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This paper examines the use of algorithms and non-algorithmic models in mathematics and science, especially in biology, during the past century by summarizing the gradual development of a conceptual rationale for non-algorithmic models in biology. First, beginning a century ago, mathematicians found it impossible to constrain mathematics in an algorithmic straitjacket via öö's Incompleteness Theorems, so how would it be possible in biology? By the 1930s, biology was resolutely imitating classical physics, with biologists enforcing a reductionist agenda to expunge function, purpose, teleology, and vitalism from biology. Interestingly, physicists and mathematicians often understood better than biologists that mathematical representations of living systems required different approaches than those of dead matter. Nicolas Rashevsky, the Father of Mathematical Biology, and Robert Rosen, his student, pointed out that the complex systems of life cannot be reduced to machines or mechanisms as per the Newtonian paradigm. Robert Rosen concluded that living systems are not amenable to algorithmic models that are primarily syntactical. Life requires semantics for its description. Rashevsky and Rosen pioneered Relational Biology, initially using Graph Theory to model living systems. Later, Rosen created a metabolic–repair model (M, R)-system using Category Theory to encode the basic entailments of life itself. Although reductionism still dominates in current biology, several subsequent authors have built upon the Rashevsky–Rosen intellectual foundation and have explained, extended, and explored its ramifications. Algorithmic formulations have become increasingly inadequate for investigating and modeling living systems. Biology is shifting from a science of simple systems to complex ones. This transition will only be successful once mathematics fully depicts what it means to be alive. This paper is a call to mathematicians from biologists asking for help in doing this. [ABSTRACT FROM AUTHOR]

Causal multivariate time-series analysis, combined with network theory, provide a powerful tool for studying complex ecological interactions. However, these methods have limitations often underestimated when used in graphical modelling of ecological systems. In this opinion article, I examine the relationship between formal logic methods used to describe causal networks and their inherent statistical and epistemological limitations. I argue that while these methods offer valuable insights, they are restricted by axiomatic assumptions, statistical constraints and the incompleteness of our knowledge. To prove that, I first consider causal networks as formal systems, define causality and formalize their axioms in terms of modal logic and use ecological counterexamples to question the axioms. I also highlight the statistical limitations when using multivariate time-series analysis and Granger causality to develop ecological networks, including the potential for spurious correlations among other data characteristics. Finally, I draw upon Gödel's incompleteness theorems to highlight the inherent limits of fully understanding complex networks as formal systems and conclude that causal ecological networks are subject to initial rules and data characteristics and, as any formal system, will never fully capture the intricate complexities of the systems they represent. This article is part of the theme issue 'Connected interactions: enriching food web research by spatial and social interactions'. [ABSTRACT FROM AUTHOR]

In the theory of recursive functions, a recursively enumerable (but not recursive) set K = {x | x ∈Wx} is obtained by Cantor's diagonal method. Based on Diophantine sets, K is expressed by some polynomial that has positive roots. On the contrary, the set K ¯ = x x ∉ W x is not recursively enumerable. None of the computable functions can enumerate all the elements of the set K ¯ . As a result of the productivity of the set K ¯ , a parameter exists, for which the polynomial has no positive roots. However, it is impossible to prove their absence since this parameter does not belong to any recursively enumerable set. [ABSTRACT FROM AUTHOR]

This volume is one of two special issues collecting articles by invited speakers of the conference 'Celebrating 90 Years of Gödel's Incompleteness Theorems' held in Nürtingen (Germany) in July 2021. The conference was organized by the Carl Friedrich von Weizsäcker Center at the University of Tübingen with support by the ERC-funded project Gödel Enigma: Rediscovering Kurt Gödel through his unpublished works at the University of Helsinki and by the Kurt Gödel Society in Vienna. [ABSTRACT FROM AUTHOR]

Gödel's Incompleteness Theorems suggest that no single formal system can capture the entirety of one's mathematical beliefs, while pointing at a hierarchy of systems of increasing logical strength that make progressively more explicit those implicit assumptions. This notion of implicit commitment motivates directly or indirectly several research programmes in logic and the foundations of mathematics; yet there hasn't been a direct logical analysis of the notion of implicit commitment itself. In a recent paper, we carried out an initial assessment of this project by studying necessary conditions for implicit commitments; from seemingly weak assumptions on implicit commitments of an arithmetical system |$S$|⁠ , it can be derived that a uniform reflection principle for |$S$| —stating that all numerical instances of theorems of |$S$| are true—must be contained in |$S$| 's implicit commitments. This study gave rise to unexplored research avenues and open questions. This paper addresses the main ones. We generalize this basic framework for implicit commitments along two dimensions: in terms of iterations of the basic implicit commitment operator, and via a study of implicit commitments of theories in arbitrary first-order languages, not only couched in an arithmetical language. [ABSTRACT FROM AUTHOR]

In this paper, we study the metamathematics of consistent arithmetical theories |$T$| (containing |$\textsf {I}\varSigma _{1}$|⁠); we investigate numerical properties based on proof predicates that depend on numerations of the axioms. Numeral Completeness. For every true (in |$\mathbb {N}$|⁠) sentence |$\vec {Q}\vec {x}.\varphi (\vec {x})$|⁠ , with |$\varphi (\vec {x})$| a |$\varSigma _{1}(\textsf {I}\varSigma _1)$| -formula, there is a numeration |$\tau $| of the axioms of |$T$| such that |$\textsf {I}\varSigma _1\vdash \vec {Q}\vec {x}. \texttt {Pr}_{\tau }(\ulcorner \varphi (\overset {\text{.} }{\vec {x}})\urcorner)$|⁠ , where |$\texttt {Pr}_{\tau }$| is the provability predicate for the numeration |$\tau $|⁠. Numeral Consistency. If |$T$| is consistent, there is a |$\varSigma _{1}(\textsf {I}\varSigma _1)$| -numeration |$\tau $| of the axioms of |$\textsf {I}\varSigma _{1}$| such that |$\textsf {I}\varSigma _1\vdash \forall\, x. \texttt {Pr}_{\tau }(\ulcorner \neg \textit {Prf}(\ulcorner \perp \urcorner , \overset {\text{.}}{x})\urcorner)$|⁠ , where |$\textit {Prf}(x,y)$| denotes a |$\varDelta _{1}(\textsf {I}\varSigma _1)$| -definition of ' |$y$| is a |$T$| -proof of |$x$| '. Finitist consistency is addressed by generalizing a result of Artemov: Partial finitism. If |$T$| is consistent, there is a primitive recursive function |$f$| such that, for all |$n\in \mathbb {N}$|⁠ , |$f(n)$| is the code of an |$\textsf {I}\varSigma _{1}$| -proof of |$\neg\, \textit{Prf}(\ulcorner \perp \urcorner ,\overline {n})$|⁠. These results are not in conflict with Gödel's Incompleteness Theorems. Rather, they allow to extend their usual interpretation and show a deep connection to reflections in Hilbert's last papers of 1931. [ABSTRACT FROM AUTHOR]

We unify Gödel 's first incompleteness theorem (1931), Tarski 's undefinability theorem (1933), Gödel-Carnap 's diagonal lemma (1934) and Rosser 's (strengthening of Gödel 's first) incompleteness theorem (1936), whose proofs resemble much and use almost the same technique. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Cyberspace Studies (JCSS) is the property of University of Tehran and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We redefine a system of varieties definable by a schema of equations to include finite dimensions. Then we present a technique using ultraproducts enabling one to lift results proved for every finite dimension to the transfinite. Let Ord denote the class of all ordinals. Let ⟨Kα : α ∈ Ord⟩ be a system of varieties definable by a schema. Given any ordinal α, we define an operator Nrα that acts on Kβ for any β > α giving an algebra in Kα, as an abstraction of taking α-neat reducts for cylindric algebras. We show that for any positive k, and any infinite ordinal α that SNrαKα+k+1 cannot be axiomatized by a finite schema over SNrαKα+k given that the result is valid for all finite dimensions greater than some fixed finite ordinal. We apply our results to cylindric algebras and Halmos quasipolyadic algebras with equality. As an application to our algebraic result we obtain a strong incompleteness theorem (in the sense that validitities are not captured by finitary Hilbert style axiomatizations) for an algebraizable extension of Lω,ω. [ABSTRACT FROM AUTHOR]

The literature dealing with Gödel's legacy is largely preoccupied with challenging his philosophical views, regarding them as outdated. We believe that such an approach prevents us from seeing Gödel's views in the right light and understanding their rationale. In this article, his views are discussed in the philosophical realm in which he himself understood them. We explore the consequences of Gödel's incompleteness theorems for the question of the objectivity of mathematics and its epistemology. Taking set theory as the paradigm of formal mathematical theories, we examine the relationship between its incompleteness and extensionality. We argue, based on his philosophical views, that Gödel believed incompleteness can be overcome only by some intensional considerations about concepts from the basis of mathematical theories. These considerations should eventually lead to founding the so-called logic of concepts. [ABSTRACT FROM AUTHOR]

This study furthers the utilisation of the parametric group method of data handling (GMDH) in assessing the possibility of rainfall modelling and prediction, using publicly available temperature and rainfall data. In using ordinary GMDH approaches, the modelling is inconclusive with no clear consistency demonstrated through coefficients of determination and analysis of variance. Hence, an empirical assessment has been undertaken to provide an explanation of the inconsistency. In doing so, state variable distribution, their classification within the fuzzy context, and the need to integrate the principle of incompatibility into the GMDH modelling format are all assessed. The mathematical foundations of GMDH are discussed within the heuristic framework of data partitioning, partial description synthesis, the limitations of the least-squares coefficient of determination, incompleteness theorem, and the necessity for an external criterion in the selection procedure for polynomials. Methods for modelling improvement include the potential for hybridisation with least square support vector machines (LSSVM), the application of filters for parameter estimation, and the combination with signal processing techniques, ensemble empirical mode decomposition (EEMD), wavelet transformation (WT), and wavelet packet transformation (WPT). These have been investigated in addition to the implementation of enhanced GMDH (eGMDH) and fuzzy GMDH (FGMDH). The inclusion of exogenous data and its application within the GMDH modelling paradigm are also discussed. The study concludes with recommendations to enhance the potential for future rainfall modelling study success using parametric GMDH. [ABSTRACT FROM AUTHOR]

We point out that some questions in quantum field theory are undecidable in a precise mathematical sense. More concretely, it will be demonstrated that there is no algorithm answering whether a given 2d supersymmetric Lagrangian theory breaks supersymmetry or not. It will also be shown that there is a specific 2d supersymmetric Lagrangian theory which breaks supersymmetry if and only if the standard Zermelo-Fraenkel set theory with the axiom of choice is consistent, which can never be proved or disproved as the consequence of Gödel’s second incompleteness theorem. The article includes a brief and informal introduction to the phenomenon of undecidability and its previous appearances in theoretical physics. A 15-minute video presentation of the content is also available at . [ABSTRACT FROM AUTHOR]

The authors analyze Diophantine sets and show that all recursively enumerable sets are Diophantine. Based on the classical results from the theory of recursive functions, a simple version of the theorem on the incompleteness of arithmetic is provided: there is a polynomial that has no positive integer solutions, and for which it is impossible to prove the absence of positive roots. [ABSTRACT FROM AUTHOR]

In contrast to dominant approaches to human reason involving essentially a logical and instrumental conception of rationality easily modeled by artificial intelligence mechanisms, I argue that the specific capacities of the human mind are meta-analytic in nature, understood as irreducible to the analytic or the logical, or else the computational. Firstly, the assumption of a meta-analytical level of rationality is derived from key insights developed in various branches of the social sciences. This meta-analytical level is then inferred from Gödel's incompleteness theorem. On these bases, and with the help of psychological approaches to consciousness, I argue that human rationality may be characterized as the capacity to make meaningful use of signs. [ABSTRACT FROM AUTHOR]

In his paper on the incompleteness theorems, Gödel seemed to say that a direct way of constructing a formula that says of itself that it is unprovable might involve a faulty circularity. In this note, it is proved that 'direct' self-reference can actually be used to prove his result. [ABSTRACT FROM AUTHOR]

Shortly after Kurt Gödel had announced an early version of the 1st incompleteness theorem, John von Neumann wrote a letter to inform him of a remarkable discovery, i.e. that the consistency of a formal system containing arithmetic is unprovable, now known as the 2nd incompleteness theorem. Although today von Neumann's proof of the theorem is considered lost, recent literature has explored many of the issues surrounding his discovery. Yet, one question still awaits a satisfactory answer: how did von Neumann achieve his result, knowing as little as he seemingly did about the 1st incompleteness theorem? In this article, I shall advance a conjectural argument to answer this question, after having rejected the argument widely shared in the literature and having analyzed the relevant documents surrounding his discovery. The argument I shall advance strictly links two of the three letters written by von Neumann to Gödel in the late 1930 and early 1931 (i.e. respectively that of November 20, 1930 and that of January 12, 1931) and finds the key for von Neumann's discovery in his prompt understanding of the Gödel sentence A – as the documents refer to it – as expressing consistency for a formal system that contains arithmetic. [ABSTRACT FROM AUTHOR]

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]

Natalie Alana Ashton, in "Extended Rationality and Epistemic Relativism", takes up the issue of the relation of hinge epistemology to epistemic relativism. In response, Coliva argues that a transcendental hinge epistemology would face the difficult question of explaining why the self-stultifying character of doubts about inescapable hinges should give us any reason to think that those hinges are I true i . Ashton criticizes Coliva's arguments against epistemic relativism, and furthermore suggests that Coliva's own Extended Rationality can be adapted into a plausible form of epistemic relativism. Coliva takes issue with Ashton's construal of what a plausible form of relativism would look like. [Extracted from the article]

Copyright of Kultura i Wychowanie is the property of Akademii Humanistyczno-Ekonomicznej and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

1936 was a watershed year for computability. Debates among Gödel, Church and others over the correct analysis of the intuitive concept "human effectively computable", an analysis at the heart of the Incompleteness Theorems, the Entscheidungsproblem, the question of what a finite computation is, and most urgently—for Gödel—the generality of the Incompleteness Theorems, were definitively set to rest with the appearance, in that year, of the Turing Machine. The question I explore here is, do the mathematical facts exhaust what is to be said about the thinking behind the "confluence of ideas in 1936"? I will argue for a cultural role in Gödel's, and, by extension, the larger logical community's absorption of Turing's 1936 model. As scaffolding I employ a conceptual framework due to the critic Leo Marx of the technological sublime; I also make use of the distinction within the technological sublime due to Caroline Jones, between its iconic and performative modes—a distinction operating within the conceptual art of the 1960s, but serving the history of computability equally well. [ABSTRACT FROM AUTHOR]

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

In this paper, we prove that no consistent finitely axiomatized theory one‐dimensionally interprets its own extension with predicative comprehension. This constitutes a result with the flavor of the Second Incompleteness Theorem whose formulation is completely arithmetic‐free. Probably the most important novel feature that distinguishes our result from the previous results of this kind is that it is applicable to arbitrary weak theories, rather than to extensions of some base theory. The methods used in the proof of the main result yield a new perspective on the notion of sequential theory, in the setting of forcing‐interpretations. [ABSTRACT FROM AUTHOR]

For several decades, Carnap's philosophy of mathematics used to be either dismissed or ignored. It was perceived as a form of linguistic conventionalism and thus taken to rely on the bankrupt notion of truth by convention. However, recent scholarship has revealed a more subtle picture. It has been forcefully argued that Carnap is not a linguistic conventionalist in any straightforward sense, and that supposedly decisive objections against his position target a straw man. This raises two questions. First, how exactly should we characterise Carnap's actual philosophy of mathematics? Secondly, is his position an attractive alternative to established views? I will tackle these issues by looking at Carnap's response to the incompleteness theorems. Drawing on arguments put forward by Gödel and Beth, I show that some crucial aspects of Carnap's positive account have remained underdeveloped. Suggestions on what a full evaluation of Carnap's position requires are made. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Gödel argued that the incompleteness theorems entail that the mind is not a machine, or that certain arithmetical propositions are absolutely undecidable. His view was that the mind is not a machine, and that no arithmetical propositions are absolutely undecidable. I argue that his position presupposes that the idealized mathematician has an ability which I call the recursive-ordinal recognition ability. I show that we have this ability if, and only if, there are no absolutely undecidable arithmetical propositions. I argue that there are such propositions, but that no recognizable example of one can be identified, even in principle. [ABSTRACT FROM AUTHOR]

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]

The article focuses on analysis of case law using the example of headscarf bans and discrimination based on the Basic Law. It mentions incompleteness to work out the dogmatics in the area of ​​multidimensional discrimination and categories are socially constructed categorizations solidified by language, which lead to the establishment of supposedly homogeneous groups. It also mentions European and international law multi-dimensional constellations not always seen.

Kurt Gödel behind the rational mathematics: Stephen Budiansky: Journey to the edge of reason: the life of Kurt Gödel. Therefore, Budiansky can only claim to take the story of Gödel's life to another level by using previously inaccessible, or at least not transcribed, shorthand material by Gödel. For the generations that possibly enjoy the t-shirt version of Gödel's work, I would hope that the variety of his work would spark more diversified Gödel merchandise. [Extracted from the article]

We use Gödel's incompleteness theorems as a case study for investigating mathematical depth. We examine the philosophical question of what the depth of Gödel's incompleteness theorems consists in. We focus on the methodological study of the depth of Gödel's incompleteness theorems, and propose three criteria to account for the depth of the incompleteness theorems: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of Gödel's incompleteness theorems. [ABSTRACT FROM AUTHOR]

We prove the equivalence of the semantic version of Tarski's theorem on the undefinability of truth with the semantic version of the diagonal lemma and also show the equivalence of a syntactic version of Tarski's undefinability theorem with a weak syntactic diagonal lemma. We outline two seemingly diagonal-free proofs for these theorems from the literature and show that the syntactic version of Tarski's theorem can deliver Gödel–Rosser's incompleteness theorem. [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 the Handbook of Mathematical Logic, the Paris-Harrington variant of Ramsey's theorem is celebrated as the first result of a long 'search' for a purely mathematical incompleteness result in first-order Peano arithmetic. This paper questions the existence of any such search and the status of the Paris-Harrington result as the first mathematical incompleteness result. In fact, I argue that Gentzen gave the first such result, and that it was restated by Goodstein in a number-theoretic form. [ABSTRACT FROM AUTHOR]

The basic axioms or formal conditions of decision theory, especially the ordering condition put on preferences and the axioms underlying the expected utility (EU) formula, are subject to a number of counter-examples, some of which can be endowed with normative value and thus fall within the ambit of a philosophical reflection on practical rationality. Against such counter-examples, a defensive strategy has been developed which consists in redescribing the outcomes of the available options in such a way that the threatened axioms or conditions continue to hold. We examine how this strategy performs in three major cases: Sen's counterexamples to the binariness property of preferences, the Allais paradox of EU theory under risk, and the Ellsberg paradox of EU theory under uncertainty. We find that the strategy typically proves to be lacking in several major respects, suffering from logical triviality, incompleteness, and theoretical insularity (i.e., being cut off from the methods and results of decision theory). To give the strategy more structure, philosophers have developed "principles of individuation"; but we observe that these do not address the aforementioned defects. Instead, we propose the method of checking whether the strategy can overcome its typical defects once it is given a proper theoretical expansion (i.e., it is duly developed using the available tools of decision theory). We find that the strategy passes the test imperfectly in Sen's case and not at all in Allais's. In Ellsberg's case, however, it comes close to meeting our requirement. But even the analysis of this more promising application suggests that the strategy ought to address the decision problem as a whole, rather than just the outcomes, and that it should extend its revision process to the very statements it is meant to protect. Thus, by and large, the same cautionary tale against redescription practices runs through the analysis of all three cases. A more general lesson, simply put, is that there is no easy way out from the paradoxes of decision theory. [ABSTRACT FROM AUTHOR]

Copyright of Paradigma is the property of Universidad Pedagogica Experimental Libertador and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

Copyright of Procesos de Mercado is the property of Procesos de Mercado. Revista Europea de Economia Politica and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Interesting as they are by themselves in philosophy and mathematics, paradoxes can be made even more fascinating when turned into proofs and theorems. For example, Russell's paradox, which overthrew Frege's logical edifice, is now a classical theorem in set theory, to the effect that no set contains all sets. Paradoxes can be used in proofs of some other theorems—thus Liar's paradox has been used in the classical proof of Tarski's theorem on the undefinability of truth in sufficiently rich languages. This paradox (as well as Richard's paradox) appears implicitly in Gödel's proof of his celebrated first incompleteness theorem. In this paper, we study Yablo's paradox from the viewpoint of first- and second-order logics. We prove that a formalization of Yablo's paradox (which is second order in nature) is non-first-orderizable in the sense of George Boolos (1984). [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.

An analysis of the undecidability of Diophantine equations showed that problems of recognition of the properties of the NP class are decidable, i.e., a non-deterministic algorithm or exhaustive search at the problem input gives a positive or negative answer. For polynomial Diophantine equations, such a non-deterministic algorithm does not exist. A simple version of Gödel's theorem on the incompleteness of arithmetic follows from the undecidability of Diophantine equations. [ABSTRACT FROM AUTHOR]

In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately monotonic curvature singularities can dynamically form along a spacelike hypersurface. In this article, we study the Cauchy problem and give a rigorous proof of this phenomenon in sufficiently high dimensions, thereby providing the first constructive proof of stable curvature-blowup (without symmetry assumptions) along a spacelike hypersurface as an effect of pure gravity. Our proof applies to an open subset of regular initial data satisfying the assumptions of Hawking's celebrated "singularity" theorem, which shows that the solution is geodesically incomplete but does not reveal the nature of the incompleteness. Specifically, our main result is a proof of the dynamic stability of the Kasner curvature singularity for a subset of Kasner solutions whose metrics exhibit only moderately (as opposed to severely) spatially anisotropic behavior. Of independent interest is our method of proof, which is more robust than earlier approaches in that (i) it does not rely on approximate monotonicity identities and (ii) it accommodates the possibility that the solution develops very singular high-order spatial derivatives, whose blowup-rates are allowed to be, within the scope of our bootstrap argument, much worse than those of the base-level quantities driving the fundamental blowup. For these reasons, our approach could be used to obtain similar blowup-results for various Einstein-matter systems in any number of spatial dimensions for solutions corresponding to an open set of moderately spatially anisotropic initial data, thus going beyond the nearly spatially isotropic regime treated in earlier works. [ABSTRACT FROM AUTHOR]

Sign-based semantics and embodied semantics are argued to be mutually beneficial to one another. However, while the body does shape our cognitive activities to a great extent, this does not entail that cognition can be reduced to sensorimotor simulation, i.e that the mind can be reduced to the body. Language itself bears testimony to this, as the mind is construed in ordinary discourse as having the incredible capacity of being free to travel beyond the limits of present time and current spatial location. Nagel has argued famously that mind is a fundamental datum of nature that the materialist version of evolutionary biology is unable to account for, as consciousness has an essentially subjective character to it, a ‘what it is like for the conscious organism itself’ aspect, that cannot be reduced to the matter of which the organism is constituted. Two modern scientific developments refute the contention that the human mind can be explained as a purely material machine: quantum theory in physics and Gödel’s Incompleteness Theorem in mathematics. Just because the mind works through the body does not entail that the mind can be reduced to the body. [ABSTRACT FROM AUTHOR]