Relevant bibliographies by topics / Mathematical incompleteness

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Mathematical incompleteness'

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Published: 4 June 2025
Last updated: 23 June 2025

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Journal articles on the topic "Mathematical incompleteness"

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Warnez, Br Matthew T. "Mathematical Incompleteness and Divine Ineffability." Logos: A Journal of Catholic Thought and Culture 23, no. 3 (2020): 49–74. http://dx.doi.org/10.1353/log.2020.0026.

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TIESZEN, RICHARD. "Mathematical Realism and Gödel's Incompleteness Theorems." Philosophia Mathematica 2, no. 3 (1994): 177–201. http://dx.doi.org/10.1093/philmat/2.3.177.

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Longo, Giuseppe. "Reflections on Concrete Incompleteness." Philosophia Mathematica 19, no. 3 (2011): 255–80. https://doi.org/10.1093/philmat/nkr016.

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How do we prove true, but unprovable propositions?  Godel produced a statement whose undecidability derives from its "ad hoc" construction.  Concrete or mathematical incompleteness results, instead, are interesting unprovable statements of Formal Arithmetic.  We point out where exactly lays the unprovability along the ordinary mathematical proofs of two (very) interesting formally unprovable propositions, Kruskal-Friedman theorem on trees and Girard's Normalization Theorem in Type Theory.  Their validity is based on robust cognitive performances, which ground mathematics on our relation to space and time, such as symmetries and order, or on the generality of Herbrand's notion of prototype proof.
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Feferman, Solomon. "Reflecting on incompleteness." Journal of Symbolic Logic 56, no. 1 (1991): 1–49. http://dx.doi.org/10.2307/2274902.

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To what extent can mathematical thought be analyzed in formal terms? Gödel's theorems show the inadequacy of single formal systems for this purpose, except in relatively restricted parts of mathematics. However at the same time they point to the possibility of systematically generating larger and larger systems whose acceptability is implicit in acceptance of the starting theory. The engines for that purpose are what have come to be called reflection principles. These may be iterated into the constructive transfinite, leading to what are called recursive progressions of theories. A number of informative technical results have been obtained about such progressions (cf. Feferman [1962], [1964], [1968] and Kreisel [1958], [1970]). However, for some years I had hoped to give a more realistic and perspicuous finite generation procedure. This was first done in a rather special way in Feferman [1979] for the characterization of predicativity, which may be regarded as that part of mathematical thought implicit in our acceptance of elementary number theory. What is presented here is a new and simple notion of the reflective closure of a schematic theory which can be applied quite generally.Two examples of schematic theories in the sense used here are versions of Peano arithmetic and Zermelo set theory.
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Kostic, Jovana, and Slobodan Vujosevic. "Kurt Gödel and the logic of concepts." Publications de l'Institut Math?matique (Belgrade) 115, no. 129 (2024): 1–19. http://dx.doi.org/10.2298/pim2429001k.

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The literature dealing with G?del?s legacy is largely preoccupied with challenging his philosophical views, regarding them as outdated. We believe that such an approach prevents us from seeing G?del?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 G?del?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 G?del 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.
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Paseau, A. C. "Letter Games: a metamathematical taster." Mathematical Gazette 100, no. 549 (2016): 442–49. http://dx.doi.org/10.1017/mag.2016.109.

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Metamathematics is the mathematical study of mathematics itself. Two of its most famous theorems were proved by Kurt Gödel in 1931. In a simplified form, Gödel's first incompleteness theorem states that no reasonable mathematical system can prove all the truths of mathematics. Gödel's second incompleteness theorem (also simplified) in turn states that no reasonable mathematical system can prove its own consistency. Another famous undecidability theorem is that the Continuum Hypothesis is neither provable nor refutable in standard set theory. Many of us logicians were first attracted to the field as students because we had heard something of these results. All research mathematicians know something of them too, and have at least a rough sense of why ‘we can't prove everything we want to prove’.
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Shapiro, Stewart. "Incompleteness, Mechanism, and Optimism." Bulletin of Symbolic Logic 4, no. 3 (1998): 273–302. http://dx.doi.org/10.2307/421032.

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§1. Overview. Philosophers and mathematicians have drawn lots of conclusions from Gödel's incompleteness theorems, and related results from mathematical logic. Languages, minds, and machines figure prominently in the discussion. Gödel's theorems surely tell us something about these important matters. But what?A descriptive title for this paper would be “Gödel, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility”. Adding “God and the Devil” would probably be redundant. Despite the breath-taking, whirlwind tour, I have the modest aim of forging connections between different parts of this literature and clearing up some confusions, together with the less modest aim of not introducing any more confusions.I propose to focus on three spheres within the literature on incompleteness. The first, and primary, one concerns arguments that Gödel's theorem refutes the mechanistic thesis that the human mind is, or can be accurately modeled as, a digital computer or a Turing machine. The most famous instance is the much reprinted J. R. Lucas [18]. To summarize, suppose that a mechanist provides plans for a machine, M, and claims that the output of M consists of all and only the arithmetic truths that a human (like Lucas), or the totality of human mathematicians, will ever or can ever know. We assume that the output of M is consistent.
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Dimou, Argyris, Panos Argyrakis, and Raoul Kopelman. "Tumor Biochemical Heterogeneity and Cancer Radiochemotherapy: Network Breakdown Zone-Model." Entropy 24, no. 8 (2022): 1069. http://dx.doi.org/10.3390/e24081069.

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Breakdowns of two-zone random networks of the Erdős–Rényi type are investigated. They are used as mathematical models for understanding the incompleteness of the tumor network breakdown under radiochemotherapy, an incompleteness that may result from a tumor’s physical and/or chemical heterogeneity. Mathematically, having a reduced node removal probability in the network’s inner zone hampers the network’s breakdown. The latter is described quantitatively as a function of reduction in the inner zone’s removal probability, where the network breakdown is described in terms of the largest remaining clusters and their size distributions. The effects on the efficacy of radiochemotherapy due to the tumor micro-environment (TME)’s chemical make-up, and its heterogeneity, are discussed, with the goal of using such TME chemical heterogeneity imaging to inform precision oncology.
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Bydzan, Vitalii. "POSSIBLE SOLUTION TO GÖDEL'S INCOMPLETENESS THEOREM AND GÖDEL'S SECOND THEOREM." Grail of Science, no. 30 (August 16, 2023): 197–209. http://dx.doi.org/10.36074/grail-of-science.04.08.2023.032.

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Plan
 
 Statement of the problem
 Find information to solve the problem
 Clarification of information to solve the problem
 Formulation of the lemma to solve the problem
 Search for a principle to solve the problem
 Proving the lemma that every mathematical system needs an observer
 Proving the lemma that every mathematical system needs an observer, whose existence only he can know, because whose existence cannot be proved
 
 8.Mathematical record of problem solving
 
 Confirmation of the consistency and completeness of the formal system for one observer.
 Necessary and sufficient conditions for the formation of a consistent system for society (group of observers)
 Solving the liar paradox as a byproduct of solving the problem
 Using observer’s view on The Ship of Theseus
 The unexpected hanging paradox
 The sorites paradox
 The philosophical basis of the theorem proof
 Some reasonable conclusions from this work that can be applied in other scientific
 Conclusions of solving the problem
 My sincere thanks for the provided knowledge / information sources
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Longo, Giuseppe, and Arnaud Viarouge. "Mathematical Intuition and the Cognitive Roots of Mathematical Concepts." Topoi 29, no. 1 (2010): 15–27. https://doi.org/10.1007/s11245-009-9063-6.

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The foundation of Mathematics is both a logico-formal issue and an epistemological one.  By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata.  By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper.  This "genealogy of concepts", so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis by this too often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction.  For the purposes of our investigation, we will hint here to the philosophical frame as well as to the some recent advances in Cognition that support our claim, the cognitive origin and the constitutive role of mathematical intuition.
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Dissertations / Theses on the topic "Mathematical incompleteness"

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Lindman, Phillip A. (Phillip Anthony). "Intuition versus Formalization: Some Implications of Incompleteness on Mathematical Thought." Thesis, University of North Texas, 1994. https://digital.library.unt.edu/ark:/67531/metadc277970/.

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This paper describes the tension between intuition about number theory and attempts to formalize it. I will first examine the root of the dilemma, Godel's First Incompleteness Theorem, which demonstrates that in any reasonable formalization of number theory, there will be independent statements. After proving the theorem, I consider some of its consequences on intuition, focusing on Freiling's "Dart Experiment" which is based on our usual notion of the real numbers as a line. This experiment gives an apparent refutation of the Axiom of Choice and the Continuum Hypothesis; however, it also leads to an equally apparent paradox. I conclude that such paradoxes are inevitable as the formalization of mathematics takes us further from our initial intuitions.
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Jitsuchon, Somchai. "Three applications of market incompleteness and market imperfection." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape7/PQDD_0026/NQ38906.pdf.

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Crawley, Karen. "Limited ink : interpreting and misinterpreting GÜdel's incompleteness theorem in legal theory." Thesis, McGill University, 2006. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=101814.

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This thesis explores the significance of Godel's Theorem for an understanding of law as rules, and of legal adjudication as rule-following. It argues that Godel's Theorem, read through Wittgenstein's understanding of rules and language as a contextual activity, and through Derrida's account of 'undecidability,' offers an alternative account of the relationship of judging to justice. Instead of providing support for the 'indeterminacy' claim, Godel's Theorem illuminates the predicament of undecidability that structures any interpretation and every legal decision, and which constitutes the opening to justice. The first argument in this thesis examines Godel's proof, Wittgenstein's views on rules, and Derrida's undecidability, as manifestations of a common concern with the limits of what can be formalized. The meta-argument examines their misinterpretation and misappropriation within legal theory as a case study of just what they mean about meaning, context, and justice as necessarily co-implicated.
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Souba, Matthew. "From the Outside Looking In: Can mathematical certainty be secured without being mathematically certain that it has been?" The Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1574777956439624.

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Tanswell, Fenner Stanley. "Proof, rigour and informality : a virtue account of mathematical knowledge." Thesis, University of St Andrews, 2017. http://hdl.handle.net/10023/10249.

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This thesis is about the nature of proofs in mathematics as it is practiced, contrasting the informal proofs found in practice with formal proofs in formal systems. In the first chapter I present a new argument against the Formalist-Reductionist view that informal proofs are justified as rigorous and correct by corresponding to formal counterparts. The second chapter builds on this to reject arguments from Gödel's paradox and incompleteness theorems to the claim that mathematics is inherently inconsistent, basing my objections on the complexities of the process of formalisation. Chapter 3 looks into the relationship between proofs and the development of the mathematical concepts that feature in them. I deploy Waismann's notion of open texture in the case of mathematical concepts, and discuss both Lakatos and Kneebone's dialectical philosophies of mathematics. I then argue that we can apply work from conceptual engineering to the relationship between formal and informal mathematics. The fourth chapter argues for the importance of mathematical knowledge-how and emphasises the primary role of the activity of proving in securing mathematical knowledge. In the final chapter I develop an account of mathematical knowledge based on virtue epistemology, which I argue provides a better view of proofs and mathematical rigour.
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Batistela, Rosemeire de Fátima [UNESP]. "O Teorema da Incompletude de Gödel em cursos de Licenciatura em Matemática." Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/148797.

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Submitted by ROSEMEIRE DE FATIMA BATISTELA null (rosebatistela@hotmail.com) on 2017-02-11T02:22:43Z No. of bitstreams: 1 tese finalizada 10 fevereiro 2017 com a capa.pdf: 2263896 bytes, checksum: 413948c6a47fb47a21e1587275d29c03 (MD5)<br>Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2017-02-15T16:56:58Z (GMT) No. of bitstreams: 1 batistela_rf_dr_rcla.pdf: 2263896 bytes, checksum: 413948c6a47fb47a21e1587275d29c03 (MD5)<br>Made available in DSpace on 2017-02-15T16:56:58Z (GMT). No. of bitstreams: 1 batistela_rf_dr_rcla.pdf: 2263896 bytes, checksum: 413948c6a47fb47a21e1587275d29c03 (MD5) Previous issue date: 2017-02-02<br>Apresentamos nesta tese uma proposta de inserção do tema teorema da incompletude de Gödel em cursos de Licenciatura em Matemática. A interrogação norteadora foi: como sentidos e significados do teorema da incompletude de Gödel podem ser atualizados em cursos de Licenciatura em Matemática? Na busca de elaborarmos uma resposta para essa questão, apresentamos o cenário matemático presente à época do surgimento deste teorema, expondo-o como a resposta negativa para o projeto do Formalismo que objetivava formalizar toda a Matemática a partir da aritmética de Peano. Além disso, trazemos no contexto, as outras duas correntes filosóficas, Logicismo e Intuicionismo, e os motivos que impossibilitaram o completamento de seus projetos, que semelhantemente ao Formalismo buscaram fundamentar a Matemática sob outras bases, a saber, a Lógica e os constructos finitistas, respectivamente. Assim, explicitamos que teorema da incompletude de Gödel aparece oferecendo resposta negativa à questão da consistência da aritmética, que era um problema para a Matemática na época, estabelecendo uma barreira intransponível para a demonstração dessa consistência, da qual dependia o sucesso do Formalismo e, consequentemente, a fundamentação completa da Matemática no ideal dos formalistas. Num segundo momento, focamos na demonstração deste teorema expondo-a em duas versões distintas, que para nós se nos mostraram apropriadas para serem trabalhadas em cursos de Licenciatura em Matemática. Uma, como possibilidade de conduzir o leitor pelos meandros da prova desenvolvida por Gödel em 1931, ilustrando-a, bem como, as ideias utilizadas nela, aclarando a sua compreensão. Outra, como opção que valida o teorema da incompletude apresentando-o de maneira formal, portanto, com endereçamentos e objetivos distintos, por um lado, a experiência com a numeração de Gödel e a construção da sentença indecidível, por outro, com a construção formal do conceito de método de decisão de uma teoria. Na sequência, apresentamos uma discussão focada na proposta de Bourbaki para a Matemática, por compreendermos que a atitude desse grupo revela a forma como o teorema da incompletude de Gödel foi acolhido nessa ciência e como ela continuou após este resultado. Nessa exposição aparece que o grupo Bourbaki assume que o teorema da incompletude não impossibilita que a Matemática prossiga em sua atividade, ele apenas sinaliza que o aparecimento de proposições indecidíveis, até mesmo na teoria dos números naturais, é inevitável. Finalmente, trazemos a proposta de como atualizar sentidos e significados do teorema da incompletude de Gödel em cursos de Licenciatura em Matemática, aproximando o tema de conteúdos agendados nas ementas, propondo discussão de aspectos desse teorema em diversos momentos, em disciplinas que julgamos apropriadas, culminando no trabalho com as duas demonstrações em disciplinas do último semestre do curso. A apresentação é feita tomando como exemplar um curso de Licenciatura em Matemática. Consideramos por fim, a importância do trabalho com um resultado tão significativo da Lógica Matemática que requer atenção da comunidade da Educação Matemática, dado que as consequências deste teorema se relacionam com a concepção de Matemática ensinada em todos os níveis escolares, que, muito embora não tenham relação com conteúdos específicos, expõem o alcance do método de produção da Matemática.<br>In this thesis we present a proposal to insert Gödel's incompleteness theorem in Mathematics Education undergraduate courses. The main research question guiding this investigation is: How can the senses and meanings of Gödel's incompleteness theorem be updated in Mathematics Education undergraduate courses? In answering the research question, we start by presenting the mathematical scenario from the time when the theorem emerged; this scenario proposed a negative response to the project of Formalism, which aimed to formalize all Mathematics based upon Peano’s arithmetic. We also describe Logicism and Intuitionism, focusing on reasons that prevented the completion of these two projects which, in similarly to Formalism, were sought to support mathematics under other bases of Logic and finitists constructs. Gödel's incompleteness theorem, which offers a negative answer to the issue of arithmetic consistency, was a problem for Mathematics at that time, as the Mathematical field was passing though the challenge of demonstrating its consistency by depending upon the success of Formalism and upon the Mathematics’ rationale grounded in formalists’ ideal. We present the proof of Gödel's theorem by focusing on its two different versions, both being accessible and appropriate to be explored in Mathematics Education undergraduate courses. In the first one, the reader will have a chance to follow the details of the proof as developed by Gödel in 1931. The intention here is to expose Gödel’ ideas used at the time, as well as to clarify understanding of the proof. In the second one, the reader will be familiarized with another proof that validates the incompleteness theorem, presenting it in its formal version. The intention here is to highlight Gödel’s numbering experience and the construction of undecidable sentence, and to present the formal construction of the decision method concept from a theory. We also present a brief discussion of Bourbaki’s proposal for Mathematics, highlighting Bourbaki’s group perspective which reveals how Gödel’s incompleteness theorem was important and welcome in science, and how the field has developed since its result. It seems to us that Bourbaki’s group assumes that the incompleteness theorem does not preclude Mathematics from continuing its activity. Thus, from Bourbaki’s perspective, Gödel’s incompleteness theorem only indicates the arising of undecidable propositions, which are inevitable, occurring even in the theory of natural numbers. We suggest updating the senses and the meanings of Gödel's incompleteness theorem in Mathematics Education undergraduate courses by aligning Gödel's theorem with secondary mathematics school curriculum. We also suggest including discussion of this theorem in different moments of the secondary mathematics school curriculum, in which students will have elements to build understanding of the two proofs as a final comprehensive project. This study contributes to the literature by setting light on the importance of working with results of Mathematical Logic such as Gödel's incompleteness theorem in secondary mathematics courses and teaching preparation. It calls the attention of the Mathematical Education community, since its consequences are directly related to the design of mathematics and how it is being taught at all grade levels. Although some of these mathematics contents may not be related specifically to the theorem, the understanding of the theorem shows the broad relevance of the method in making sense of Mathematics.
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Ng, Yui-kin, and 吳銳堅. "Computers, Gödel's incompleteness theorems and mathematics education: a study of the implications of artificialintelligence for secondary school mathematics." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1994. http://hub.hku.hk/bib/B31957419.

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Walinsky, Clifford. "Constructive negation in logic programs /." Full text open access at:, 1987. http://content.ohsu.edu/u?/etd,149.

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St, John Gavin. "On formally undecidable propositions of Zermelo-Fraenkel set theory." Youngstown State University / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1369657108.

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"On incompleteness in modal logic. An account through second-order logic." Tulane University, 1998.

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The dissertation gives a second-order-logic-based explanation of modal incompleteness. The leading concept is that modal incompleteness is to be explained in terms of the incompleteness of standard second-order logic, since modal language is basically a second-order language. The development of Kripke-style semantics for modal logic has been underpinned by the conjecture that (maybe) all modal systems are characterizable by classes of frames defined by first-order conditions on a binary (accessibility) relation. However, the discovery of certain incomplete (uncharacterizable) modal systems has undermined the all-encompassing feature of Kripke-style approach. There are logics not determined by any class of Kripke frames at all In the dissertation I investigate a normal incomplete sentential modal system due to J. F. A. K. Van Benthem, and I address both the formal and the philosophical facets of modal incompleteness from the vantage point that modal logic is essentially second-order in its nature. Modal systems can be analyzed in terms of structures with a domain of second-order individuals (subsets) that are assigned under an interpretation to propositional variables within languages of sentential modal logic. Hence, the phenomenon of there being incomplete modal systems can be accounted for through the incompleteness of standard second-order logic Since second-order logic plays a crucial role in the explanation, Quine's animadversions upon second-order logic and in particular his views that second-order logic is 'Set Theory in Sheep's Clothing' are examined. Against Quine's stance I will seek to show that a vindication of second-order logic can be gotten provided a proper due is given to the sharp distinction between the logical (Fregean) notion of set which is the concern of second-order logic and the iterative notion of set which lies within the realm of set theory<br>acase@tulane.edu
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Books on the topic "Mathematical incompleteness"

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Smullyan, Raymond M. Gödel's incompleteness theorems. Oxford University Press, 1992.

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Smullyan, Raymond M. Gödel's incompleteness theorems. Oxford University Press, 1992.

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Goldstern, Martin. The incompleteness phenomenon: A new course in mathematical logic. A K Peters, 1995.

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Carbonell-Nicolau, Oriol. Testing out contractual incompleteness: Evidence from soccer. National Bureau of Economic Research, 2005.

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Carbonell-Nicolau, Oriol. Testing out contractual incompleteness: Evidence from soccer. National Bureau of Economic Research, 2005.

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Franzén, Torkel. Inexhaustability: A non-exhaustive treatment. AK Peters, 2004.

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Chaitim, Gregory J. Thinking about Gödel and Turing: Essays on complexity 1970-2007. World Scientific, 2007.

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Cowan, Daniel A. Seeing negation as always dependent frees mathematical logic from paradox, incompleteness, and undecidability-- and opens the door to its positive possibilities. Joseph Publishing Company, 2008.

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Ilʹin, V. P. Iterative incomplete factorization methods. World Scientific, 1992.

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Berto, Francesco. There's something about Gödel: The complete guide to the incompleteness theorem. Wiley-Blackwell, 2009.

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Book chapters on the topic "Mathematical incompleteness"

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Pollard, Stephen. "Peano Arithmetic, Incompleteness." In A Mathematical Prelude to the Philosophy of Mathematics. Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-05816-0_2.

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Dawson, John W. "Facets of Incompleteness." In Mathematical Logic and Its Applications. Springer US, 1987. http://dx.doi.org/10.1007/978-1-4613-0897-3_2.

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Rautenberg, Wolfgang. "Incompleteness and Undecidability." In A Concise Introduction to Mathematical Logic. Springer New York, 2009. http://dx.doi.org/10.1007/978-1-4419-1221-3_6.

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Manin, Yu I. "Gödel’s Incompleteness Theorem." In A Course in Mathematical Logic for Mathematicians. Springer New York, 2009. http://dx.doi.org/10.1007/978-1-4419-0615-1_7.

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Srivastava, Shashi Mohan. "Representability and Incompleteness Theorems." In A Course on Mathematical Logic. Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-5746-6_7.

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Friedman, Harvey M. "Concrete Mathematical Incompleteness: Basic Emulation Theory." In Outstanding Contributions to Logic. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-96274-0_12.

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Goranko, Valentin. "Completeness and Incompleteness in the Bimodal Base ℒ(R,−R)." In Mathematical Logic. Springer US, 1990. http://dx.doi.org/10.1007/978-1-4613-0609-2_22.

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Dawson, John W. "The Reception of Gödel’s Incompleteness Theorems." In Perspectives on the History of Mathematical Logic. Birkhäuser Boston, 1985. http://dx.doi.org/10.1007/978-0-8176-4769-8_7.

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Markose, Sheri. "The Gödelian Foundations of Self-Reference,the Liar and Incompleteness: Arms Racein Complex Strategic Innovation." In Trends in Mathematical Economics. Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-32543-9_11.

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Markose, Sheri. "Erratum to: The Gödelian Foundations of Self-Reference,the phLiar and Incompleteness: Arms Racein Complex Strategic Innovation." In Trends in Mathematical Economics. Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-32543-9_20.

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Conference papers on the topic "Mathematical incompleteness"

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Ton Alias, Mohd Aswadi, Nurul Asni Mohamed, M. Iskandar Bakeri, et al. "Application of Artificial Intelligence in Corrosion Management." In CONFERENCE 2024. AMPP, 2024. https://doi.org/10.5006/c2024-20711.

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Abstract Corrosion management system encompassing the various stages of an asset life starting from design, construction, through to operation and decommissioning remains the key focus in ensuring integrity and safe operation of the asset. Corrosion study is conducted during the initial design phase, followed by multiple reviews during the operational stage as part of the overall corrosion management process. These studies aim to identify all damage mechanisms that can be present, including both non-age-related and age-related mechanisms. Currently in the oil and gas industry, corrosion rate predictions for age-related mechanisms are generated via mathematical equations or correlations as outcome from laboratory testing and analyses which may not be representative of the actual operating condition. These predictions impose limitations with regards to utilizing inputs produced from big data. Application of artificial intelligence to predict corrosion rate offers advantages where real high frequency data streams from IoT sensors are analyzed via machine learning algorithm thus providing prediction based on historical experience of specific asset. Data preprocessing is an important step in machine learning that involves transforming raw data from various parameters so that issues owing to the incompleteness, inconsistency, and/or lack of appropriate representation of trends are resolved to arrive at a data set that is in an understandable format. Feature engineering will then be performed which analyze the parameter correlation to obtain the most suitable combination and the best features and data characteristics. For corrosion rate prediction, the supervised learning algorithm is applicable such as logistic regression, naive bayes, support vector machines, artificial neural networks, and random forests. The final step of the machine learning modelling is the model validation. The predicted corrosion rates will be verified with actual thickness measurement at site. To date, we have covered 30 process units which includes different trains, 120 corrosion groups selected from a total of about 3800 corrosion groups for the whole facility. 700 customized machine learning models were developed. Success is defined by best highest accuracy (&amp;gt;80%) with an optimum model run time. Recent validation has shown the ability to predict an anomaly in future trend which coincides with actual increase in corrosion rate.
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Osetrov, Oleksandr, and Rainer Haas. "Simulation of Hydrogen Combustion in Spark Ignition Engines Using a Modified Wiebe Model." In 2024 Stuttgart International Symposium. SAE International, 2024. http://dx.doi.org/10.4271/2024-01-3016.

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&lt;div class="section abstract"&gt;&lt;div class="htmlview paragraph"&gt;Due to its physical and chemical properties, hydrogen is an attractive fuel for internal combustion engines, providing grounds for studies on hydrogen engines. It is common practice to use a mathematical model for basic engine design and an essential part of this is the simulation of the combustion cycle, which is the subject of the work presented here. One of the most widely used models for describing combustion in gasoline and diesel engines is the Wiebe model. However, for cases of hydrogen combustion in DI engines, which are characterized by mixture stratification and in some cases significant incomplete combustion, practically no data can be found in the literature on the application of the Wiebe model. Based on Wiebe’s formulas, a mathematical model of hydrogen combustion has been developed. The model allows making computations for both DI and PFI hydrogen engines. The parameters of the Wiebe model were assessed for three different engines in a total of 26 operating modes. The modified base model considers the significant incompleteness of hydrogen combustion, which occurs at high air/fuel equivalence ratio. For PFI and DI hydrogen engines, equations and numerical values for the Wiebe model coefficients were determined to describe the dynamic and duration of combustion. Based on our simulation results we suggest using the sum of two Wiebe curves to describe combustion in zones with a lean mixture in DI engines. This allows a more accurate characterization of the combustion dynamics and pressure curves. In order to model a double hydrogen injection, we suggest using the sum of three Wiebe curves representing the combustion of the first injection in the flame front, the diffusion combustion of the second injection, and the relatively slow combustion in lean mixture zones. In the paper, we present a method for selecting the coefficients of each of the Wiebe curves.&lt;/div&gt;&lt;/div&gt;
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Solov'eva, Anastasiya, Sergey Solov'ev, Leonid Shevcov, and Valeriya Piven'. "ANALYSIS OF RELIABILITY OF FLAT TROUSERS BASED ON P-BLOCKS." In PROBLEMS OF APPLIED MECHANICS. Bryansk State Technical University, 2020. http://dx.doi.org/10.30987/conferencearticle_5fd1ed0352ef87.51750998.

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The paper considers an approach to the probabilistic analysis of the reliability of flat trusses based on&#x0D; p-boxes (probability boxes, p-boxes). Modeling of stochastic parameters in the form of p-blocks is justified for building pavement structures due to significant variability of climatic loads, variations in the physical and mechanical properties of coating materials, installation tolerances and other uncertainties. The advantage of this method is the possibility of using it with incomplete (limited) statistical information - when it is difficult to determine the probability distribution law or the parameters of a random variable. Variants of constructing p-blocks are illustrated for various types of incompleteness of statistical information: for an unknown distribution law using Chebyshev's inequality, for interval estimates of the parameters of random variables, etc. Information is given on the possibility of performing algebraic operations on p-blocks. The probability of no-failure operation with such approaches will be presented as an interval of values. If the interval is too wide (uninformative), the quality of statistical information should be improved by conducting additional tests. The paper presents mathematical models of limiting states taking into account the variability of the basic random variables. The possibility of using the proposed approach in the framework of most practical problems in the construction industry for assessing the safety of statically definable farms is shown. As a result, a formula is given for assessing the reliability of a truss as a conditional mechanical sequential system (in terms of the theory of reliability), taking into account the lack of information about the dependence of its elements. The algorithm for analyzing reliability is considered on a numerical example. The developed approach can be used for other types of statically definable hinge-rod systems.
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Sabeghi, Maryam, Warren Smith, Janet K. Allen, and Farrokh Mistree. "Solution Space Exploration in Model-Based Realization of Engineered Systems." In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-46457.

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George Box a British mathematician and professor of statistics wrote that “essentially, all models are wrong, but some are useful.” In keeping with George Box’s observation we suggest that in the model-based realization of complex systems, the decision maker must be able to work constructively with decision models of varying fidelity, completeness and accuracy in order to make defendable decisions under uncertainty. The models, and the search algorithms that use these models, will never be perfect and the inherent inaccuracy and incompleteness of analysis models and solvers manifest as uncertainties in the projected outcomes. Therefore, a significant and desirable step in any model-based application is to find stable and robust solutions in which variation of the (input) variables and parameters within manageable tolerances has the minimum effect on delivering favorable, system performance. In this paper we present a method for visualizing and exploring the solution space using the compromise Decision Support Problem (cDSP) as a decision model to aid a decision maker in finding these stable and robust solutions. The efficacy of the method is illustrated using the design of a shell and tube heat exchanger as an example. The method is generalizable to other decision constructs. Our emphasis is on the method rather than the results per se.
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Dimitrakopoulou, Georgia. "WILLIAM BLAKE�S AESTHETICS IN THE MYTH OF THE ANCIENT BRITONS." In 9th SWS International Scientific Conferences on ART and HUMANITIES - ISCAH 2022. SGEM WORLD SCIENCE, 2022. http://dx.doi.org/10.35603/sws.iscah.2022/s10.16.

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William Blake�s aesthetic vision of the secular world is based on divine inspiration. In the myth of The Ancient Britons, he discusses the three aesthetic categories of the sublime, the beautiful and the ugly. These establish his theory of art, which is based on Jesus the Imagination. The sublime, the beautiful and the ugly are forms indicative of gradations of divine influx in every individual. In this sense, the myth explicitly describes and distinguishes the three aesthetic categories that shape the secular and eternal human existence. It also concerns art and the role of the artist. Art is imagination and communication, and the artist is the inhabitant of that happy country of Eden. [1]. The artist is motivated by creative imagination, whose aesthetic quest starts from divine inspiration and ends in eternity. The true artist is the man of imagination, the poetic genius and the visionary aesthete. For Blake, imagination is a sublime force, the major aesthetic category of his vision and relates to the Strong man of The Ancient Britons. In addition, beauty is a distinct aesthetic category essential and supplementary to the formation of the Sublime of Imagination, Jesus incarnated, the archetype of Blake�s Strong man. He [Blake] distinguishes the three aesthetic categories of the sublime, the beautiful and the ugly by their actions, which define man. As he claims: �The Beautiful man acts from duty, whereas The Strong man acts from conscious superiority, and The Ugly man acts from love of carnage.� [2]. Starting from the conviction that antiquity and classical art provided obsolete models for emulation, Blake concluded that since the mathematic form is not art, it should not be the rule of the English eighteenth century art. Gothic, which is the living form, represents the union of the secular and divine worlds. The gothic artistic style is the incarnated Jesus, the Sublime of Imagination, Blake�s aesthetic apex, that is the supreme aesthetic category of his vision. The sublime and the beautiful are not contraries. They are supplementary aesthetic forms which contribute to the understanding of art. Beauty and intellectuality identify. Moreover, beauty is the power, the energizer of the true artist. Who is the human sublime? He is �The Strong man� who acts from conscious superiority, according to the divine decrees and the inspired, prophetic mind. Who is �The Beautiful man�? He is the man who acts from duty. Lastly, �The Ugly Man� is the man of war, aggressive, he/she acts from love of carnage, approaching to the beast in features and form, with a unique characteristic, that is the incapability of intellect. [3]. Undoubtedly, Blake�s aesthetic vision presents many difficulties in interpretation. In my opinion, the sublime is not an aesthetic category and/or a mere value that Blake uses randomly without artistic reference. His aestheticism is secular and aspiring to perfection. The secular sublime, which describes the fallen human state, suggests the masculine and feminine experience of the Fall. Consequently, the human situation appears doomed and irredeemable. If the sublime is the masculine and pathos the feminine forms, Blake assumes that the inevitability of reasoning and suppression of desire, whose origin is energy, brings about their separation and incompleteness. In a non-communicative intercourse, the sublime (masculine) and the beautiful (pathos) are apart. These are the fallen state�s consequences. As the masculine and feminine are not contraries but supplementary forms, so the sublime and pathos are potentially integrated entities. In eternity, the sublime and pathos are joined in an intellectual androgynous form. This theoretical idea is the core of Blake�s aesthetic �theory�. In fact, his aesthetic realism does not overlap his aesthetic idealism. He is optimistic, despite Urizen�s - reason�s predominance. The artist is the model of human salvation. Imagination is a redemptive force, �Exuberance is Beauty�, and the incapability of intellect is �The Ugly Man�. The three classes of men, the elect, the redeemed and the reprobate juxtapose to the sublime, beautiful and ugly. These are restored to their true forms and their qualities are reinstated in infinity. All human forms are redeemable states, not static but progressive, even if their fulfilment on earth is improbable.
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