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Journal articles on the topic 'Axioma de Peano'
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1
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.
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Hrubeš, Pavel. "Kreisel's Conjecture with minimality principle." Journal of Symbolic Logic 74, no. 3 (September 2009): 976–88. http://dx.doi.org/10.2178/jsl/1245158094.
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AbstractWe prove that Kreisel's Conjecture is true, if Peano arithmetic is axiomatised using minimality principle and axioms of identity (theory PAM). The result is independent on the choice of language of PAM. We also show that if infinitely many instances of A(x) are provable in a bounded number of steps in PAM then there exists . The results imply that PAM does not prove scheme of induction or identity schemes in a bounded number of steps.
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Linnebo, Øystein. "Predicative Fragments of Frege Arithmetic." Bulletin of Symbolic Logic 10, no. 2 (June 2004): 153–74. http://dx.doi.org/10.2178/bsl/1082986260.
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AbstractFrege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume's Principle, which says that the number of Fs is identical to the number of Gs if and only if the Fs and the Gs can be one-to-one correlated. According to Frege's Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA—one having to do with Hume's Principle, the other, with the underlying second-order logic—and investigates how much of Frege's Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension.
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Quine, W. V. "Free Logic, Description, and Virtual Classes." Dialogue 36, no. 1 (1997): 101–8. http://dx.doi.org/10.1017/s001221730000932x.
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ResuméOn montre ici que la theorie des classes virtuelles constitue une application de la logique libre. Il s'agit de la théorie classique de la quantification qui ne fait usage que d'un prédicat à deux places : l'epsilon de Peano, et qui ne comporte qu'un axiome d'extensionnalité. Elle n'introduit aucune présupposition d'existence. Les individus y sont identifiés à leur singleton, et elle accommode les termes qui ne designent rien. On montre aussi comment cette théorie peut fructifier et devenir une théorie des ensembles plus élaborée à l'aide de définitions et de conventions notationnelles.
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Hardin, Christopher S., and Daniel J. Velleman. "The mean value theorem in second order arithmetic." Journal of Symbolic Logic 66, no. 3 (September 2001): 1353–58. http://dx.doi.org/10.2307/2695111.
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This paper is a contribution to the project of determining which set existence axioms are needed to prove various theorems of analysis. For more on this project and its history we refer the reader to [1] and [2].We work in a weak subsystem of second order arithmetic. The language of second order arithmetic includes the symbols 0, 1, =, <, +, ·, and ∈, together with number variables x, y, z, … (which are intended to stand for natural numbers), set variables X, Y, Z, … (which are intended to stand for sets of natural numbers), and the usual quantifiers (which can be applied to both kinds of variables) and logical connectives. We write ∀x < t φ and ∃x < t φ as abbreviations for ∀x(x < t → φ) and ∃x{x < t ∧ φ) respectively; these are called bounded quantifiers. A formula is said to be if it has no quantifiers applied to set variables, and all quantifiers applied to number variables are bounded. It is if it has the form ∃xθ and it is if it has the form ∀xθ, where in both cases θ is .The theory RCA0 has as axioms the usual Peano axioms, with the induction scheme restricted to formulas, and in addition the comprehension scheme, which consists of all formulas of the formwhere φ is , ψ is , and X does not occur free in φ(n). (“RCA” stands for “Recursive Comprehension Axiom.” The reason for the name is that the comprehension scheme is only strong enough to prove the existence of recursive sets.) It is known that this theory is strong enough to allow the development of many of the basic properties of the real numbers, but that certain theorems of elementary analysis are not provable in this theory. Most relevant for our purposes is the fact that it is impossible to prove in RCA0 that every continuous function on the closed interval [0, 1] attains maximum and minimum values (see [1]).Since the most common proof of the Mean Value Theorem makes use of this theorem, it might be thought that the Mean Value Theorem would also not be provable in RCA0. However, we show in this paper that the Mean Value Theorem can be proven in RCA0. All theorems stated in this paper are theorems of RCA0, and all of our reasoning will take place in RCA0.
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WEBER, ZACH. "TRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY." Review of Symbolic Logic 3, no. 1 (January 14, 2010): 71–92. http://dx.doi.org/10.1017/s1755020309990281.
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This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.
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López Palma, Helena. "La relación entre el número gramatical y el número léxico." Revista Española de Lingüística 2, no. 50 (December 2020): 49–81. http://dx.doi.org/10.31810/rsel.50.2.3.
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Estudiamos las semejanzas y las diferencias de dos categorías que expresan número en español: la categoría gramatical de número y la categoría léxica de numeral cardinal. Aplicamos un modelo comparativo basado en Bosque 1989. El sistema de número gramatical y el sistema de número léxico comparten la función aditiva mínima que los genera. Difieren en (a) la naturaleza de las unidades que construyen sus sistemas: unidad natural en los nombres; card(n) unidad axiomática en los números y (b) las propiedades del dominio: Semirretícula de uniones (conjunto parcialmente ordenado); secuencia de números naturales ⟨1,2,...,n⟩ ∈ ℕ (conjunto totalmente ordenado). Proponemos un modelo que generaliza los Axiomas de Peano para las funciones que construyen el sistema.
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Dzierzgowski, Daniel. "Models of intuitionistic TT and NF." Journal of Symbolic Logic 60, no. 2 (June 1995): 640–53. http://dx.doi.org/10.2307/2275855.
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AbstractLet us define the intuitionistic part of a classical theory T as the intuitionistic theory whose proper axioms are identical with the proper axioms of T. For example, Heyting arithmetic HA is the intuitionistic part of classical Peano arithmetic PA.It's a well-known fact, proved by Heyting and Myhill, that ZF is identical with its intuitionistic part.In this paper, we mainly prove that TT, Russell's Simple Theory of Types, and NF, Quine's “New Foundations,” are not equal to their intuitionistic part. So, an intuitionistic version of TT or NF seems more naturally definable than an intuitionistic version of ZF.In the first section, we present a simple technique to build Kripke models of the intuitionistic part of TT (with short examples showing bad properties of finite sets if they are defined in the usual classical way).In the remaining sections, we show how models of intuitionistic NF2 and NF can be obtained from well-chosen classical ones. In these models, the excluded middle will not be satisfied for some non-stratified sentences.
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LEACH-KROUSE, GRAHAM. "GENERALIZING BOOLOS’ THEOREM." Review of Symbolic Logic 10, no. 1 (October 17, 2016): 80–91. http://dx.doi.org/10.1017/s1755020316000332.
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AbstractIt’s well known that it’s possible to extract, from Frege’s Grudgesetze, an interpretation of second-order Peano Arithmetic in the theory $H{P^2}$, whose sole axiom is Hume’s principle. What’s less well known is that, in Die Grundlagen Der Arithmetic §82–83 Boolos (2011), George Boolos provided a converse interpretation of $H{P^2}$ in $P{A^2}$. Boolos’ interpretation can be used to show that the Frege’s construction allows for any model of $P{A^2}$ to be recovered from some model of $H{P^2}$. So the space of possible arithmetical universes is precisely characterized by Hume’s principle.In this paper, I show that a large class of second-order theories admit characterization by an abstraction principle in this sense. The proof makes use of structural abstraction principles, a class of abstraction principles of considerable intrinsic interest, and categories of interpretations in the sense of Visser (2003).
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Japaridze, Giorgi. "Arithmetics based on computability logic." Logical Investigations 25, no. 2 (December 23, 2019): 61–74. http://dx.doi.org/10.21146/2074-1472-2019-25-2-61-74.
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This paper is a brief survey of number theories based on em computability logic (CoL) a game-semantically conceived logic of computational tasks of resources. Such theories, termed em clarithmetics, are conservative extensions of first-order Peano arithmetic.
The first section of the paper lays out the conceptual basis of CoL and describes the relevant fragment of its formal language, with so called parallel connectives, choice connectives and quantifiers, and blind quantifiers. Both syntactically and semantically, this is a conservative generalization of the language of classical logic.
Clarithmetics, based on the corresponding fragment of CoL in the same sense as Peano
arithmetic is based on classical logic, are discussed in the second section. The axioms and inference rules of the system of clarithmetic named ${\bf CLA11}$ are presented, and the main results on this system are stated: constructive soundness, extensional completeness, and intensional completeness.
In the final section two potential applications of clarithmetics are addressed: clarithmetics as declarative programming languages in an extreme sense, and as tools for separating computational complexity classes. When clarithmetics or similar CoL-based theories are viewed as programming languages, programming reduces to proof-search, as programs can be mechanically extracted from proofs; such programs also serves as their own formal verifications, thus fully neutralizing the notorious (and generally undecidable) program verification problem. The second application reduces the problem of separating various computational complexity classes to separating the corresponding versions of clarithmetic, the potential benefits of which stem from the belief that separating theories should generally be easier than separating complexity classes directly.
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Rahn, Alexander, Eldar Sultanow, Max Henkel, Sourangshu Ghosh, and Idriss J. Aberkane. "An Algorithm for Linearizing the Collatz Convergence." Mathematics 9, no. 16 (August 9, 2021): 1898. http://dx.doi.org/10.3390/math9161898.
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The Collatz dynamic is known to generate a complex quiver of sequences over natural numbers for which the inflation propensity remains so unpredictable it could be used to generate reliable proof-of-work algorithms for the cryptocurrency industry; it has so far resisted every attempt at linearizing its behavior. Here, we establish an ad hoc equivalent of modular arithmetics for Collatz sequences based on five arithmetic rules that we prove apply to the entire Collatz dynamical system and for which the iterations exactly define the full basin of attractions leading to any odd number. We further simulate these rules to gain insight into their quiver geometry and computational properties and observe that they linearize the proof of convergence of the full rows of the binary tree over odd numbers in their natural order, a result which, along with the full description of the basin of any odd number, has never been achieved before. We then provide two theoretical programs to explain why the five rules linearize Collatz convergence, one specifically dependent upon the Axiom of Choice and one on Peano arithmetic.
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Ressayre, J. P. "Formal languages defined by the underlying structure of their words." Journal of Symbolic Logic 53, no. 4 (December 1988): 1009–26. http://dx.doi.org/10.1017/s0022481200027894.
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Abstracti) We show for each context-free language L that by considering each word of L as a structure in a natural way, one turns L into a finite union of classes which satisfy a finitary analog of the characteristic properties of complete universal first order classes of structures equipped with elementary embeddings. We show this to hold for a much larger class of languages which we call free local languages, ii) We define local languages, a class of languages between free local and context-sensitive languages. Each local language L has a natural extension L∞ to infinite words, and we prove a series of “pumping lemmas”, analogs for each local language L of the “uvxyz theorem” of context free languages: they relate the existence of large words in L or L∞ to the existence of infinite “progressions” of words included in L, and they imply the decidability of various questions about L or L∞. iii) We show that the pumping lemmas of ii) are independent from strong axioms, ranging from Peano arithmetic to ZF + Mahlo cardinals.We hope that these results are useful for a model-theoretic approach to the theory of formal languages.
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KOLMAKOV, EVGENY, and LEV BEKLEMISHEV. "AXIOMATIZATION OF PROVABLE n-PROVABILITY." Journal of Symbolic Logic 84, no. 02 (February 8, 2019): 849–69. http://dx.doi.org/10.1017/jsl.2018.82.
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AbstractA formula φ is called n-provable in a formal arithmetical theory S if φ is provable in S together with all true arithmetical ${{\rm{\Pi }}_n}$-sentences taken as additional axioms. While in general the set of all n-provable formulas, for a fixed $n > 0$, is not recursively enumerable, the set of formulas φ whose n-provability is provable in a given r.e. metatheory T is r.e. This set is deductively closed and will be, in general, an extension of S. We prove that these theories can be naturally axiomatized in terms of progressions of iterated local reflection principles. In particular, the set of provably 1-provable sentences of Peano arithmetic $PA$ can be axiomatized by ${\varepsilon _0}$ times iterated local reflection schema over $PA$. Our characterizations yield additional information on the proof-theoretic strength of these theories (w.r.t. various measures of it) and on their axiomatizability. We also study the question of speed-up of proofs and show that in some cases a proof of n-provability of a sentence can be much shorter than its proof from iterated reflection principles.
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Adamowicz, Zofia. "On maximal theories." Journal of Symbolic Logic 56, no. 3 (September 1991): 885–90. http://dx.doi.org/10.2307/2275057.
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Let S be a recursive theory. Let a theory T* consisting of Σ1 sentences be called maximal (with respect to S) if T* is maximal consistent with S, i.e. there is no Σ1 sentence consistent with T* + S which is not in T*.A maximal theory with respect to IΔ0 was considered by Wilkie and Paris in [WP] in connection with the end-extension problem.Let us recall that IΔ0 is the fragment of Peano arithmetic consisting of the finite collection of algebraic axioms PA− together with the induction scheme restricted to bounded formulas.The main open problem concerning the end-extendability of models of IΔ0 is the following:(*) Does every model of IΔ0 + BΣ1 have a proper end-extension to a model of IΔ0?Here BΣ1 is the following collection scheme:where φ runs over bounded formulas and may contain parameters.It is well known(see [KP]) that if I is a proper initial segment of a model M of IΔ0, then I satisfies IΔ0 + BΣ1.For a wide discussion of the problem (*) see [WP]. Wilkie and Paris construct in [WP] a model M of IΔ0 + BΣ1 which has no proper end-extension to a model of IΔ0 under the assumption IΔ0 ⊢¬Δ0H (see [WP] for an explanation of this assumption). Their model M is a model of a maximal theory T* where S = IΔ0.Moreover, T*, which is the set Σ1(M) of all Σ1 sentences true in M, is not codable in M.
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Yang, Yue. "The thickness lemma from P− + IΣ1 + ¬BΣ2." Journal of Symbolic Logic 60, no. 2 (June 1995): 505–11. http://dx.doi.org/10.2307/2275845.
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Let P− denote the Peano axioms minus the induction scheme. Let IΣn, (I∏n), BΣn (B∏n), LΣn (L∏n denote the induction scheme, the collection scheme, and the least number principle for Σn-(∏n-) formulas respectively. Paris and Kirby [3] studied the relative proof-theoretic strengths of those schemes. The general theorem states that IΣn, I∏n, LΣn, and L∏n are equivalent; IΣn implies BΣn implies IΣn–1; but not conversely.In recent years, people have been interested in doing recursion theory on fragments of arithmetic. One of the purposes of this study is to understand the priority methods. Much work has been done in this area. For example, M. Mytilinaios [5] showed that the Sacks splitting theorem can be proven in P− + IΣ1. Later, J. Mourad showed that the Sacks splitting theorem is indeed equivalent to IΣ1 [4]. M. Groszek and M. Mytilinaios [1] showed that P− + IΣ2 is sufficient to prove the existence of a high incomplete r.e. set. On the other hand, M. Mytilinaios and T. Slaman [6] showed that P− + IΣ1 is too weak to prove the existence of such a set. A natural question to ask is if the existence of such a set implies IΣ2. In this paper, we will show the answer is negative by constructing a model of P− + IΣ1 + ¬BΣ2 which has a high incomplete r.e. set. Notice that, as shown by M. Groszek and T. Slaman in [2], P− + IΣ1 is too weak to show the transitivity of weak Turing reducibility on Σ2-sets.
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Enayat, Ali. "Conservative extensions of models of set theory and generalizations." Journal of Symbolic Logic 51, no. 4 (December 1986): 1005–21. http://dx.doi.org/10.2307/2273912.
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An attempt to answer the following question gave rise to the results of the present paper. Let be an arbitrary model of set theory. Does there exist an elementary extension of satisfying the two requirements: (1) contains an ordinal exceeding all the ordinals of ; (2) does not enlarge any (hyper) integer of ? Note that a trivial application of the ordinary compactness theorem produces a model satisfying condition (1); and an internal ultrapower modulo an internal ultrafilter produces a model satisfying condition (2) (but not (1), because of the axiom of replacement). Also, such a satisfying both conditions (1) and (2) exists if the external cofinality of the ordinals of is countable, since by [KM], would then have an elementary end extension.Using a class of models constructed by M. Rubin using in [RS], and already employed in [E1], we prove that our question in general has a negative answer (see Theorem 2.3). This result generalizes the results of M. Kaufmann and the author (appearing respectively in [Ka] and [E1]) concerning models of set theory with no elementary end extensions.In the course of the proof it was necessary to establish that all conservative extensions (see Definition 2.1) of models of ZF must be cofinal. This is in direct contrast with the case of Peano arithmetic where all conservative extensions are end extensional (as observed by Phillips in [Ph1]). This led the author to introduce two useful weakenings of the notion of a conservative end extension which, as shown by the “completeness” theorems in §3, can exist.
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Meyer, Robert, and Chris Mortensen. "Inconsistent Models for Relevant Arithmetics." Australasian Journal of Logic 18, no. 5 (July 21, 2021): 380–400. http://dx.doi.org/10.26686/ajl.v18i5.6908.
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This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6] (see also Routley [10] and Asenjo [11]). In those works, Peano’s axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R♯, was absolutely consistent. It was pointed out that such a result escapes incau- tious formulations of Goedel’s second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as a model arithmetic modulo two. Modulo arithmetics are not or- dinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3 (see e.g. [1] or [8]), which has the advantage that while it is an extension of R, it is finite valued and so easier to handle.
The resulting model-theoretic structure (called in this paper RM32) is interesting in its own right in that the set of sentences true therein consti- tutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R♯. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P♯. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli (called RM3i here). We first study the properties of these arithmetics in this paper. The study is then generalised by vary- ing the logical base, to give the arithmetics RMni, of logical base RMn and modulus i. Not all of these exist, however, as arithmetical properties and logical properties interact, as we will show. The arithmetics RMni give rise, on intersection, to an inconsistent arithmetic RMω which is not of modulo i for any i. We also study its properties, and, among other results, we show by finitistic means that the more natural relevant arithmetics R♯ and R♯♯ are incomplete (whether or not consistent and recursively enumerable). In the rest of the paper we apply these techniques to several topics, particularly relevant quantum arithmetic in which we are able to show (unlike classical quantum arithmetic) that the law of distribution remains unprovable. Aside from its intrinsic interest, we regard the present exercise as a demonstration that inconsistent theories and models are of mathematical worth and interest.
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Paris, J. B., A. J. Wilkie, and A. R. Woods. "Provability of the pigeonhole principle and the existence of infinitely many primes." Journal of Symbolic Logic 53, no. 4 (December 1988): 1235–44. http://dx.doi.org/10.1017/s0022481200028061.
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In this note we shall be interested in the following problems.Problem 1. Can IΔ0 ⊢ ∀x∃y > x(y is prime)?Here I Δ0 is Peano arithmetic with the induction axiom restricted to bounded (i.e. Δ0) formulae.Problem 2. Can IΔ0 ⊢ Δ0 PHP?Here Δ0 PHP (Δ0 pigeonhole principle) is the schemafor θ ∈ Δ0, or equivalently in IΔ0, for a Δ0 formula F(x,y)written .By obtaining partial solutions to Problem 2 we shall show that Problem 1 has a positive solution if IΔ0 is replaced by IΔ0 + ∀xxlog(x) exists.Our notation will be entirely standard (see for example [3] and [4]). In particular all logarithms will be to the base 2 and in expressions like log(x), (1 + ε)x, etc. we shall always mean the integer part of these quantities.Concerning Problem 2 we remark that it is shown in [5] that for k ∈ N and F ∈ Δ0,As far as we know this is the best result of this form, in that we do not know how to replace log(z)k by anything larger. However, as we shall show in Theorem 1, we can do much better if we increase the difference between the sizes of the domain and range of F.In what follows let M be a countable nonstandard model of IΔ0, and let be those subsets of M defined by Δ0 formulae with parameters from M.Theorem 1. For k ∈ N andF ∈ Δ0,Here log0(x) = x, logk + 1(x) = log(logk(x)).Proof. To simplify matters, consider first the case k = 1. So assume M ⊨ alog(a) exists and with and a > 1. The idea of the proof is the following.
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Andreas, Holger, and Georg Schiemer. "Modal Structuralism with Theoretical Terms." Erkenntnis, May 8, 2021. http://dx.doi.org/10.1007/s10670-021-00378-w.
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AbstractIn this paper, we aim to explore connections between a Carnapian semantics of theoretical terms and an eliminative structuralist approach in the philosophy of mathematics. Specifically, we will interpret the language of Peano arithmetic by applying the modal semantics of theoretical terms introduced in Andreas (Synthese 174(3):367–383, 2010). We will thereby show that the application to Peano arithmetic yields a formal semantics of universal structuralism, i.e., the view that ordinary mathematical statements in arithmetic express general claims about all admissible interpretations of the Peano axioms. Moreover, we compare this application with the modal structuralism by Hellman (Mathematics without numbers: towards a modal-structural interpretation. Oxford University Press: Oxford, 1989), arguing that it provides us with an easier epistemology of statements in arithmetic.
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D’Aquino, Paola, Jamshid Derakhshan, and Angus Macintyre. "Truncations of ordered abelian groups." Algebra universalis 82, no. 2 (April 6, 2021). http://dx.doi.org/10.1007/s00012-021-00717-6.
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AbstractWe give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a $$+$$ + , but the addition of a TOAG is not necessarily even a cancellative semigroup. The main examples are initial segments $$[0, \tau ]$$ [ 0 , τ ] of an ordered abelian group, with a truncation of the addition. We prove that any model of these axioms (i.e. a truncated ordered abelian group) is an initial segment of an ordered abelian group. We define Presburger TOAG’s, and give a criterion for a TOAG to be a Presburger TOAG, and for two Presburger TOAG’s to be elementarily equivalent, proving analogues of classical results on Presburger arithmetic. Their main interest for us comes from the model theory of certain local rings which are quotients of valuation rings valued in a truncation [0, a] of the ordered group $${\mathbb {Z}}$$ Z or more general ordered abelian groups, via a study of these truncations without reference to the ambient ordered abelian group. The results are used essentially in a forthcoming paper (D’Aquino and Macintyre, The model theory of residue rings of models of Peano Arithmetic: The prime power case, 2021, arXiv:2102.00295) in the solution of a problem of Zilber about the logical complexity of quotient rings, by principal ideals, of nonstandard models of Peano arithmetic.
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Kohlenbach, Ulrich. "On the Uniform Weak König’s Lemma." BRICS Report Series 6, no. 11 (January 11, 1999). http://dx.doi.org/10.7146/brics.v6i11.20068.
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The so-called weak K¨onig's lemma WKL asserts the existence of an infinite<br />path b in any infinite binary tree (given by a representing function f). Based on<br />this principle one can formulate subsystems of higher-order arithmetic which<br />allow to carry out very substantial parts of classical mathematics but are PI^0_2-conservative<br />over primitive recursive arithmetic PRA (and even weaker fragments of arithmetic). In [10] we established such conservation results relative to finite type extensions PRA^omega of PRA (together with a quantifier-free axiom of choice schema). In this setting one can consider also a uniform version UWKL of WKL which asserts the existence of a functional Phi which selects uniformly in a given infinite binary tree f an infinite path Phi f of that tree.<br />This uniform version of WKL is of interest in the context of explicit mathematics as developed by S. Feferman. The elimination process in [10] actually can be used to eliminate even this uniform weak K¨onig's lemma provided that PRA^omega only has a quantifier-free rule of extensionality QF-ER instead of the full axioms (E) of extensionality for all finite types. In this paper we show that in the presence of (E), UWKL is much stronger than WKL: whereas WKL remains to be Pi^0_2 -conservative over PRA, PRA^omega +(E)+UWKL contains (and is conservative over) full Peano arithmetic PA.
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Guzelian, Christopher P. "Testing Economic Theory." REVISTA PROCESOS DE MERCADO, October 15, 2018, 303–13. http://dx.doi.org/10.52195/pm.v15i2.54.
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Two years ago, Bob Mulligan and I empirically tested whether the Bank of Amsterdam, a prototypical central bank, had caused a boom-bust cycle in the Amsterdam commodities markets in the 1780s owing to the bank’s sudden initiation of low-fractional-re-serve banking (Guzelian & Mulligan 2015).1 Widespread criticism came quickly after we presented our data findings at that year’s Austrian Economic Research Conference. Walter Block representa-tively responded: «as an Austrian, I maintain you cannot «test» apodictic theories, you can only illustrate them».2
Non-Austrian, so-called «empirical» economists typically have no problem with data-driven, inductive research. But Austrians have always objected strenuously on ontological and epistemolog-ical grounds that such studies do not produce real knowledge (Mises 1998, 113-115; Mises 2007). Camps of economists are talking past each other in respective uses of the words «testing» and «eco-nomic theory». There is a vital distinction between «testing» (1) an economic proposition, praxeologically derived, and (2) the rele-vance of an economic proposition, praxeologically derived. The former is nonsensical; the latter may be necessary to acquire eco-nomic theory and knowledge. Clearing up this confusion is this note’s goal.
Rothbard (1951) represents praxeology as the indispensible method for gaining economic knowledge. Starting with a Aristote-lian/Misesian axiom «humans act» or a Hayekian axiom of «humans think», a voluminous collection of logico-deductive eco-nomic propositions («theorems») follows, including theorems as sophisticated and perhaps unintuitive as the one Mulligan and I examined: low-fractional-reserve banking causes economic cycles.
There is an ontological and epistemological analog between Austrian praxeology and mathematics. Much like praxeology, we «know» mathematics to be «true» because it is axiomatic and deductive. By starting with Peano Axioms, mathematicians are able by a long process of creative deduction, to establish the real number system, or that for the equation an + bn = cn, there are no integers a, b, c that satisfy the equation for any integer value of n greater than 2 (Fermat’s Last Theorem).
But what do mathematicians mean when they then say they have mathematical knowledge, or that they have proven some-thing «true»? Is there an infinite set of rational numbers floating somewhere in the physical universe? Naturally no. Mathemati-cians mean that they have discovered an apodictic truth — some-thing unchangeably true without reference to physical reality because that truth is a priori.
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Ratsa, M. F. "A formal reduction of the general problem of expressibility of formulas in the Gödel-Löb provability logic." Discrete Mathematics and Applications 12, no. 3 (January 1, 2002). http://dx.doi.org/10.1515/dma-2002-0308.
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AbstractIt is a well-known idea to embed the intuitionistic logic into the modal logic in order to interprete the modality 'provable' as the deducibility in the Peano arithmetics with also well-known difficulties. P. M. Solovay and A. V. Kuznetsov introduced into consideration the Gödel-Löb provability logic whose formulas are constructed from propositional variables with the use of the connectives &, v, ⊃, ¬, and Δ (Gödelised provability). This logic is defined by the classic propositional calculus complemented by the three Δ-axiomsΔ (p ⊃ q) ⊃ (Δp ⊃ Δq), Δ(Δp ⊃ Δp) Δp, Δp ⊃ ΔΔp,and the extension rule (the Gödel rule). A formula F is called (functionally) expressible in terms of a system of formulas ∑ in logic L if, on the base of ∑ and variables, it is possible to obtain F with the use of the weakened substitution rule and the rule of change by equivalent in L. The general problem of expressibility in a logic L requires to give an algorithm which for any formula F and any finite system of formulas ∑ recognises whether F is expressible in terms of ∑ in L.In this paper, it is proved that in the Gödel-Löb provability logic and in many of extensions of this logic there is no algorithm which recognises the expressibility. In other words, the general expressibility problem is algorithmically undecidable in these logics.