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Google Scholar OpenURL Placeholder Text WorldCat 11 Gödel, Kurt [ 1934 ]: ' On undecidable propositions of formal mathematical systems ', in [Davis, 1965], pp. 39 - 74. Google Scholar OpenURL Placeholder Text WorldCat 9 Gödel, Kurt [ 1931a ]: ' Diskussion zur Grundlegung der Mathematik ', Erkenntnis 2, 147 - 151. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC 6 Gödel, Kurt [ 1930 ]: ' Die Vollstandigkeit der Axiome des logischen Funktionenkalküls ', Monatshefte für Mathematik und Physik 37, 349 - 360. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC 19 Wang Hao [ 1996 ]; A Logical Journey, from Gödel to Philosophy. [Extracted from the article]

An in-depth examination of the foundations of mathematics reveals how its treatment is centered around the topic of "unique foundation vs. no need for a foundation" in a traditional setting. In this paper, I show that by applying Shelah's stability procedures to mathematics, we confine ourselves to a certain section that manages to escape the Gödel phenomenon and can be classified. We concentrate our attention on this mainly because of its tame nature. This result makes way for a new approach in foundations through model-theoretic methods. We then cover Penelope Maddy's "foundational virtues" and what it means for a theory to be foundational. Having explored what a tame foundation can amount to, we argue that it can fulfil some of Maddy's foundational qualities. In the last part, we will examine the consequences of this new paradigm -- some philosophical in nature -- on topics like philosophy of mathematical practice, the incompleteness theorems and others. [ABSTRACT FROM AUTHOR]

We extend both the Hawking–Penrose theorem and its generalisation due to Galloway and Senovilla to Lorentzian metrics of regularity C 1 . For metrics of such low regularity, two main obstacles have to be addressed. On the one hand, the Ricci tensor now is distributional, and on the other hand, unique solvability of the geodesic equation is lost. To deal with the first issue in a consistent way, we develop a theory of tensor distributions of finite order, which also provides a framework for the recent proofs of the theorems of Hawking and of Penrose for C 1 -metrics (Graf in Commun Math Phys 378(2):1417–1450, 2020). For the second issue, we study geodesic branching and add a further alternative to causal geodesic incompleteness to the theorem, namely a condition of maximal causal non-branching. The genericity condition is re-cast in a distributional form that applies to the current reduced regularity while still being fully compatible with the smooth and C 1 , 1 -settings. In addition, we develop refinements of the comparison techniques used in the proof of the C 1 , 1 -version of the theorem (Graf in Commun Math Phys 360:1009–1042, 2018). The necessary results from low regularity causality theory are collected in an appendix. [ABSTRACT FROM AUTHOR]

In this paper we present a study that had as a guiding question: What is the beauty in mathematics and the beauty in Gödel's incompleteness theorem? We seek to present identifiable content for what we understand as beauty in Gödel's incompleteness theorem. For this, a bibliographic study was made and different notions of beauty in mathematics are brought and articulated. We also present our understanding of beauty in Gödel's theorem. We understand that the meaning of mathematical beauty of a theorem is that of an illumination that evidences the result. Moreover, we understand that this light allows itself to be seen insofar as one is familiar with the theory and with the tools used in the demonstration, to the point of being able to perceive the axioms used, the conciseness of the proof, the originality of the articulation of ideas, the possibilities of generalizing the result and the opening of new research fronts. We also recognize that the knowledge built by Gödel in the elaboration of his completeness theorem was fundamental in the view of the problem of the consistency of arithmetic and in the approach chosen to demonstrate the consistency of arithmetic, which became the demonstration of the incompleteness of the theory of arithmetic. [ABSTRACT FROM AUTHOR]

This paper deals with the issue of interdisciplinarity in literature, tracing the mathematical influences — and particularly those of the incompleteness theorems by Kurt Gödel — on the literary works of Hans Magnus Enzensberger. Our attention is focused on Enzensberger’s publications in mathematics, on his poem Hommage à Gödel, and his montage epos The Sinking of the Titanic. This paper submits a new, interdisciplinary interpretation of the poem Hommage à Gödel as a kind of art manifesto, and comes to the conclusion that the reason Enzensberger finds the incompleteness theorems so interesting is that they offer a scientific basis for undecidable problems and paradoxes. He applies Gödel’s theorems to literature, where they enable a creative, playful oscillation between two positions or aspects of an undecidable problem and self-reference in literature.

This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Gödel's incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have a place of pride. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, etc. Perhaps, by Gödel's incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find the solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Gödel's incompleteness theorems will have ramifications in other areas involving logic. This paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics. [ABSTRACT FROM AUTHOR]

This paper deals with the issue of interdisciplinarity in literature, tracing the mathematical influences -- and particularly those of the incompleteness theorems by Kurt Gödel -- on the literary works of Hans Magnus Enzensberger. Our attention is focused on Enzensberger's publications in mathematics, on his poem Hommage à Gödel, and his montage epos The Sinking of the Titanic. This paper submits a new, interdisciplinary interpretation of the poem Hommage à Gödel as a kind of art manifesto, and comes to the conclusion that the reason Enzensberger finds the incompleteness theorems so interesting is that they offer a scientific basis for undecidable problems and paradoxes. He applies Gödel's theorems to literature, where they enable a creative, playful oscillation between two positions or aspects of an undecidable problem and self-reference in literature. [ABSTRACT FROM AUTHOR]

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 Gödel'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 Gödel'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'. [ABSTRACT FROM AUTHOR]

In Gödel’s Incompleteness Theorems, for every mathematical system there are correct statements that cannot be proven to be correct within that system. We here extend this to address the question of axiomatic statements that are perceived (or known) to be correct but which mathematics, as presently constituted, cannot prove. We refer to these as perceptual axioms. [ABSTRACT FROM AUTHOR]

This paper argues that incompleteness of intertemporal financial markets has little effect (on welfare, prices, or consumption) in an economy with a single consumption good, provided that traders are long-lived and patient, a riskless bond is traded, shocks are transitory, and there is no aggregate risk. In an economy with aggregate risk, a similar conclusion holds, provided traders share the same CRRA utility function and the right assets are traded. Examples demonstrate that these conclusions need not hold if the wrong assets are traded or if the economy has multiple consumption goods. KEYWORDS: Infinite horizon economies, incomplete markets, law of large numbers, consumption smoothing, permanent income hypothesis.

Without introducing either debt constraints or transversality conditions to avoid the possibility of Ponzi schemes, we show the existence of equilibrium in an infinite horizon incomplete markets economy with a collateral structure. KEYWORDS: Exogenous collateral, incomplete markets, infinite horizon, equilibrium, Ponzi schemes.

Can the so-called first incompleteness theorem refer to itself? Many or maybe even all the paradoxes in mathematics are connected with some kind of self-reference. Gödel built his proof on the ground of self-reference: a statement which claims its unprovability. So, he demonstrated that undecidable propositions exist in any enough rich axiomatics (i.e. such one which contains Peano arithmetic in some sense). What about the decidability of the very first incompleteness theorem? We can display that it fulfills its conditions. That's why it can be applied to itself, proving that it is an undecidable statement. It seems to be a too strange kind of proposition: its validity implies its undecidability. If the validity of a statement implies its untruth, then it is either untruth (reductio ad absurdum) or an antinomy (if also its negation implies its validity). A theory that contains a contradiction implies any statement. Appearing of a proposition, whose validity implies its undecidability, is due to the statement that claims its unprovability. Obviously, it is a proposition of self-referential type. By Gödel'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 a proposition of that special kind is used in a proof? In our opinion, the price is analogous to «applying» of a contradictory in a theory: any statement turns out to be undecidable. If the first incompleteness theorem is an undecidable theorem, then it is impossible to prove that the very completeness of Peano arithmetic is also an undecidable statement (the second incompleteness theorem). Hilbert's program for an arithmetical self-foundation of mathematics is partly rehabilitated: only partly, because it is not decidable and true, but undecidable; that's why both it and its negation may be accepted as true, however not simultaneously true. The first incompleteness theorem gains the statute of axiom of a very special, semi-philosophical kind: it divides mathematics as whole into two parts: either Gödel mathematics or Hilbert mathematics. Hilbert's program of self-foundation of mathematic is valid only as to the latter. [ABSTRACT FROM AUTHOR]

This paper will apply post-structural semiotic theories to study the texts of Gödel's first incompleteness theorem. I will study the texts' own articulations of concepts of 'meaning', analyze the mechanisms they use to sustain their senses of validity, and point out how the texts depend (without losing their mathematical rigor) on sustaining some shifts of meaning. I will demonstrate that the texts manifest semiotic effects, which we usually associate with poetry and everyday speech. I will conclude with an analysis of how the picture I paint relates to an ethics of mathematical production. [ABSTRACT FROM AUTHOR]

The article profiles the Russian mathematician Andrei Andreevich Markov Jr. It includes information on Markov's personal and educational background, as well as his early interest in abstract mathematics indicated by his series of papers on geometry, algebra, topology and analysis. Markov, however, is most well-known for founding the Russian school of constructive mathematics during the late 1940s and early 1950s. His constructive mathematics is considered as one of the 20th century's most important and coherent constructivist trends.

We give some information about new proofs of the incompleteness theorems, found in 1990s. Some of them do not require the diagonal lemma as a method of construction of an independent statement. [Copyright &y& Elsevier]

Presents a biographical information on mathematician Emil Post and his approach to Kurt Gödel's incompleteness theorem. Chronological summary of the main events of Post's life and career; Post's proof for the completeness and consistency of propositional logic; Application of the diagonal argument; Problem of showing that all computation is reflected in normal systems.

The partial ordering of Mcdvcdcv reducibility restricted to the family of Π[sub0]¹ classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a Π[sub0][sub1] class, which we call a "c.c. separating class". We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublatticc generated by the c.c. separating classes. [ABSTRACT FROM AUTHOR]

Let T be the standard Veblen 1908 ordinal notation system for Γ0 as defined, for example, in Schütte's 1977 textbook [13] on Proof Theory. We define a slight modification of the standard assignment of fundamental sequences for the limit ordinals in T and prove that Γ0 is subrecursively inaccessible for this assignment, i.e. the induced slow and fast growing hierarchy match up at Γ0 for the first time.The results of this paper also indicate that ϕℇ00 may be considered as a new slow growing ordinal of PA in the sense that the induced slow growing hierarchy up to ϕℇ00 classifies the PA-provablyrecursive functions.We show how the results of this paper can be used to bui d a subrecursive hierarchyover the predicative ordinals in the spirit of Wainer's 1970 abstract [16 ]. [ABSTRACT FROM AUTHOR]

The subject matter of this research is Kant's apriorism underlying Hilbert's formalism in the philosophical grounding of mathematics as a self-sufficing system. The research aim is the invention of such a logically formalized axiomatic epistemology system, in which it is possible to construct formal deductive inferences of formulae—modeling the formalism ideal of Hilbert—from the assumption of Kant's apriorism in relation to mathematical knowledge. The research method is hypothetical–deductive (axiomatic). The research results and their scientific novelty are based on a logically formalized axiomatic system of epistemology called Σ + C, constructed here for the first time. In comparison with the already published formal epistemology systems Ξ and Σ, some of the axiom schemes here are generalized in Σ + C, and a new symbol is included in the object-language alphabet of Σ + C, namely, the symbol representing the perfection modality, C: "it is consistent that...". The meaning of this modality is defined by the system of axiom schemes of Σ + C. A deductive proof of the consistency of Σ + C is submitted. For the first time, by means of Σ + C, it is deductively demonstrated that, from the conjunction of Σ + C and either the first or second version of Gödel's theorem of incompleteness of a formal arithmetic system, the formal arithmetic investigated by Gödel is a representation of an empirical knowledge system. Thus, Kant's view of mathematics as a self-sufficient, pure, a priori knowledge system is falsified. [ABSTRACT FROM AUTHOR]

In the paper some applications of Gödel‘s incompleteness theorems to discussions of problems of computer science are presented. In particular the problem of relations between the mind and machine (arguments by J.J.C. Smart and J.R. Lucas) is discussed. Next Gödel‘s opinion on this issue is studied. Finally some interpretations of Gödel‘s incompleteness theorems from the point of view of the information theory are presented. [ABSTRACT FROM AUTHOR]

The article provides other proofs of old results of the studies related to mathematical logics. It transform the proof of the second incompleteness theorem given in the study of H. Kotlarski to a proof-theoretic version, which avoids the use of the arithmetized completeness theorem. Other old results that were given new proofs are the Arithmetical Hierarchy Theorem and Tarski's Theorem on undefinability of truth.

The article discusses the compatibility of a principal ideal to the construction of a coideal of a ring in Bishop's constructive mathematics. It presents notation, terminology, elementary definitions, and basic concepts about the algebraic structures of Bishop's constructive mathematics. Moreover, it shows a theorem on the existence of a coideal of a ring compatible with the diversity relation in congruence of principal ideal.

This article focuses on the range of order statistics. It presents several mathematical formulas regarding the statistics. It further says that P.A. Samuelson has shown that one of the given theorems yields an inequality sharper than that of Chebyshev. It is of interest to note that there is a discrepancy between Samuelson's second theorem and the corresponding theorem of this article.