wolpert but have not found a single citation and so I present this very brief summary. Wolpert proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv dot org) on the limits to inference (computation) that are so general they are independent of the device doing (...) the computation, and even independent of the laws of physics, so they apply across computers, physics, and human behavior. They make use of Cantor's diagonalization, the liar paradox and worldlines to provide what may be the ultimate theorem in Turing Machine Theory, and seemingly provide insights into impossibility, incompleteness, the limits of computation, and the universe as computer, in all possible universes and all beings or mechanisms, generating, among other things, a non- quantum mechanical uncertainty principle and a proof of monotheism. There are obvious connections to the classic work of Chaitin, Solomonoff, Komolgarov and Wittgenstein and to the notion that no program (and thus no device) can generate a sequence (or device) with greater complexity than it possesses. One might say this body of work implies atheism since there cannot be any entity more complex than the physical universe and from the Wittgensteinian viewpoint, ‘more complex’ is meaningless (has no conditions of satisfaction, i.e., truth-maker or test). Even a ‘God’ (i.e., a ‘device’with limitless time/space and energy) cannot determine whether a given ‘number’ is ‘random’, nor find a certain way to show that a given ‘formula’, ‘theorem’ or ‘sentence’ or ‘device’ (all these being complex language games) is part of a particular ‘system’. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 2nd ed (2019) and Suicidal Utopian Delusions in the 21st Century 4th ed (2019) . (shrink)

I have read many recent discussions of the limits of computation and the universe as computer, hoping to find some comments on the amazing work of polymath physicist and decision theorist David Wolpert but have not found a single citation and so I present this very brief summary. Wolpert proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv.org) on the limits to inference (computation) that are so general they are independent of the device doing the computation, (...) and even independent of the laws of physics, so they apply across computers, physics, and human behavior. They make use of Cantor's diagonalization, the liar paradox and worldlines to provide what may be the ultimate theorem in Turing Machine Theory, and seemingly provide insights into impossibility, incompleteness, the limits of computation,and the universe as computer, in all possible universes and all beings or mechanisms, generating, among other things,a non- quantum mechanical uncertainty principle and a proof of monotheism. There are obvious connections to the classic work of Chaitin, Solomonoff, Komolgarov and Wittgenstein and to the notion that no program (and thus no device) can generate a sequence (or device) with greater complexity than it possesses. One might say this body of work implies atheism since there cannot be any entity more complex than the physical universe and from the Wittgensteinian viewpoint, ‘more complex’ is meaningless (has no conditions of satisfaction, i.e., truth-maker or test). Even a ‘God’ (i.e., a ‘device’ with limitless time/space and energy) cannot determine whether a given ‘number’ is ‘random’ nor can find a certain way to show that a given ‘formula’, ‘theorem’ or ‘sentence’ or ‘device’ (all these being complex language games) is part of a particular ‘system’. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my article The Logical Structure of Philosophy, Psychology, Mind and Language as Revealed in Wittgenstein and Searle 59p(2016). For all my articles on Wittgenstein and Searle see my e-book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Wittgenstein and Searle 367p (2016). Those interested in all my writings in their most recent versions may consult my e-book Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2016’ 662p (2016). -/- All of my papers and books have now been published in revised versions both in ebooks and in printed books. -/- Talking Monkeys: Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet - Articles and Reviews 2006-2017 (2017) https://www.amazon.com/dp/B071HVC7YP. -/- The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle--Articles and Reviews 2006-2016 (2017) https://www.amazon.com/dp/B071P1RP1B. -/- Suicidal Utopian Delusions in the 21st century: Philosophy, Human Nature and the Collapse of Civilization - Articles and Reviews 2006-2017 (2017) https://www.amazon.com/dp/B0711R5LGX . (shrink)

The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set (...) theory. Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering “theorem”, transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for “Hilbert mathematics” accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel’s papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part. (shrink)

There is a long-standing gap in the literature as to whether Gödelian incompleteness constitutes a challenge for Neo-Logicism, and if so how serious it is. In this paper, I articulate and address the challenge in detail. The Neo-Logicist project is to demonstrate the analyticity of arithmetic by deriving all its truths from logical principles and suitable definitions. The specific concern raised by Gödel’s first incompleteness theorem is that no single sound system of logic syntactically implies all arithmetical truths. (...) I set out some responses that initially seem appealing and explain why they are not compelling. The upshot is that Neo-Logicism either offers an epistemic route only to some truths of arithmetic; or that it has to move from a syntactic to a semantic notion of logical consequence, which risks undermining its epistemic goals. I conclude by considering Crispin Wright’s recent attempt to address Gödelian incompleteness, which I argue is not satisfactory. (shrink)

Gödel's incompleteness theorems establish the stunning result that mathematics cannot be fully formalized and, further, that any formal system containing a modicum of number or set theory cannot establish its own consistency. Wilfried Sieg and Clinton Field, in their paper Automated Search for Gödel's Proofs, presented automated proofs of Gödel's theorems at an abstract axiomatic level; they used an appropriate expansion of the strategic considerations that guide the search of the automated theorem prover AProS. (...) The representability conditions that allow the syntactic notions of the metalanguage to be represented inside the object language were taken as axioms in the automated proofs. The concrete task I am taking on in this project is to extend the search by formally verifying these conditions. Using a formal metatheory defined in the language of binary trees, the syntactic objects of the metatheory lend themselves naturally to a direct encoding in Zermelo's theory of sets. The metatheoretic notions can then be inductively defined and shown to be representable in the object-theory using appropriate inductive arguments. Formal verification of the representability conditions is the first step towards an automated proof thereof which, in turn, brings the automated verification of Gödel's theorems one step closer to completion. (shrink)

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

The mind-body problem is one of the important topics in philosophy of mind and cognitive science. Following the analytical tradition of linguistic and logical analysis, we focus on two aspects of the mind- body problem: one is around Gödel's incompleteness theorem, and the other is on cognitive logic, especially on the question of whether Epistemological Arithmetic and machines are private. In the former case, in response to the popular view that the Gödel Incompleteness Theorem supports dualism in (...) the mind-body problem, H. Putnam constructs a counter-argument that defends the computationalist agenda: the equivalence of mind and Turing machine. In this regard, the paper discusses Putnam's line of analysis and then argues that it is unsound. In the latter case, the paper examines in detail the cognitive form of the expression of privateness and concludes, together with recent results in EA, that the existence of privateness in Turing machines is logically untenable and, therefore, we argue that the mind is not equivalent to a machine in terms of cognitive ability and privateness, i.e., the computationalist agenda is not valid. (shrink)

Within the (Haskell Curry) notion of a formal system we complete Tarski's formal correctness: ∀x True(x) ↔ ⊢ x and use this finally formalized notion of Truth to refute his own Undefinability Theorem (based on the Liar Paradox), the Liar Paradox, and the (Panu Raatikainen) essence of the conclusion of the 1931 Incompleteness Theorem.

Storrs McCall continues the tradition of Lucas and Penrose in an attempt to refute mechanism by appealing to Gödel’s incompleteness theorem. That is, McCall argues that Gödel’s theorem “reveals a sharp dividing line between human and machine thinking”. According to McCall, “[h]uman beings are familiar with the distinction between truth and theoremhood, but Turing machines cannot look beyond their own output”. However, although McCall’s argumentation is slightly more sophisticated than the earlier Gödelian anti-mechanist arguments, in the end it fails (...) badly, as it is at odds with the logical facts. (shrink)

In 1956, a few writings of Wittgenstein that he didn't publish in his lifetime were revealed to the public. These writings were gathered in the book Remarks on the Foundations of Mathematics (1956). There, we can see that Wittgenstein had some discontentment with the way philosophers, logicians, and mathematicians were thinking about paradoxes, and he even registered a few polemic reasons to not accept Gödel’s incompleteness theorems.

The “four-color” theorem seems to be generalizable as follows. The four-letter alphabet is sufficient to encode unambiguously any set of well-orderings including a geographical map or the “map” of any logic and thus that of all logics or the DNA plan of any alive being. Then the corresponding maximally generalizing conjecture would state: anything in the universe or mind can be encoded unambiguously by four letters. That admits to be formulated as a “four-letter theorem”, and thus one can search for (...) a properly mathematical proof of the statement. It would imply the “four colour theorem”, the proof of which many philosophers and mathematicians believe not to be entirely satisfactory for it is not a “human proof”, but intermediated by computers unavoidably since the necessary calculations exceed the human capabilities fundamentally. It is furthermore rather unsatisfactory because it consists in enumerating and proving all cases one by one. Sometimes, a more general theorem turns out to be much easier for proving including a general “human” method, and the particular and too difficult for proving theorem to be implied as a corollary in certain simple conditions. The same approach will be followed as to the four colour theorem, i.e. to be deduced more or less trivially from the “four-letter theorem” if the latter is proved. References are only classical and thus very well-known papers: their complete bibliographic description is omitted. (shrink)

It is commonly thought that such topics as Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason are disparate scientific physical or mathematical issues having little or nothing in common. I suggest that they are largely standard philosophical problems (i.e., language games) which were resolved by Wittgenstein over 80 years ago. -/- Wittgenstein also demonstrated the fatal error in regarding mathematics or language or our behavior in general as a unitary coherent logical ‘system,’ rather than (...) as a motley of pieces assembled by the random processes of natural selection. “Gödel shows us an unclarity in the concept of ‘mathematics’, which is indicated by the fact that mathematics is taken to be a system” and we can say (contra nearly everyone) that is all that Gödel and Chaitin show. Wittgenstein commented many times that ‘truth’ in math means axioms or the theorems derived from axioms, and ‘false’ means that one made a mistake in using the definitions, and this is utterly different from empirical matters where one applies a test. Wittgenstein often noted that to be acceptable as mathematics in the usual sense, it must be useable in other proofs and it must have real world applications, but neither is the case with Godel’s Incompleteness. Since it cannot be proved in a consistent system (here Peano Arithmetic but a much wider arena for Chaitin), it cannot be used in proofs and, unlike all the ‘rest’ of PA it cannot be used in the real world either. As Rodych notes “…Wittgenstein holds that a formal calculus is only a mathematical calculus (i.e., a mathematical language-game) if it has an extra- systemic application in a system of contingent propositions (e.g., in ordinary counting and measuring or in physics) …” Another way to say this is that one needs a warrant to apply our normal use of words like ‘proof’, ‘proposition’, ‘true’, ‘incomplete’, ‘number’, and ‘mathematics’ to a result in the tangle of games created with ‘numbers’ and ‘plus’ and ‘minus’ signs etc., and with -/- ‘Incompleteness’ this warrant is lacking. Rodych sums it up admirably. “On Wittgenstein’s account, there is no such thing as an incomplete mathematical calculus because ‘in mathematics, everything is algorithm [and syntax] and nothing is meaning [semantics]…” -/- I make some brief remarks which note the similarities of these ‘mathematical’ issues to economics, physics, game theory, and decision theory. -/- Those wishing further comments on philosophy and science from a Wittgensteinian two systems of thought viewpoint may consult my other writings -- Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle 2nd ed (2019), Suicide by Democracy 4th ed (2019), The Logical Structure of Human Behavior (2019), The Logical Structure of Consciousness (2019, Understanding the Connections between Science, Philosophy, Psychology, Religion, Politics, and Economics and Suicidal Utopian Delusions in the 21st Century 5th ed (2019), Remarks on Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason in Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych, Berto, Floyd, Moyal-Sharrock and Yanofsky (2019), and The Logical Structure of Philosophy, Psychology, Sociology, Anthropology, Religion, Politics, Economics, Literature and History (2019). (shrink)

In the Appendix I of his "Remarks on the Foundations of Mathematics", Wittgenstein elaborates a different interpretation of Gödel’s First Incompleteness Theorem, which we have come to refer to as "Gödel’s Theorem" or "Incompleteness Theorem". This nomenclature arises from the recognition that the so-called "Second Incompleteness Theorem" is essentially a corollary of the primary theorem. Wittgenstein aims to reassess Gödel’s conclusion that there exist true formulas not demonstrable within formal systems capable of representing a sufficient amount of (...) arithmetic theory. Gödel’s initial reaction, as well as other commentators, was that Wittgenstein had not understood the proof. Nevertheless, recent commentators view worthy commentaries in wittgensteinian writings: some commentators, such as Juliet Floyd and Hilary Putnam, distinguish between mathematical proof and philosophical prose that surrounds the theorem, making it possible to understand Wittgenstein’s remarks. Ultimately, Wittgenstein’s observations serve as a lens through which Gödel’s theorem can be reconsidered within the realm of non-classical logics, such as paraconsistent logic. (shrink)

The present paper intends to report two results. It is shown that the formula P(x)=∀y∀z[¬G(x, y)→¬M(z)] provides the logic underlying gauge symmetry, where M denotes the predicate of being massive. For the logic of spontaneous symmetry breaking, by Higgs mechanism, we have P(x)=∀y∀z[G(x, y)→M(z)]. Notice that the above two formulas are not logically equivalent. The results are obtained by integrating four components, namely, gauge symmetry and Higgs mechanism in quantum field theory, and Gödel's incompleteness theorem and Tarski's indefinability (...) theorem in mathematical logic. Gödel numbering is the key for arithmetic modeling applied in this paper. (shrink)

We argue that Wittgenstein’s philosophical perspective on Gödel’s most famous theorem is even more radical than has commonly been assumed. Wittgenstein shows in detail that there is no way that the Gödelian construct of a string of signs could be assigned a useful function within (ordinary) mathematics. — The focus is on Appendix III to Part I of Remarks on the Foundations of Mathematics. The present reading highlights the exceptional importance of this particular set of remarks and, more specifically, emphasises (...) its refined composition and rigorous internal structure. (shrink)

This article examines the various Liar paradoxes and their near kin, Grelling’s paradox and Gödel’s Incompleteness Theorem with its self-referential Gödel sentence. It finds the family of paradoxes to be generated by circular definition–whether of statements, predicates, or sentences–a manoeuvre that generates pseudo-statements afflicted with the Liar syndrome: semantic vacuity, semantic incoherence, and predicative catalepsy. Such statements, e.g., the self-referential Liar statement, are meaningless, and hence fail to say anything, a point that invalidates the reasoning on which the various (...) paradoxes rest. The seeming plausibility of the paradoxes is due to the fact that often the sentence used to make the pseudo-statement is ambiguous in that it may also be used to make a genuine statement about the pseudo-statement. Hence, if a formal system is to avoid ambiguity and consequent paradox and contradiction, it must distinguish between the two statements the sentence may be used to make. Gödel’s Theorem presents a further complication in that the self-reference involved is sentential rather than statemental. Nevertheless, on the intended interpretation of the system as a formalization of arithmetic, the self-referential Gödel sentence can only be an ambiguous statement, one that is both a pseudo-statement and its genuine double. Consequently, the conclusions commonly drawn from Gödel’s theorem must be deemed unwarranted. Arithmetic might well be formalized in a proper system that either excludes circular definition or introduces disambiguators. (shrink)

We respond to some of the points made by Bennet and Blanck (2022) concerning a previous publication of ours (2021).

Can the so-ca\led first incompleteness theorem refer to itself? Many or maybe even all the paradoxes in mathematics are connected with some kind of self-reference. Gбdel built his proof on the ground of self-reference: а statement which claims its unprovabllity. So, he demonstrated that undecidaЬle propositions exist in any enough rich axiomatics (i.e. such one which contains Peano arithmetic in some sense). What about the decidabllity of the very first incompleteness theorem? We can display that it fulfills its (...) conditions. That's why it can Ье applied to itself, proving that it is an undecidaЬle statement. It seems to Ье а too strange kind of proposition: its validity implies its undecidabllity. If the validity of а statement implies its untruth, then it is either untruth (reductio ad absurdum) or an antinomy (if also its negation implies its validity). А theory that contains а contradiction implies any statement. Appearing of а proposition, whose validity implies its undecidabllity, is due to the statement that claims its unprovability. Obviously, it is а proposition of self-referential type. Ву Gбdel's words, it is correlative with Richard's or liar paradox, or even with any other semantic or mathematical one. What is the cost, if а proposition of that special kind is used in а proof? ln our opinion, the price is analogous to «applying» of а contradictory in а theory: any statement turns out to Ье undecidaЬ!e. Ifthe first incompleteness theorem is an undecidaЬ!e theorem, then it is impossiЬle to prove that the very completeness of Peano arithmetic is also an tmdecidaЬle statement (the second incompleteness theorem). Hilbert's program for ап arithmetical self-foundation of matheшatics is partly rehabllitated: only partly, because it is not decidaЬ!e and true, but undecidaЬle; that's wby both it and its negation шау Ье accepted as true, however not siшultaneously true. The first incompleteness theoreш gains the statute of axiom of а very special, semi-philosophical kind: it divides mathematics as whole into two parts: either Godel шathematics or Нilbert matheшatics. Нilbert's program of self-foundation ofmatheшatic is valid only as to the latter. (shrink)

(This is for the Cambridge Handbook of Analytic Philosophy, edited by Marcus Rossberg) In this handbook entry, I survey the different ways in which formal mathematical methods have been applied to philosophical questions throughout the history of analytic philosophy. I consider: formalization in symbolic logic, with examples such as Aquinas’ third way and Anselm’s ontological argument; Bayesian confirmation theory, with examples such as the fine-tuning argument for God and the paradox of the ravens; foundations of mathematics, with examples such as (...) Hilbert’s programme and Gödel’s incompleteness theorems; social choice theory, with examples such as Condorcet’s paradox and Arrow’s theorem; ‘how possibly’ results, with examples such as Condorcet’s jury theorem and recent work on intersectionality theory; and the application of advanced mathematics in philosophy, with examples such as accuracy-first epistemology. (shrink)

Bertrand Russell was one of the best-known proponents of logicism: the theory that mathematics reduces to, or is an extension of, logic. Russell argued for this thesis in his 1903 The Principles of Mathematics and attempted to demonstrate it formally in Principia Mathematica (PM 1910–1913; with A. N. Whitehead). Russell later described his work as a further “regressive” step in understanding the foundations of mathematics made possible by the late 19th century “arithmetization” of mathematics and Frege’s logical definitions of arithmetical (...) concepts. The logical system of PM sought to improve on earlier attempts by solving the contradictions found in, e.g., Frege’s system, by employing a theory of types. In this article, I also consider and critically evaluate the most common objections to Russell’s logicism, including the claim that it is undermined by Gödel’s incompleteness results, and Putnam's charge of “if-thenism”. I suggest that if we are willing to accept a slightly revisionist account of what counts as a mathematical truth, these criticisms do not obviously refute Russell’s claim to have established that mathematical truths generally are a species of logical truth. (shrink)

In the paper it is demonstrated that Bell's theorem is unproveable.

Einstein wrote his famous sentence "God does not play dice with the universe" in a letter to Max Born in 1920. All experiments have confirmed that quantum mechanics is neither wrong nor “incomplete”. One can says that God does play dice with the universe. Let quantum mechanics be granted as the rules generalizing all results of playing some imaginary God’s dice. If that is the case, one can ask how God’s dice should look like. God’s dice turns out to be (...) a qubit and thus having the shape of a unit ball. Any item in the universe as well the universe itself is both infinitely many rolls and a single roll of that dice for it has infinitely many “sides”. Thus both the smooth motion of classical physics and the discrete motion introduced in addition by quantum mechanics can be described uniformly correspondingly as an infinite series converges to some limit and as a quantum jump directly into that limit. The second, imaginary dimension of God’s dice corresponds to energy, i.e. to the velocity of information change between two probabilities in both series and jump. (shrink)

The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of Hilbert (...) mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat’s last theorem proved by Andrew Wiles; Poincaré’s conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved “machine-likely” by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat’s last theorem is shown as a Gödel insoluble statement by means of Yablo’s paradox. Thus, Wiles’s proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in “Fermat arithmetic” introduced by “epoché to infinity” (following the pattern of Husserl’s original “epoché to reality”) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a “Wittgenstein ladder”. Poincaré’s conjecture can be reinterpreted physically by Minkowski space and thus reduced to the “nonstandard homeomorphism” of a bit of information mathematically. Perelman’s proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré’s conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions. (shrink)

It is quite well-known from Kurt G¨odel’s (1931) ground-breaking 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 are not rudimentary. We present a simple and elementary proof of this fact in the first part of the 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. (shrink)