Results for 'models of arithmetic'

954 found
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  1.  79
    A recursive model for arithmetic with weak induction.Zofia Adamowicz & Guillermo Morales-Luna - 1985 - Journal of Symbolic Logic 50 (1):49-54.
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  2.  38
    On Constructing Models for Arithmetic.Dana Scott - 1973 - Journal of Symbolic Logic 38 (2):336-337.
  3.  51
    Inconsistent Models (and Infinite Models) for Arithmetics with Constructible Falsity.Thomas Macaulay Ferguson - 2019 - Logic and Logical Philosophy 28 (3):389-407.
    An earlier paper on formulating arithmetic in a connexive logic ended with a conjecture concerning C♯, the closure of the Peano axioms in Wansing’s connexive logic C. Namely, the paper conjectured that C♯ is Post consistent relative to Heyting arithmetic, i.e., is nontrivial if Heyting arithmetic is nontrivial. The present paper borrows techniques from relevant logic to demonstrate that C♯ is Post consistent simpliciter, rendering the earlier conjecture redundant. Given the close relationship between C and Nelson’s paraconsistent (...)
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  4.  20
    (1 other version)Models for Substructural Arithmetics.Greg Restall - 2010 - Australasian Journal of Logic 8:82-99.
    This paper explores models for arithmetic in substructural logics. In the existing literature on substructural arithmetic, frame semantics for substructural logics are absent. We will start to fill in the picture in this paper by examining frame semantics for the substructural logics C (linear logic plus distribution), R (relevant logic) and CK (C plus weakening). The eventual goal is to find negation complete models for arithmetic in R.
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  5.  44
    Structures elementarily closed relative to a model for arithmetic.Eugene W. Madison - 1968 - Journal of Symbolic Logic 33 (1):101-104.
  6.  84
    On certain types and models for arithmetic.Andreas Blass - 1974 - Journal of Symbolic Logic 39 (1):151-162.
    There is an analogy between concepts such as end-extension types and minimal types in the model theory of Peano arithmetic and concepts such as P-points and selective ultrafilters in the theory of ultrafilters on N. Using the notion of conservative extensions of models, we prove some theorems clarifying the relation between these pairs of analogous concepts. We also use the analogy to obtain some model-theoretic results with techniques originally used in ultrafilter theory. These results assert that every countable (...)
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  7.  50
    A model for intuitionistic non-standard arithmetic.Ieke Moerdijk - 1995 - Annals of Pure and Applied Logic 73 (1):37-51.
    This paper provides an explicit description of a model for intuitionistic non-standard arithmetic, which can be formalized in a constructive metatheory without the axiom of choice.
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  8.  96
    (1 other version)Inconsistent models for relevant arithmetics.Robert Meyer & Chris Mortensen - 1984 - Journal of Symbolic Logic 49 (3):917-929.
    This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6]. 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 (...)
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  9.  24
    Arithmetic Models for Formal Systems.Hao Wang - 1955 - Journal of Symbolic Logic 20 (1):76-77.
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  10. Model-theoretic properties characterizing Peano arithmetic.Richard Kaye - 1991 - Journal of Symbolic Logic 56 (3):949-963.
    Let= {0,1, +,·,<} be the usual first-order language of arithmetic. We show that Peano arithmetic is the least first-order-theory containingIΔ0+ exp such that every complete extensionTof it has a countable modelKsatisfying(i)Khas no proper elementary substructures, and(ii) wheneverL≻Kis a countable elementary extension there isandsuch that.Other model-theoretic conditions similar to (i) and (ii) are also discussed and shown to characterize Peano arithmetic.
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  11.  15
    Model Theory and Non-Standard Arithmetic.A. Robinson - 1970 - Journal of Symbolic Logic 35 (1):149-149.
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  12. Some models for intuitionistic finite type arithmetic with Fan functional.A. S. Troelstra - 1977 - Journal of Symbolic Logic 42 (2):194-202.
    In this note we shall assume acquaintance with [T4] and the parts of [T1] which deal with intuitionistic arithmetic in all finite types. The bibliography just continues the bibliography of [T4].The principal purpose of this note is the discussion of two models for intuitionistic finite type arithmetic with fan functional. The first model is needed to correct an oversight in the proof of Theorem 6 [T4, §5]: the model ECF+as defined there cannot be shown to have the (...)
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  13.  47
    The arithmetic of cuts in models of arithmetic.Richard Kaye - 2013 - Mathematical Logic Quarterly 59 (4-5):332-351.
    We present a number of results on the structure of initial segments of models of Peano arithmetic with the arithmetic operations of addition, subtraction, multiplication, division, exponentiation and logarithm. Each of the binary operations introduced is defined in two dual ways, often with quite different results, and we attempt to systematise the issues and show how various calculations may be carried out. To understand the behaviour of addition and subtraction we introduce a notion of derivative on cuts, (...)
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  14. The Intended Model of Arithmetic. An Argument from Tennenbaum's Theorem.Paula Quinon & Konrad Zdanowski - 2006
     
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  15.  21
    Neutrally expandable models of arithmetic.Athar Abdul‐Quader & Roman Kossak - 2019 - Mathematical Logic Quarterly 65 (2):212-217.
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  16. Weak Arithmetics and Kripke Models.Morteza Moniri - 2002 - Mathematical Logic Quarterly 48 (1):157-160.
    In the first section of this paper we show that i Π1 ≡ W⌝⌝lΠ1 and that a Kripke model which decides bounded formulas forces iΠ1 if and only if the union of the worlds in any path in it satisflies IΠ1. In particular, the union of the worlds in any path of a Kripke model of HA models IΠ1. In the second section of the paper, we show that for equivalence of forcing and satisfaction of Πm-formulas in a linear (...)
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  17. Real closed fields and models of arithmetic (vol 75, pg 1, 2010).P. D'Aquino, J. F. Knight & S. Starchenko - 2012 - Journal of Symbolic Logic 77 (2).
  18.  64
    Hyper-Slingshot. Is Fact-Arithmetic Possible?Wojciech Krysztofiak - 2015 - Foundations of Science 20 (1):59-76.
    The paper presents a new argument supporting the ontological standpoint according to which there are no mathematical facts in any set theoretic model of arithmetical theories. It may be interpreted as showing that it is impossible to construct fact-arithmetic. The importance of this conclusion arises in the context of cognitive science. In the paper, a new type of slingshot argument is presented, which is called hyper-slingshot. The difference between meta-theoretical hyper-slingshots and conventional slingshots consists in the fact that the (...)
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  19.  52
    Discernible elements in models for peano arithmetic.Andrzej Ehrenfeucht - 1973 - Journal of Symbolic Logic 38 (2):291-292.
  20.  47
    A model of ZF + there exists an inaccessible, in which the dedekind cardinals constitute a natural non-standard model of arithmetic.Gershon Sageev - 1981 - Annals of Mathematical Logic 21 (2):221-281.
  21.  28
    Wang Hao. Arithmetic models for formal systems. Methodos, vol. 3 , pp. 217–232.Solomon Feferman - 1955 - Journal of Symbolic Logic 20 (1):76-77.
  22. Nonstandard Models of Peano Arithmetic.S. Kochen & Saul A. Kripke - 1982 - L’Enseignement Mathematique (3-4):211-231.
     
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  23.  46
    Non-standard numbers: a semantic obstacle for modelling arithmetical reasoning.Anderson De Araújo & Walter Carnielli - 2012 - Logic Journal of the IGPL 20 (2):477-485.
    The existence of non-standard numbers in first-order arithmetics is a semantic obstacle for modelling our arithmetical skills. This article argues that so far there is no adequate approach to overcome such a semantic obstacle, because we can also find out, and deal with, non-standard elements in Turing machines.
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  24.  20
    Models of Propositional Calculi in Recursive Arithmetic.R. L. Goodstein - 1963 - Journal of Symbolic Logic 28 (4):291-291.
  25.  82
    Do dynamical reduction models imply that arithmetic does not apply to ordinary macroscopic objects?G. C. Ghirardi & A. Bassi - 1999 - British Journal for the Philosophy of Science 50 (1):49-64.
    We analyse a recent paper in which an alleged devastating criticism of the so called GRW proposal to account for the objectification of the properties of macroscopic systems has been presented and we show that the author has not taken into account the precise implications of the GRW theory. This fact makes his conclusions basically wrong. We also perform a survey of measurement theory aimed to focus better on the physical and the conceptual aspects of the so-called macro-objectification problem.
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  26.  62
    Corrigendum to“Weak Arithmetics and Kripke Models”.Morteza Moniri - 2004 - Mathematical Logic Quarterly 50 (6):637-638.
    We give a corrected proof of the main result in the paper [2] mentioned in the title.
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  27. Comparing Peano arithmetic, Basic Law V, and Hume’s Principle.Sean Walsh - 2012 - Annals of Pure and Applied Logic 163 (11):1679-1709.
    This paper presents new constructions of models of Hume's Principle and Basic Law V with restricted amounts of comprehension. The techniques used in these constructions are drawn from hyperarithmetic theory and the model theory of fields, and formalizing these techniques within various subsystems of second-order Peano arithmetic allows one to put upper and lower bounds on the interpretability strength of these theories and hence to compare these theories to the canonical subsystems of second-order arithmetic. The main results (...)
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  28. Semantic Arithmetic: A Preface.John Corcoran - 1995 - Agora 14 (1):149-156.
    SEMANTIC ARITHMETIC: A PREFACE John Corcoran Abstract Number theory, or pure arithmetic, concerns the natural numbers themselves, not the notation used, and in particular not the numerals. String theory, or pure syntax, concems the numerals as strings of «uninterpreted» characters without regard to the numbe~s they may be used to denote. Number theory is purely arithmetic; string theory is purely syntactical... in so far as the universe of discourse alone is considered. Semantic arithmetic is a broad (...)
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  29.  15
    An Arithmetically Complete Predicate Modal Logic.Yunge Hao & George Tourlakis - 2021 - Bulletin of the Section of Logic 50 (4):513-541.
    This paper investigates a first-order extension of GL called \. We outline briefly the history that led to \, its key properties and some of its toolbox: the \emph{conservation theorem}, its cut-free Gentzenisation, the ``formulators'' tool. Its semantic completeness is fully stated in the current paper and the proof is retold here. Applying the Solovay technique to those models the present paper establishes its main result, namely, that \ is arithmetically complete. As expanded below, \ is a first-order modal (...)
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  30.  23
    Securing Arithmetical Determinacy.Sebastian G. W. Speitel - 2024 - Ergo: An Open Access Journal of Philosophy 11.
    The existence of non-standard models of first-order Peano-Arithmetic (PA) threatens to undermine the claim of the moderate mathematical realist that non-mysterious access to the natural number structure is possible on the basis of our best arithmetical theories. The move to logics stronger than FOL is denied to the moderate realist on the grounds that it merely shifts the indeterminacy “one level up” into the meta-theory by, illegitimately, assuming the determinacy of the notions needed to formulate such logics. This (...)
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  31.  77
    Arithmetical definability over finite structures.Troy Lee - 2003 - Mathematical Logic Quarterly 49 (4):385.
    Arithmetical definability has been extensively studied over the natural numbers. In this paper, we take up the study of arithmetical definability over finite structures, motivated by the correspondence between uniform AC0 and FO. We prove finite analogs of three classic results in arithmetical definability, namely that < and TIMES can first-order define PLUS, that < and DIVIDES can first-order define TIMES, and that < and COPRIME can first-order define TIMES. The first result sharpens the equivalence FO =FO to FO = (...)
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  32.  56
    (1 other version)The Bass-milnor-serre theorem for nonstandard models in peano arithmetic.Anatole Khelif - 1993 - Journal of Symbolic Logic 58 (4):1451-1458.
  33.  91
    Arithmetical and specular self-reference.Damjan Bojadžiev - 2004 - Acta Analytica 19 (33):55-63.
    Arithmetical self-reference through diagonalization is compared with self-recognition in a mirror, in a series of diagrams that show the structure and main stages of construction of self-referential sentences. A Gödel code is compared with a mirror, Gödel numbers with mirror images, numerical reference to arithmetical formulas with using a mirror to see things indirectly, self-reference with looking at one’s own image, and arithmetical provability of self-reference with recognition of the mirror image. The comparison turns arithmetical self-reference into an idealized model (...)
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  34.  93
    Compositionality in language and arithmetic.Carlos Montemayor & Fuat Balci - 2007 - Journal of Theoretical and Philosophical Psychology 27 (1):53-72.
    The lack of conceptual analysis within cognitive science results in multiple models of the same phenomena. However, these models incorporate assumptions that contradict basic structural features of the domain they are describing. This is particularly true about the domain of mathematical cognition. In this paper we argue that foundational theoretic aspects of psychological models for language and arithmetic should be clarified before postulating such models. We propose a means to clarify these foundational concepts by analyzing (...)
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  35. Modeling ancient and modern arithmetic practices: Addition and multiplication with Arabic and Roman numerals.Dirk Schlimm & Hansjörg Neth - 2008 - In B. C. Love, K. McRae & V. M. Sloutsky, Proceedings of the 30th Annual Conference of the Cognitive Science Society. Cognitive Science Society. pp. 2097--2102.
    To analyze the task of mental arithmetic with external representations in different number systems we model algorithms for addition and multiplication with Arabic and Roman numerals. This demonstrates that Roman numerals are not only informationally equivalent to Arabic ones but also computationally similar—a claim that is widely disputed. An analysis of our models' elementary processing steps reveals intricate tradeoffs between problem representation, algorithm, and interactive resources. Our simulations allow for a more nuanced view of the received wisdom on (...)
     
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  36.  16
    Model theory and its applications.Ralph Kopperman - 1972 - Boston,: Allyn & Bacon.
    In order to understand many of the proofs in this text the reader should have a one-semester background in set theory, including discussions of ordinal numbers, general a Cartesian products, cardinal arithmetic, and several forms of the axiom choice.
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  37. Model-checking CTL* over flat Presburger counter systems.Stéphane Demri, Alain Finkel, Valentin Goranko & Govert van Drimmelen - 2010 - Journal of Applied Non-Classical Logics 20 (4):313-344.
    This paper concerns model-checking of fragments and extensions of CTL* on infinite-state Presburger counter systems, where the states are vectors of integers and the transitions are determined by means of relations definable within Presburger arithmetic. In general, reachability properties of counter systems are undecidable, but we have identified a natural class of admissible counter systems (ACS) for which we show that the quantification over paths in CTL* can be simulated by quantification over tuples of natural numbers, eventually allowing translation (...)
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  38.  51
    Non‐Standard Models of Ordinal Arithmetics.E. A. Sonenberg - 1979 - Mathematical Logic Quarterly 25 (1-2):5-27.
  39.  36
    Forcing and satisfaction in Kripke models of intuitionistic arithmetic.Maryam Abiri, Morteza Moniri & Mostafa Zaare - 2019 - Logic Journal of the IGPL 27 (5):659-670.
    We define a class of first-order formulas $\mathsf{P}^{\ast }$ which exactly contains formulas $\varphi$ such that satisfaction of $\varphi$ in any classical structure attached to a node of a Kripke model of intuitionistic predicate logic deciding atomic formulas implies its forcing in that node. We also define a class of $\mathsf{E}$-formulas with the property that their forcing coincides with their classical satisfiability in Kripke models which decide atomic formulas. We also prove that any formula with this property is an (...)
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  40.  86
    Classical and Intuitionistic Models of Arithmetic.Kai F. Wehmeier - 1996 - Notre Dame Journal of Formal Logic 37 (3):452-461.
    Given a classical theory T, a Kripke model K for the language L of T is called T-normal or locally PA just in case the classical L-structure attached to each node of K is a classical model of T. Van Dalen, Mulder, Krabbe, and Visser showed that Kripke models of Heyting Arithmetic (HA) over finite frames are locally PA, and that Kripke models of HA over frames ordered like the natural numbers contain infinitely many PA-nodes. We show (...)
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  41.  80
    Internal Categoricity in Arithmetic and Set Theory.Jouko Väänänen & Tong Wang - 2015 - Notre Dame Journal of Formal Logic 56 (1):121-134.
    We show that the categoricity of second-order Peano axioms can be proved from the comprehension axioms. We also show that the categoricity of second-order Zermelo–Fraenkel axioms, given the order type of the ordinals, can be proved from the comprehension axioms. Thus these well-known categoricity results do not need the so-called “full” second-order logic, the Henkin second-order logic is enough. We also address the question of “consistency” of these axiom systems in the second-order sense, that is, the question of existence of (...)
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  42.  12
    Encoding true second‐order arithmetic in the real‐algebraic structure of models of intuitionistic elementary analysis.Miklós Erdélyi-Szabó - 2021 - Mathematical Logic Quarterly 67 (3):329-341.
    Based on the paper [4] we show that true second‐order arithmetic is interpretable over the real‐algebraic structure of models of intuitionistic analysis built upon a certain class of complete Heyting algebras.
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  43.  17
    (1 other version)The Recursively Saturated Part of Models of Peano Arithmetic.Henryk Kotlarski - 1986 - Mathematical Logic Quarterly 32 (19‐24):365-370.
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  44.  50
    The property “arithmetic-is-recursive” on a cone.Uri Andrews, Matthew Harrison-Trainor & Noah Schweber - 2021 - Journal of Mathematical Logic 21 (3):2150021.
    We say that a theory [Formula: see text] satisfies arithmetic-is-recursive if any [Formula: see text]-computable model of [Formula: see text] has an [Formula: see text]-computable copy; that is, the models of [Formula: see text] satisfy a sort of jump inversion. We give an example of a theory satisfying arithmetic-is-recursive non-trivially and prove that the theories satisfying arithmetic-is-recursive on a cone are exactly those theories with countably many [Formula: see text]-back-and-forth types.
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  45.  50
    Recursive Differentiation Arithmetic.Denys Spirin - manuscript
    This paper introduces Recursive Differentiation Arithmetic (RDA), a formal system that redefines the foundations of arithmetic, geometry, and computation in terms of ontological differentiation rather than set-theoretic or numerical primitives. Instead of assuming numbers, space, or time as given, RDA constructs these structures from stabilized differences within a field of potentiality. The basic elements of RDA are differentiation nodes, which emerge through recursive operations of unfolding and composition. We show how natural numbers arise as a special case of (...)
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  46.  19
    Alien Intruders in Relevant Arithmetic.Robert Meyer & Chris Mortensen - 2021 - Australasian Journal of Logic 18 (5):401-425.
    This paper explores the model theory of relevant arithmetic, emphasizing the structure of nonstandard natural numbers in the relevant arithmetic R#. In particular, the authors prove the “Alien Intruder Theorem” guaranteeing the existence of a model of R# including the rational numbers in which each rational acts as a nonstandard natural number. The authors conclude by considering some consequences of and open questions about the construction used in the theorem.
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  47. Does set theory really ground arithmetic truth?Alfredo Roque Freire - manuscript
    We consider the foundational relation between arithmetic and set theory. Our goal is to criticize the construction of standard arithmetic models as providing grounds for arithmetic truth (even in a relative sense). Our method is to emphasize the incomplete picture of both theories and treat models as their syntactical counterparts. Insisting on the incomplete picture will allow us to argue in favor of the revisability of the standard model interpretation. We then show that it is (...)
     
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  48.  53
    Richard Kaye. Models of Peano arithmetic. Oxford logic guides, no. 15. Clarendon Press, Oxford University Press, Oxford and New York1991, x + 292 pp. [REVIEW]C. Dimitracopoulos - 1993 - Journal of Symbolic Logic 58 (1):357-358.
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  49.  27
    On witnessed models in fuzzy logic III - witnessed Gödel logics.Petr Häjek - 2010 - Mathematical Logic Quarterly 56 (2):171-174.
    Gödel logics with truth sets being countable closed subsets of the unit real interval containing 0 and 1 are studied under their usual semantics and under the witnessed semantics, the latter admitting only models in which the truth value of each universally quantified formula is the minimum of truth values of its instances and dually for existential quantification and maximum. An infinite system of such truth sets is constructed such that under the usual semantics the corresponding logics have pairwise (...)
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  50.  94
    Models and Computability.W. Dean - 2014 - Philosophia Mathematica 22 (2):143-166.
    Computationalism holds that our grasp of notions like ‘computable function’ can be used to account for our putative ability to refer to the standard model of arithmetic. Tennenbaum's Theorem has been repeatedly invoked in service of this claim. I will argue that not only do the relevant class of arguments fail, but that the result itself is most naturally understood as having the opposite of a reference-fixing effect — i.e., rather than securing the determinacy of number-theoretic reference, Tennenbaum's Theorem (...)
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