Academic literature on the topic 'Axioma de Peano'
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Journal articles on the topic "Axioma de Peano"
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Heck, Richard G. "The development of arithmetic in Frege's Grundgesetze der arithmetik." Journal of Symbolic Logic 58, no. 2 (June 1993): 579–601. http://dx.doi.org/10.2307/2275220.
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AbstractFrege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system—Axiom V—which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does prove each of the now standard Dedekind-Peano axioms, his proofs are devoted primarily to the derivation of his own axioms for arithmetic, which are somewhat different (though of course equivalent). These axioms, which may be yet more intuitive than the Dedekind-Peano axioms, may be taken to be “The Basic Laws of Cardinal Number”, as Frege understood them.Though the axioms of arithmetic have been known to be derivable from Hume's Principle for about ten years now, it has not been widely recognized that Frege himself showed them so to be; nor has it been known that Frege made use of any axiomatization for arithmetic whatsoever. Grundgesetze is thus a work of much greater significance than has often been thought. First, Frege's use of the inconsistent Axiom V may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establish may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establishment of Logicism), but it should not be allowed to obscure his mathematical accomplishments and his contribution to our understanding of arithmetic. Second, Frege's knowledge that arithmetic is derivable from Hume's Principle raises important sorts of questions about his philosophy of arithmetic. For example, “Why did Frege not simply abandon Axiom V and take Hume's Principle as an axiom?”
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Willard, Dan E. "How to extend the semantic tableaux and cut-free versions of the second incompleteness theorem almost to Robinson's arithmetic q." Journal of Symbolic Logic 67, no. 1 (March 2002): 465–96. http://dx.doi.org/10.2178/jsl/1190150055.
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AbstractLet us recall that Raphael Robinson's Arithmetic Q is an axiom system that differs from Peano Arithmetic essentially by containing no Induction axioms [13], [18]. We will generalize the semantic-tableaux version of the Second Incompleteness Theorem almost to the level of System Q. We will prove that there exists a single rather long Π1 sentence, valid in the standard model of the Natural Numbers and denoted as V. such that if α is any finite consistent extension of Q + V then α will be unable to prove its Semantic Tableaux consistency. The same result will also apply to axiom systems α with infinite cardinality when these infinite-sized axiom systems satisfy a minor additional constraint, called the Conventional Encoding Property.Our formalism will also imply that the semantic-tableaux version of the Second Incompleteness Theorem generalizes for the axiom system IΣ0, as well as for all its natural extensions. (This answers an open question raised twenty years ago by Paris and Wilkie [15].)
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Yukami, Tsuyoshi. "Taking out LK parts from a proof in Peano arithmetic." Journal of Symbolic Logic 51, no. 3 (September 1986): 682–700. http://dx.doi.org/10.2307/2274022.
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Let PA be Peano arithmetic with function symbols′, + and ·. The length of a proof P, denoted by lh(P), is the maximum length of threads of P (for the term ‘thread’, see [T, p. 14]). For a formula A and a natural number m, PA ⊢mA denotes the fact that there is a proof in PA of A whose length is less than or equal to m. PA ⊢ A denotes the fact that there is a proof in PA of A.G. Kreisel conjectured that the following proposition holds.“Let m be a natural number and A(a) be a formula. If for each natural number n, then PA ⊢ ∀xA(x)”.Let PA1 be the corresponding system with + and · replaced by ternary predicates and axioms saying that these predicates represent functions. Parikh [P] proved the following proposition which is obtained from Kreisel's conjecture by replacing PA by PA1.Proposition. Let A(a) be a formula in PA1and m be a natural number. Assume thatfor each natural number n. Then PA1 ⊢ ∀xA(x).The reason why Parikh's method succeeds is the fact that the only function symbol ′ in PA1 is unary. So his method fails for PA.To solve this conjecture for PA, we must make syntactical investigation into proofs in PA of formulas of the form A() with length ≤ m. Even if lengths of proofs are less than or equal to m, depths of occurrences of bound variables in induction axiom schemata or equality axiom schemata in proofs are not always bounded.
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Decock, Lieven. "Neo-Fregeanism naturalized: The role of one-to-one correspondence in numerical cognition." Behavioral and Brain Sciences 31, no. 6 (December 2008): 648–49. http://dx.doi.org/10.1017/s0140525x08005645.
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AbstractRips et al. argue that the construction of math schemas roughly similar to the Dedekind/Peano axioms may be necessary for arriving at arithmetical skills. However, they neglect the neo-Fregean alternative axiomatization of arithmetic, based on Hume's principle. Frege arithmetic is arguably a more plausible start for a top-down approach in the psychological study of mathematical cognition than Peano arithmetic.
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Changat, Manoj, and Joseph Mathew. "Induced path transit function, monotone and Peano axioms." Discrete Mathematics 286, no. 3 (September 2004): 185–94. http://dx.doi.org/10.1016/j.disc.2004.02.017.
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Lubarsky, Robert S. "An introduction to γ-recursion theory (or what to do in KP – Foundation)." Journal of Symbolic Logic 55, no. 1 (March 1990): 194–206. http://dx.doi.org/10.2307/2274962.
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The program of reverse mathematics has usually been to find which parts of set theory, often used as a base for other mathematics, are actually necessary for some particular mathematical theory. In recent years, Slaman, Groszek, et al, have given the approach a new twist. The priority arguments of recursion theory do not naturally or necessarily lead to a foundation involving any set theory; rather, Peano Arithmetic (PA) in the language of arithmetic suffices. From this point, the appropriate subsystems to consider are fragments of PA with limited induction. A theorem in this area would then have the form that certain induction axioms are independent of, necessary for, or even equivalent to a theorem about the Turing degrees. (See, for examples, [C], [GS], [M], [MS], and [SW].)As go the integers so go the ordinals. One motivation of α-recursion theory (recursion on admissible ordinals) is to generalize classical recursion theory. Since induction in arithmetic is meant to capture the well-foundedness of ω, the corresponding axiom in set theory is foundation. So reverse mathematics, even in the context of a set theory (admissibility), can be changed by the influence of reverse recursion theory. We ask not which set existence axioms, but which foundation axioms, are necessary for the theorems of α-recursion theory.When working in the theory KP – Foundation Schema (hereinafter called KP−), one should really not call it α-recursion theory, which refers implicitly to the full set of axioms KP. Just as the name β-recursion theory refers to what would be α-recursion theory only it includes also inadmissible ordinals, we call the subject of study here γ-recursion theory. This answers a question by Sacks and S. Friedman, “What is γ-recursion theory?”
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Chong, C. T. "Maximal sets and fragments of Peano arithmetic." Nagoya Mathematical Journal 115 (September 1989): 165–83. http://dx.doi.org/10.1017/s0027763000001604.
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This work is inspired by the recent paper of Mytilinaios and Slaman [9] on the infinite injury priority method. It may be considered to fall within the general program of the study of reverse recursion theory: What axioms of Peano arithmetic are required or sufficient to prove theorems in recursion theory? Previous contributions to this program, especially with respect to the finite and infinite injury priority methods, can be found in the works of Groszek and Mytilinaios [4], Groszek and Slaman [5], Mytilinaios [8], Slaman and Woodin [10]. Results of [4] and [9], for example, together pinpoint the existence of an incomplete, nonlow r.e. degree to be provable only within some fragment of Peano arithmetic at least as strong as P- + IΣ2. Indeed an abstract principle on infinite strategies, such as that on the construction of an incomplete high r.e. degree, was introduced in [4] and shown to be equivalent to Σ2 induction over the base theory P- + IΣ0 of Peano arithmetic.
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Read, Dwight. "Learning natural numbers is conceptually different than learning counting numbers." Behavioral and Brain Sciences 31, no. 6 (December 2008): 667–68. http://dx.doi.org/10.1017/s0140525x08005840.
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AbstractHow children learn number concepts reflects the conceptual and logical distinction between counting numbers, based on a same-size concept for collections of objects, and natural numbers, constructed as an algebra defined by the Peano axioms for arithmetic. Cross-cultural research illustrates the cultural specificity of counting number systems, and hence the cultural context must be taken into account.
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White, Jonathan J. "The Peano Axioms: An IBL Unit Constructing the Natural Numbers." PRIMUS 27, no. 7 (July 14, 2016): 725–35. http://dx.doi.org/10.1080/10511970.2016.1199619.
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Kahle, Reinhard. "Dedekinds Sätze und Peanos Axiomata." Philosophia Scientae, no. 25-1 (February 25, 2021): 69–93. http://dx.doi.org/10.4000/philosophiascientiae.2846.
Full textDissertations / Theses on the topic "Axioma de Peano"
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Sousa, Pedro SÃrgio Sales de. "A construÃÃo dos nÃmeros naturais: um foco nas quatro operaÃÃes fundamentais." Universidade Federal do CearÃ, 2014. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=13230.
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O presente trabalho tem como objetivo apresentar a construÃÃo dos nÃmeros naturais e a definiÃÃo axiomÃtica no que diz respeito Ãs quatro operaÃÃes fundamentais para alunos e professores do ensino fundamental. Para isso foi apresentado uma sequÃncia abordando inicialmente as consideraÃÃes sobre o estudo da MatemÃtica, o conceito de MatemÃtica, o saber matemÃtico e um breve histÃrico matemÃtico para se perceber como teorias e prÃticas matemÃticas foram criadas, desenvolvidas e utilizadas num contexto especÃfico de cada Ãpoca. No segundo momento foi descrita a construÃÃo dos nÃmeros naturais atravÃs dos axiomas de Peano, prosseguindo com a definiÃÃo rigorosa de cada operaÃÃo e finalizando com a relaÃÃo de ordem no conjunto dos nÃmeros naturais.This paper aims to present the construction of the natural numbers and the axiomatic definition with respect to the four fundamental operations for students and teachers of elementary school.To this was presented a sequence initially addressing on the study of mathematics, the concept of mathematics, mathematical knowledge and a mathematical brief history to see how mathematical theories and practices are designed, developed and used in a specific context of each era. The second moment was described the construction of natural numbers through the Peano axioms, continuing with the rigorous definition of each operation and ending with the order relation in the set of natural numbers.
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Sousa, Pedro Sérgio Sales de. "A construção dos números naturais: um foco nas quatro operações fundamentais." reponame:Repositório Institucional da UFC, 2014. http://www.repositorio.ufc.br/handle/riufc/10444.
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SOUSA, Pedro Sérgio Sales de. A construção dos números naturais: um foco nas quatro operações fundamentais. 2014. 40 f. Dissertação (Mestrado em Matemática em Rede Nacional) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2014.Submitted by Erivan Almeida (eneiro@bol.com.br) on 2015-01-12T16:34:08Z No. of bitstreams: 1 2014_dis_psssousa.pdf: 1314798 bytes, checksum: d58f97e2efdcfc7a1a8d07e1edb49b0b (MD5)
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This paper aims to present the construction of the natural numbers and the axiomatic definition with respect to the four fundamental operations for students and teachers of elementary school.To this was presented a sequence initially addressing on the study of mathematics, the concept of mathematics, mathematical knowledge and a mathematical brief history to see how mathematical theories and practices are designed, developed and used in a specific context of each era. The second moment was described the construction of natural numbers through the Peano axioms, continuing with the rigorous definition of each operation and ending with the order relation in the set of natural numbers.
O presente trabalho tem como objetivo apresentar a construção dos números naturais e a definição axiomática no que diz respeito às quatro operações fundamentais para alunos e professores do ensino fundamental. Para isso foi apresentado uma sequência abordando inicialmente as considerações sobre o estudo da Matemática, o conceito de Matemática, o saber matemático e um breve histórico matemático para se perceber como teorias e práticas matemáticas foram criadas, desenvolvidas e utilizadas num contexto específico de cada época. No segundo momento foi descrita a construção dos números naturais através dos axiomas de Peano, prosseguindo com a definição rigorosa de cada operação e finalizando com a relação de ordem no conjunto dos números naturais.
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Oliveira, Wesley Sidney Santos. "A construção ortodoxa dos números : dos números naturais aos complexos." Universidade Federal de Sergipe, 2017. https://ri.ufs.br/handle/riufs/6522.
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In this work, we investigated the construction of natural, integer, rational, real, complex, quaternion
and Octonion numbers. More precisely, the set of real numbers was achieved by applying
two methods: Dedekind Cuts and Equivalence Classes of Cauchy Sequences. Our study is only
based on using Peano Axioms, which are directly related to the natural numbers, in order to get
the basic properties satis ed by these numbers. In addition, we carefully proved the elementary
results involving real numbers. This process in question was developed constructively throughout
of the concepts of the integer and rational numbers. Next, we show that it is possible to establish
the existence of complex numbers along with their more usual arithmetic properties. Finally, we
nish each chapter of our work showing some possible applications in each set worked.No presente trabalhos, investigamos, cuidadosamente, a construção do números Naturais, inteiros, Racionais, Reais e Complexos. Sendo que, o conjunto dos números reais foi obtido através dos conhecidos métodos: Cortes de Dedekind e Classes de Equivalência por sequência de Cauchy. O estudo consistiu em utilizar os famosos Axiomas de Peano, ps quais estão relacionados aos números naturais, em ordem a obter as em conhecidas propriedades elementares, satisfeitas para todos esses números. E, a partir deste conhecimento, encontramos rigorosamente as provas dos resultados básicos envolvendo os números reais. Este processo em questão, foi desenvolvida de maneira construtiva através dos números inteiros e racionais. Em seguida, mostramos que é possível estabelecer a existência de números complexos, juntamente com suas propriedades aritméticas mais usuais. Por fim, terminamos cada capítulo do nosso trabalho, mostrando algumas possíveis aplicações em cada conjunto trabalhado.
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Blot, Valentin. "Game semantics and realizability for classical logic." Thesis, Lyon, École normale supérieure, 2014. http://www.theses.fr/2014ENSL0945/document.
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Cette thèse étudie deux modèles de réalisabilité pour la logique classique construits sur la sémantique des jeux HO, interprétant la logique, l'arithmétique et l'analyse classiques directement par des programmes manipulant un espace de stockage d'ordre supérieur.La non-innocence en jeux HO autorise les références d'ordre supérieur, et le non parenthésage révèle la CPS des jeux HO et fournit une catégorie de continuations dans laquelle interpréter le lambda-mu calcul de Parigot. Deux modèles de réalisabilité sont construits sur cette interprétation calculatoire directe des preuves classiques.Le premier repose sur l'orthogonalité, comme celui de Krivine, mais il est simplement typé et au premier ordre. En l'absence de codage de l'absurdité au second ordre, une mu-variable libre dans les réaliseurs permet l'extraction. Nous définissons un bar-récurseur et prouvons qu'il réalise l'axiome du choix dépendant, utilisant deux conséquences de la structure de CPO du modèle de jeux: toute fonction sur les entiers (même non calculable) existe dans le modèle, et toute fonctionnelle sur des séquences est Scott-continue. La bar-récursion est habituellement utilisée pour réaliser intuitionnistiquement le « double negation shift » et en déduire la traduction négative de l'axiome du choix. Ici, nous réalisons directement l'axiome du choix dans un cadre classique.Le second, très spécifique au modèle de jeux, repose sur des conditions de gain: des ensembles de positions d'un jeu munis de propriétés de cohérence. Un réaliseur est alors une stratégie dont les positions sont toutes gagnantesThis thesis investigates two realizability models for classical logic built on HO game semantics. The main motivation is to have a direct computational interpretation of classical logic, arithmetic and analysis with programs manipulating a higher-order store.Relaxing the innocence condition in HO games provides higher-order references, and dropping the well-bracketing of strategies reveals the CPS of HO games and gives a category of continuations in which we can interpret Parigot's lambda-mu calculus. This permits a direct computational interpretation of classical proofs from which we build two realizability models.The first model is orthogonality-based, as the one of Krivine. However, it is simply-typed and first-order. This means that we do not use a second-order coding of falsity, and extraction is handled by considering realizers with a free mu-variable. We provide a bar-recursor in this model and prove that it realizes the axiom of dependent choice, relying on two consequences of the CPO structure of the games model: every function on natural numbers (possibly non computable) exists in the model, and every functional on sequences is Scott-continuous. Usually, bar-recursion is used to intuitionistically realize the double negation shift and consequently the negative translation of the axiom of choice. Here, we directly realize the axiom of choice in a classical setting.The second model relies on winning conditions and is very specific to the games model. A winning condition is a set of positions in a game which satisfies some coherence properties, and a realizer of a formula is then a strategy which positions are all winning
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Almeida, João Paulo da Cruz [UNESP]. "Indução finita, deduções e máquina de Turing." Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/151718.
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Este trabalho apresenta uma proposta relacionada ao ensino e prática do pensamento dedutivo formal em Matemática. São apresentados no âmbito do conjunto dos números Naturais três temas essencialmente interligados: indução/boa ordem, dedução e esquemas de computação representados pela máquina teórica de Turing. Os três temas se amalgamam na teoria lógica de dedução e tangem os fundamentos da Matemática, sua própria indecidibilidade e extensões / limites de tudo que pode ser deduzido utilizando a lógica de Aristóteles, caminho tão profundamente utilizado nos trabalhos de Gödel, Church, Turing, Robinson e outros. São apresentadas inúmeros esquemas de dedução referentes às “fórmulas” e Teoremas que permeiam o ensino fundamental e básico, com uma linguagem apropriada visando treinar os alunos (e professores) para um enfoque mais próprio pertinente à Matemática.
This work deals with the teaching and practice of formal deductive thinking in Mathematics. Three essentially interconnected themes are presented within the set of Natural Numbers: induction, deduction and computation schemes represented by the Turing theoretical machine. The three themes are put together into the logical theory of deduction and touch upon the foundations of Mathematics, its own undecidability and the extent / limits of what can be deduced by using Aristotle's logic, that is the subject in the works of Gödel, Church, Turing, Robinson, and others. There are a large number of deduction schemes referring to the "formulas" and Theorems that are usual subjects in elementary and basic degrees of the educational field, with an appropriate language in order to train students (and teachers) for a more pertinent approach to Mathematics.
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Santos, Rafael Messias. "Fundamentos de lógica, conjuntos e números naturais." Universidade Federal de Sergipe, 2015. https://ri.ufs.br/handle/riufs/6488.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPESThe present work has as main objective to approach the fundaments of logic and the notions of sets in a narrow and elementary way, culminating in the construction of natural numbers. We present and advance, as far as possible, natural and intuitively, the concepts of propositions and open propositions, and the use of these in the speci cation sets, according with the axiom of the speci cation. We also present the logic connectives of open propositions and logic equivalences, relating them to the sets. We showed the concept of Theorem, as well as some forms of writing and demonstrations in the scope of the sets, and we used properties and relations of sets in the demonstration techniques. Our study ended with the construction of natural numbers and some of its properties, for example, the Relation Order.
O presente trabalho tem como principal objetivo abordar os fundamentos de lógica e as noções de conjuntos de maneira estreita e elementar, culminando na constru- ção dos números naturais. Apresentamos, e progredimos na medida do possível, de forma natural e/ou intuitiva, os conceitos de proposições e proposições abertas, e o uso destes nas especi cações de conjuntos, de acordo com o axioma da especi cação. Apresentamos também os conectivos lógicos de proposições abertas e as equivalências lógicas, relacionando-os aos conjuntos. Mostramos o conceito de Teorema, bem como algumas formas de escritas e demonstrações no âmbito dos conjuntos, e utilizamos propriedades e relações de conjuntos nas técnicas de demonstração. Encerramos nosso estudo com a construção dos números naturais e algumas das suas principais propriedades, como por exemplo, a Relação de Ordem.
Books on the topic "Axioma de Peano"
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Button, Tim, and Sean Walsh. Internal categoricity and the natural numbers. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198790396.003.0010.
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The simple conclusion of the preceding chapters is that moderate modelism fails. But this leaves us with a choice between abandoning moderation and abandoning modelism. The aim of this chapter, and the next couple of chapters, is to outline a speculative way to save moderation by abandoning modelism. The idea is to do metamathematics without semantics, by working deductively in a higher-order logic. In this chapter, the focus is on the internal categoricity of arithmetic. After formalising an internal notion of a model of the Peano axioms, we show how to internalise Dedekind’s Categority Theorem. The resulting “intolerance” of Peano arithmetic provides internalists with a way to draw the distinction between algebraic and univocal theories. In the appendices, we discuss how this relates to Parsons’ important work, and establish a certain dependence of the internal categoricity theorem on higher-order logic.
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Linnebo, Øystein. The Natural Numbers. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780199641314.003.0010.
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How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.
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Button, Tim, and Sean Walsh. Categoricity and the natural numbers. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198790396.003.0007.
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This chapter focuses on modelists who want to pin down the isomorphism type of the natural numbers. This aim immediately runs into two technical barriers: the Compactness Theorem and the Löwenheim-Skolem Theorem (the latter is proven in the appendix to this chapter). These results show that no first-order theory with an infinite model can be categorical; all such theories have non-standard models. Other logics, such as second-order logic with its full semantics, are not so expressively limited. Indeed, Dedekind's Categoricity Theorem tells us that all full models of the Peano axioms are isomorphic. However, it is a subtle philosophical question, whether one is entitled to invoke the full semantics for second-order logic — there are at least four distinct attitudes which one can adopt to these categoricity result — but moderate modelists are unable to invoke the full semantics, or indeed any other logic with a categorical theory of arithmetic.
Book chapters on the topic "Axioma de Peano"
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Forster, Otto. "Die Peano-Axiome." In Algorithmische Zahlentheorie, 1–8. Wiesbaden: Vieweg+Teubner Verlag, 1996. http://dx.doi.org/10.1007/978-3-663-09239-1_1.
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Forster, Otto. "Die Peano-Axiome." In Algorithmische Zahlentheorie, 1–8. Wiesbaden: Springer Fachmedien Wiesbaden, 2014. http://dx.doi.org/10.1007/978-3-658-06540-9_1.
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Dasgupta, Abhijit. "The Dedekind–Peano Axioms." In Set Theory, 29–46. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-8854-5_2.
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Halbeisen, Lorenz, and Regula Krapf. "Models of Peano Arithmetic." In Gödel's Theorems and Zermelo's Axioms, 199–202. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-52279-7_16.
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Halbeisen, Lorenz, and Regula Krapf. "Arithmetic in Peano Arithmetic." In Gödel's Theorems and Zermelo's Axioms, 79–88. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-52279-7_8.
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Halbeisen, Lorenz, and Regula Krapf. "Gӧdelisation of Peano Arithmetic." In Gödel's Theorems and Zermelo's Axioms, 89–108. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-52279-7_9.
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Halbeisen, Lorenz, and Regula Krapf. "Countable Models of Peano Arithmetic." In Gödel's Theorems and Zermelo's Axioms, 73–78. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-52279-7_7.
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Smullyan, Raymond M. "The Incompleteness of Peano Arithmetic with Exponentiation." In Gödel's Incompleteness Theorems. Oxford University Press, 1992. http://dx.doi.org/10.1093/oso/9780195046724.003.0006.
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We shall now turn to a formal axiom system which we call Peano Arithmetic with Exponentiation and which we abbreviate “P.E.”. We take certain correct formulas which we call axioms and provide two inference rules that enable us to prove new correct formulas from correct formulas already proved. The axioms will be infinite in number, but each axiom will be of one of nineteen easily recognizable forms; these forms are called axiom schemes. It will be convenient to classify these nineteen axiom schemes into four groups (cf. discussion that follows the display of the schemes). The axioms of Groups I and II are the so-called logical axioms and constitute a neat formalization of first-order logic with identity due to Kalish and Montague [1965], which is based on an earlier system due to Tarski [1965]. The axioms of Groups III and IV are the so-called arithmetic axioms. In displaying these axiom schemes, F, G and H are any formulas, vi and vj are any variables, and t is any term. For example, the first scheme L1 means that for any formulas F and G, the formula (F ⊃ (G ⊃ F)) is to be taken as an axiom; axiom scheme L4 means that for any variable Vi and any formulas F and G, the formula . . . (∀vi (F ⊃ G) ⊃ (∀vi (F ⊃ ∀vi G) . . . is to be taken as an axiom.
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"The Peano Axioms." In Algebra from A to Z, 100–105. World Scientific Publishing Company, 2002. http://dx.doi.org/10.1142/9789814291903_0005.
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"THE PEANO AXIOMS." In Arithmetic and Ontology, 103–28. Brill | Rodopi, 2006. http://dx.doi.org/10.1163/9789004333680_009.
Full textConference papers on the topic "Axioma de Peano"
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Cerioli, Márcia R., Vitor Krauss, and Petrucio Viana. "An Arithmetical-like Theory of Hereditarily Finite Sets." In Workshop Brasileiro de Lógica. Sociedade Brasileira de Computação, 2021. http://dx.doi.org/10.5753/wbl.2021.15774.
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This paper presents the (second-order) theory of hereditarily finite sets according to the usual pattern adopted in the presentation of the (second-order) theory of natural numbers. To this purpose, we consider three primitive concepts, together with four axioms, which are analogous to the usual Peano axioms. From them, we prove a homomorphism theorem, its converse, categoricity, and a kind of (semantical) completeness.
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Cerioli, Márcia, Hugo Nobrega, Guilherme Silveira, and Petrucio Viana. "On the (in)dependence of the Dedekind-Peano axioms for natural numbers." In CNMAC 2016 - XXXVI Congresso Nacional de Matemática Aplicada e Computacional. SBMAC, 2017. http://dx.doi.org/10.5540/03.2017.005.01.0239.
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