Search

Emergence refers to the spontaneous formation of higher level (macro) structures or patterns incomplex systems. Attempts to formalize the notion of emergence via algorithmic complexity theoryruns into the problem that the Kolmogorov complexity function is not computable. The reason for thismotivates a closer examination of the link between emergence and universal computation. FollowingWolfram’s pioneering work in the classification of cellular automata behavior, the research programsof Langton and Crutchfield, while incomplete, provide important insights to economists seeking tounderstand the relevance of emergence and universal computation to their discipline. They lead toquestions on the emergence of institutions and the concomitant changes in rule-based behavior on thepart of economic agents. 1. INTRODUCTION The twentieth century is replete with scientific and mathematical discoveriesthat have profoundly changed our world view. In physics, within a merecentury, our view of the cosmos changed from a classical (Newtonian) to arelativistic one following Einstein’s relativity theory in the early 1900s. Inmathematics, Hilbert’s faith in the closure of formal axiom systems fell apartwith Godel’s incompleteness theorem in the 1930s.Out of these ashes of lost deterministic foundations arose the sciences ofcomplex systems, first in the study of non-equilibrium thermodynamics(under the intellectual leadership of Ilya Prigogine in Brussels) and later, inbroader interdisciplinary terms, in New Mexico with the establishment of theSanta Fe Institute. Interestingly, a core element of this new paradigm—thenotion of emergence—reflects the passage of the sciences and mathematics

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 present an abstract framework in which we give simple proofs for Godel’s First and Second Incompleteness Theorems and obtain, as consequences, Davis’, Chaitin’s and Kritchman-Raz’s Theorems.

A response is given to the paper by A. M. Izmailova (Izmailova A. M. O kritike teoremy K. Gedelya o nepolnote A. V. Bessonovym [On A. V. Bessonov’s criticism of K. Gödel’s incompleteness theorem]. Studencheskii nauchnyi zhurnal “Grani nauki” [Student Scientific Journal "Facets of Science"], 2018, no. 1, p. 7-9. (in Russ.)) allegedly indicating a «serious error» in my analysis of K. Gödel’s second incompleteness theorem. It is shown that her criticism is based on gross logical errors, as well as on a misunderstanding of both the second incompleteness theorem and my results. Such a widespread misinterpretation is based on the inadmissible confusion of the proof of the consistency of formal arithmetic with the proof in it of a formula expressing its consistency. It is argued that Gödel's second theorem is not directly related to the proof of the consistency of formal arithmetic. It is proved that this theorem cannot be used in argumentation against feasibility of D. Hilbert’s finitistic program.

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

Our previous research showed that the semantic tableau deductive methodology of Fitting and Smullyan permits boundary-case exceptions to the second incompleteness theorem, if multiplication is viewed as a 3-way relation (rather than as a total function). It is known that tableau methodologies prove a schema of theorems verifying all instances of the law of the excluded middle. But if one promotes this schema of theorems into formalized logical axioms, then the meaning of the pronoun of ‘I’, used by our self-referencing engine, changes quite sharply. Our partial evasions of the second incompleteness theorem shall then come to a complete halt.

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

An attempt of the philosophical underpinning of the adoption of new standards of the fundamental physical quantities and the new International system of units (SI) is made. The measurements in the system of standards are discussed from a historical perspective with reference to the theory of scientific revolutions. The causes and effects for replacing the standards relying on material carriers by those based on seven universal physical constants are analyzed. The new SI is discussed as a basis of the non-classical science paradigm and as a preparation for a new post-classical stage.

This paper is an investigation of the relationship between G\"odel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector's that the relation $\{(A,B) \in \mathbb{R}^2 : \mathcal{O}^A \leq_H B\}$ is well-founded. We provide an alternative proof of this fact that uses G\"odel's second incompleteness theorem instead of the theory of admissible ordinals. We then derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson, from this result. Finally, we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any $A\in\mathbb{R}$, if the rank of $A$ is $\alpha$, then $\omega_1^A$ is the $(1 + \alpha)^{\text{th}}$ admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of $X$ is $\omega_1^X$., Comment: 11 pages. Corrects a mistake in the statements of two results

We take a magical tour in algebraic logic, which is the natural interface between universal algebra and mathematical logic, starting from classical results on neat embeddings due to Andreḱa, Henkin, Nemeti, Monk and Tarski, all the way to recent novel results in algebraic logic using so-called rainbow constructions. Highlighting the connections with graph theory, model theory, and finite combinatorics, this article aspires to present topics of broad interest in a way that is hopefully accessible to a large audience. Other topics dealt with include the interaction of algebraic and modal logic, the so-called (central still active) finitizability problem, Godels’s incompleteness Theorem in guarded fragments, counting the number of subvarieties of $$\textsf {RCA}_{\omega }$$ which is reminiscent of Shelah’s classification theory and the interaction of algebraic logic and descriptive set theory as means to approach Vaught’s conjecture in model theory. The paper has a survey character but it contains new results and new approaches to old ones (such as the interaction of algebraic logic and descriptive set theory).

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.

We present an abstract development of Gödel’s incompleteness theorems, performed with the help of the Isabelle/HOL proof assistant. We analyze sufficient conditions for the applicability of our theorems to a partially specified logic. In addition to the usual benefits of generality, our abstract perspective enables a comparison between alternative approaches from the literature. These include Rosser’s variation of the first theorem, Jeroslow’s variation of the second theorem, and the Świerczkowski–Paulson semantics-based approach. As part of the validation of our framework, we upgrade Paulson’s Isabelle proof to produce a mechanization of the second theorem that does not assume soundness in the standard model, and in fact does not rely on any notion of model or semantic interpretation.

In this paper, a new concept, i.e. ultra-chaos, is proposed for the first time. Unlike a normal-chaos, statistical properties such as the probability density functions (PDF) of an ultra-chaos are sensitive to tiny disturbances. We illustrate that ultra-chaos is widely existed and thus has general scientific meanings. It is found that statistical non-reproducibility is an inherent property of an ultra-chaos so that an ultra-chaos is at a higher-level of disorder than a normal-chaos. Thus, it is impossible in practice to replicate experimental/numerical results of an ultra-chaos even in statistical meanings, since random environmental noises always exist and are out of control. Thus, the ultra-chaos should be an insurmountable obstacle of reproducibility and replicability. Similar to G\"{o}del's incompleteness theorem, such kind of "incompleteness of reproducibility" reveals a limitation of our traditional scientific paradigm based on reproducible experiments, which can be traced back to Galileo. The ultra-chaos opens a new door and possibility to study chaos theory, turbulence theory, computational fluid dynamics (CFD), the statistical significance, reproducibility crisis, and so on., Comment: 17 pages, 10 figures. Accepted for publication by Advances in Applied Mathematics and Mechanics in Dec. 2021

The teachers' attitudes toward parascience, which indicate a misunderstanding of modern physics concepts, are critically analysed. The materialistic ideology has been discussed in relation to quantum mechanics, modern cosmology, and the Goedel theorem. The inconsistencies between physical inquiry methods and parascience are highlighted. Results of the research in the fields of quantum mechanics and cosmology can hardly be reconciled with atheism.

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.

© 2019 Elsevier B.V. Intensional computations are those that query the internal structure of their arguments. In a higher-order setting, such queries perform program analysis. This is beyond the expressive power of traditional term rewriting systems, such as lambda-calculus or combinatory logic, as they are extensional. In such settings it has been necessary to encode or quote the program before analysis. However, there are intensional calculi, specifically confluent term rewriting systems, that can analyse higher-order programs within the calculus proper, without quotation; there are even programs that produce the Goedel numbers of their program argument. This paper summarizes the current situation. Highlights include the following observations. We have known since 2011 that the simplest intensional calculus, SF-calculus, supports arbitrary queries of closed normal forms, including equality, pattern-matching, searching and self-interpretation. Recent work, verified using the Coq proof assistant, has shown that all recursive programs can be represented as closed normal forms in SF-calculus, and even in combinatory logic. Thus, we can here deduce that SF-calculus (but not combinatory logic) can define queries of programs. These results are compatible with direct support for lambda-abstraction. Although these results conflict with the traditional understanding of expressive power of combinatory logic and λ-calculus, as developed by Church and Kleene, our recent publication has shown that their approach is compromised by its reliance on encodings. To drive the point home, this paper uses a non-standard encoding to lambda-define a trivial solution of the Halting Problem.

In this paper, we provide a fairly general self-reference-free proof of the second incompleteness theorem from Tarski’s theorem on the undefinability of truth.

The article compares two different approaches of incorporating probability into coalition logics. One is based on the semantics of games with stochastic transitions and the other on games with the stochastic failures. The work gives an example of a non-trivial property of coalition power for the first approach and a complete axiomatization for the second approach. It turns out that the logical properties of the coalition power modality under the second approach depend on whether the modal language allows the empty coalition. The main technical results for the games with stochastic failures are a strong completeness theorem for the logical system without the empty coalition and an incompleteness theorem which shows that there is no strongly complete logical system in the language with the empty coalition.

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 Godel’s incompleteness theorems is that self-referencing leads to logical conflict or failure, as in the liar paradox or Russell’s paradox. In physics 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.

This is a modification as well as a model theoretic application of my previous paper Sd- Topology over the theory of a Model. The main intention of this article is to proof and visualize the celebrated Gödel’s completeness and incompleteness theorems in a different and lucid approach., AMS -MSC2010 No. 03B99

The chapter starts with an overview of information theory, how information is measured and how it is encoded in messages and transmitted. The ambiguity of coding with numeric alphabets is exposed with an analogy from computer science. An unambiguous coding is shown with reference to Godel’s incompleteness theorem.

In his famous article On formally undecidable propositions of Principia Mathematica and related systems I (1931), Godel established two results that marked a point of no return in the history of logic.

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.

This chapter is about the ontology and epistemology of mathematical objects. The core problem for an empiricist is that conceiving mathematical objects as existing independently of human thinking makes it impossible to understand how we can have mathematical knowledge, while the alternative, a constructivist conception, resolves the epistemological problem, but entails the identification of truth with provability. That entails that the law of excluded middle must be dismissed as a generally valid logical principle. The identificaiton of truth with provability is furthermore problematic when taking into account Godel’s first incompleteness theorem. The chapter ends with suggesting a modified constructivism, which keeps the distinction between truth and provability, thus avoiding counter arguments based on Godel’s theorem.