Results for 'models of arithmetic'

960 found
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  1.  36
    On Constructing Models for Arithmetic.Dana Scott - 1973 - Journal of Symbolic Logic 38 (2):336-337.
  2.  77
    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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  3.  46
    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.  74
    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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  5.  19
    (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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  6.  36
    Structures elementarily closed relative to a model for arithmetic.Eugene W. Madison - 1968 - Journal of Symbolic Logic 33 (1):101-104.
  7.  92
    (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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  8.  48
    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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  9. 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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  10. 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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  11.  22
    Arithmetic Models for Formal Systems.Hao Wang - 1955 - Journal of Symbolic Logic 20 (1):76-77.
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  12.  15
    Model Theory and Non-Standard Arithmetic.A. Robinson - 1970 - Journal of Symbolic Logic 35 (1):149-149.
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  13.  48
    On Automorphisms of Resplendent Models of Arithmetic.Zofia Seremet - 1984 - Mathematical Logic Quarterly 30 (19-24):349-352.
  14. 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).
  15. 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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  16.  90
    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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  17.  21
    Neutrally expandable models of arithmetic.Athar Abdul‐Quader & Roman Kossak - 2019 - Mathematical Logic Quarterly 65 (2):212-217.
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  18.  47
    Discernible elements in models for peano arithmetic.Andrzej Ehrenfeucht - 1973 - Journal of Symbolic Logic 38 (2):291-292.
  19.  42
    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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  20.  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.
  21.  75
    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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  22.  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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  23.  13
    Models of Relevant Arithmetic.John Slaney - 2022 - Australasian Journal of Logic 19 (1).
    It is well known that the relevant arithmetic R# admits finite models whose domains are the integers modulo n rather than the expected natural numbers. Less well appreciated is the fact that the logic of these models is much more subtle than that of the three-valued structure in which they are usually presented. In this paper we consider the DeMorgan monoids in which R# can be modelled, deriving a fairly complete account of those modelling the stronger (...) RM# modulo n and a partial account for the case of R# modulo a prime number. The more general case in which the modulus is arbitrary is shown to lead to infinite propositional structures even with the additional constraint that '0=1' implies everything. (shrink)
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  24.  59
    A Model of the Arithmetic of Alephs in the Equation Calculus.R. J. Plymen - 1961 - Mathematical Logic Quarterly 7 (17-18):257-258.
  25.  42
    Corrigendum to: “Real closed fields and models of arithmetic”.P. D'Aquino, J. F. Knight & S. Starchenko - 2012 - Journal of Symbolic Logic 77 (2):726-726.
  26.  51
    (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.
  27.  18
    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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  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. 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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  30.  14
    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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  31.  41
    Andreas Blass. Near coherence of filters, 1: cofinal equivalence of models of arithmetic. Notre Dame journal of formal logic, vol. 27 , pp. 579–591. [REVIEW]H. B. Enderton - 1993 - Journal of Symbolic Logic 58 (1):350.
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  32. 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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  33.  10
    Remark on Relevant Arithmetic.Chris Mortensen - 2021 - Australasian Journal of Logic 18 (5):426-427.
    This is a brief note about the history of the analysis of the collection of theories, RM3modn, in Meyer and Mortensen "Inconsistent Models for Relevant Arithmetics" Journal of Symbolic Logic 49 (1984).
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  34.  91
    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.  73
    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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  36.  62
    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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  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. Notes on polynomially bounded arithmetic.Domenico Zambella - 1996 - Journal of Symbolic Logic 61 (3):942-966.
    We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general model-theoretical investigations on fragments of bounded arithmetic.
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  39.  78
    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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  40.  44
    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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  41.  25
    (1 other version)Nonstandard models in recursion theory and reverse mathematics.C. T. Chong, Wei Li & Yue Yang - forthcoming - Association for Symbolic Logic: The Bulletin of Symbolic Logic.
    We give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models. and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey's Theorem for Pairs.
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  42.  18
    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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  43. Whither relevant arithmetic?Harvey Friedman & Robert K. Meyer - 1992 - Journal of Symbolic Logic 57 (3):824-831.
    Based on the relevant logic R, the system R# was proposed as a relevant Peano arithmetic. R# has many nice properties: the most conspicuous theorems of classical Peano arithmetic PA are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that R# is properly weaker than PA, in the sense that there is a strictly positive theorem QRF of PA which is unprovable (...)
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  44.  82
    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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  45. 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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  46.  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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  47.  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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  48.  29
    A Model–Theoretic Approach to Proof Theory.Henryk Kotlarski - 2019 - Cham, Switzerland: Springer Verlag.
    This book presents a detailed treatment of ordinal combinatorics of large sets tailored for independence results. It uses model theoretic and combinatorial methods to obtain results in proof theory, such as incompleteness theorems or a description of the provably total functions of a theory. In the first chapter, the authors first discusses ordinal combinatorics of finite sets in the style of Ketonen and Solovay. This provides a background for an analysis of subsystems of Peano Arithmetic as well as for (...)
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  49.  82
    Inconsistent nonstandard arithmetic.Chris Mortensen - 1987 - Journal of Symbolic Logic 52 (2):512-518.
    This paper continues the investigation of inconsistent arithmetical structures. In $\S2$ the basic notion of a model with identity is defined, and results needed from elsewhere are cited. In $\S3$ several nonisomorphic inconsistent models with identity which extend the (=, $\S4$ inconsistent nonstandard models of the classical theory of finite rings and fields modulo m, i.e. Z m , are briefly considered. In $\S5$ two models modulo an infinite nonstandard number are considered. In the first, it is (...)
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  50.  72
    Diagnostic Models for Procedural Bugs in Basic Mathematical Skills.John Seely Brown & Richard R. Burton - 1978 - Cognitive Science 2 (2):155-192.
    A new diagnostic modeling system for automatically synthesizing a deep‐structure model of a student's misconceptions or bugs in his basic mathematical skills provides a mechanism for explaining why a student is making a mistake as opposed to simply identifying the mistake. This report is divided into four sections: The first provides examples of the problems that must be handled by a diagnostic model. It then introduces procedural networks as a general framework for representing the knowledge underlying a skill. The challenge (...)
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