57 found
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  1.  21
    Model Theory and the Philosophy of Mathematical Practice: Formalization Without Foundationalism.John T. Baldwin - 2018 - Cambridge University Press.
    Major shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also addresses the impact of (...)
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  2.  37
    Stable generic structures.John T. Baldwin & Niandong Shi - 1996 - Annals of Pure and Applied Logic 79 (1):1-35.
    Hrushovski originated the study of “flat” stable structures in constructing a new strongly minimal set and a stable 0-categorical pseudoplane. We exhibit a set of axioms which for collections of finite structure with dimension function δ give rise to stable generic models. In addition to the Hrushovski examples, this formalization includes Baldwin's almost strongly minimal non-Desarguesian projective plane and several others. We develop the new case where finite sets may have infinite closures with respect to the dimension function δ. In (...)
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  3.  50
    Examples of non-locality.John T. Baldwin & Saharon Shelah - 2008 - Journal of Symbolic Logic 73 (3):765-782.
    We use κ-free but not Whitehead Abelian groups to constructElementary Classes (AEC) which satisfy the amalgamation property but fail various conditions on the locality of Galois-types. We introduce the notion that an AEC admits intersections. We conclude that for AEC which admit intersections, the amalgamation property can have no positive effect on locality: there is a transformation of AEC's which preserves non-locality but takes any AEC which admits intersections to one with amalgamation. More specifically we have: Theorem 5.3. There is (...)
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  4.  21
    Categoricity.John T. Baldwin - 2009 - American Mathematical Society.
    CHAPTER 1 Combinatorial Geometries and Infinitary Logics In this chapter we introduce two of the key concepts that are used throughout the text. ...
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  5.  26
    Disjoint amalgamation in locally finite aec.John T. Baldwin, Martin Koerwien & Michael C. Laskowski - 2017 - Journal of Symbolic Logic 82 (1):98-119.
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  6.  86
    Axiomatizing Changing Conceptions of the Geometric Continuum I: Euclid-Hilbert†.John T. Baldwin - 2018 - Philosophia Mathematica 26 (3):346-374.
    We give a general account of the goals of axiomatization, introducing a variant on Detlefsen’s notion of ‘complete descriptive axiomatization’. We describe how distinctions between the Greek and modern view of number, magnitude, and proportion impact the interpretation of Hilbert’s axiomatization of geometry. We argue, as did Hilbert, that Euclid’s propositions concerning polygons, area, and similar triangles are derivable from Hilbert’s first-order axioms. We argue that Hilbert’s axioms including continuity show much more than the geometrical propositions of Euclid’s theorems and (...)
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  7.  32
    The Dividing Line Methodology: Model Theory Motivating Set Theory.John T. Baldwin - 2021 - Theoria 87 (2):361-393.
    We explore Shelah's model‐theoretic dividing line methodology. In particular, we discuss how problems in model theory motivated new techniques in model theory, for example classifying theories by their potential (consistently with Zermelo–Fraenkel set theory with the axiom of choice (ZFC)) spectrum of cardinals in which there is a universal model. Two other examples are the study (with Malliaris) of the Keisler order leading to a new ZFC result on cardinal invariants and attempts to clarify the “main gap” by reducing the (...)
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  8.  33
    As an abstract elementary class.John T. Baldwin, Paul C. Eklof & Jan Trlifaj - 2007 - Annals of Pure and Applied Logic 149 (1-3):25-39.
    In this paper we study abstract elementary classes of modules. We give several characterizations of when the class of modules A with is abstract elementary class with respect to the notion that M1 is a strong submodel M2 if the quotient remains in the given class.
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  9.  32
    The amalgamation spectrum.John T. Baldwin, Alexei Kolesnikov & Saharon Shelah - 2009 - Journal of Symbolic Logic 74 (3):914-928.
    We study when classes can have the disjoint amalgamation property for a proper initial segment of cardinals. Theorem A For every natural number k, there is a class $K_k $ defined by a sentence in $L_{\omega 1.\omega } $ that has no models of cardinality greater than $ \supset _{k - 1} $ , but $K_k $ has the disjoint amalgamation property on models of cardinality less than or equal to $\mathfrak{N}_{k - 3} $ and has models of cardinality $\mathfrak{N}_{k (...)
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  10.  26
    Hanf numbers for extendibility and related phenomena.John T. Baldwin & Saharon Shelah - 2022 - Archive for Mathematical Logic 61 (3):437-464.
    This paper contains portions of Baldwin’s talk at the Set Theory and Model Theory Conference and a detailed proof that in a suitable extension of ZFC, there is a complete sentence of \ that has maximal models in cardinals cofinal in the first measurable cardinal and, of course, never again.
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  11.  25
    First-order theories of abstract dependence relations.John T. Baldwin - 1984 - Annals of Pure and Applied Logic 26 (3):215-243.
  12.  83
    The stability spectrum for classes of atomic models.John T. Baldwin & Saharon Shelah - 2012 - Journal of Mathematical Logic 12 (1):1250001-.
    We prove two results on the stability spectrum for Lω1,ω. Here [Formula: see text] denotes an appropriate notion of Stone space of m-types over M. Theorem for unstable case: Suppose that for some positive integer m and for every α μ, K is not i-stable in μ. These results provide a new kind of sufficient condition for the unstable case and shed some light on the spectrum of strictly stable theories in this context. The methods avoid the use of compactness (...)
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  13.  53
    Axiomatizing Changing Conceptions of the Geometric Continuum II: Archimedes-Descartes-Hilbert-Tarski†.John T. Baldwin - 2019 - Philosophia Mathematica 27 (1):33-60.
    In Part I of this paper we argued that the first-order systems HP5 and EG are modest complete descriptive axiomatization of most of Euclidean geometry. In this paper we discuss two further modest complete descriptive axiomatizations: Tarksi’s for Cartesian geometry and new systems for adding $$\pi$$. In contrast we find Hilbert’s full second-order system immodest for geometrical purposes but appropriate as a foundation for mathematical analysis.
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  14. (1 other version)Constructing ω-stable structures: Rank 2 fields.John T. Baldwin & Kitty Holland - 2000 - Journal of Symbolic Logic 65 (1):371-391.
    We provide a general framework for studying the expansion of strongly minimal sets by adding additional relations in the style of Hrushovski. We introduce a notion of separation of quantifiers which is a condition on the class of expansions of finitely generated models for the expanded theory to have a countable ω-saturated model. We apply these results to construct for each sufficiently fast growing finite-to-one function μ from 'primitive extensions' to the natural numbers a theory T μ of an expansion (...)
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  15.  29
    Iterated elementary embeddings and the model theory of infinitary logic.John T. Baldwin & Paul B. Larson - 2016 - Annals of Pure and Applied Logic 167 (3):309-334.
  16.  50
    Constructing ω-stable structures: model completeness.John T. Baldwin & Kitty Holland - 2004 - Annals of Pure and Applied Logic 125 (1-3):159-172.
    The projective plane of Baldwin 695) is model complete in a language with additional constant symbols. The infinite rank bicolored field of Poizat 1339) is not model complete. The finite rank bicolored fields of Baldwin and Holland 371; Notre Dame J. Formal Logic , to appear) are model complete. More generally, the finite rank expansions of a strongly minimal set obtained by adding a ‘random’ unary predicate are almost strongly minimal and model complete provided the strongly minimal set is ‘well-behaved’ (...)
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  17.  66
    K‐generic Projective Planes have Morley Rank Two or Infinity.John T. Baldwin & Masanori Itai - 1994 - Mathematical Logic Quarterly 40 (2):143-152.
    We show that K-generic projective planes have Morley rank either two or infinity. We also show give a direct argument that such planes are not Desarguesian.
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  18. Almost strongly minimal theories. I.John T. Baldwin - 1972 - Journal of Symbolic Logic 37 (3):487-493.
  19.  22
    Almost galois ω-stable classes.John T. Baldwin, Paul B. Larson & Saharon Shelah - 2015 - Journal of Symbolic Logic 80 (3):763-784.
  20.  19
    Henkin constructions of models with size continuum.John T. Baldwin & Michael C. Laskowski - 2019 - Bulletin of Symbolic Logic 25 (1):1-33.
    We describe techniques for constructing models of size continuum inωsteps by simultaneously building a perfect set of enmeshed countable Henkin sets. Such models have perfect, asymptotically similar subsets. We survey applications involving Borel models, atomic models, two-cardinal transfers and models respecting various closure relations.
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  21.  40
    (1 other version)Model Companions of for Stable T.John T. Baldwin & Saharon Shelah - 2001 - Notre Dame Journal of Formal Logic 42 (3):129-142.
    We introduce the notion T does not omit obstructions. If a stable theory does not admit obstructions then it does not have the finite cover property (nfcp). For any theory T, form a new theory by adding a new unary function symbol and axioms asserting it is an automorphism. The main result of the paper asserts the following: If T is a stable theory, T does not admit obstructions if and only if has a model companion. The proof involves some (...)
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  22.  26
    A Hanf number for saturation and omission: the superstable case.John T. Baldwin & Saharon Shelah - 2014 - Mathematical Logic Quarterly 60 (6):437-443.
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  23.  57
    Completeness and categoricity (in power): Formalization without foundationalism.John T. Baldwin - 2014 - Bulletin of Symbolic Logic 20 (1):39-79.
    We propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models). We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘categoricity in power’ has enormous structural consequences for any of the theories satisfying it. The stability hierarchy extends (...)
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  24.  29
    Trivial pursuit: Remarks on the main gap.John T. Baldwin & Leo Harrington - 1987 - Annals of Pure and Applied Logic 34 (3):209-230.
  25.  63
    Notes on quasiminimality and excellence.John T. Baldwin - 2004 - Bulletin of Symbolic Logic 10 (3):334-366.
    This paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for L ω 1 ,ω (Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) (...)
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  26.  33
    Stability theory and algebra.John T. Baldwin - 1979 - Journal of Symbolic Logic 44 (4):599-608.
  27.  52
    Formalization, primitive concepts, and purity: Formalization, primitive concepts, and purity.John T. Baldwin - 2013 - Review of Symbolic Logic 6 (1):87-128.
    We emphasize the role of the choice of vocabulary in formalization of a mathematical area and remark that this is a particular preoccupation of logicians. We use this framework to discuss Kennedy’s notion of ‘formalism freeness’ in the context of various schools in model theory. Then we clarify some of the mathematical issues in recent discussions of purity in the proof of the Desargues proposition. We note that the conclusion of ‘spatial content’ from the Desargues proposition involves arguments which are (...)
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  28.  16
    Towards a finer classification of strongly minimal sets.John T. Baldwin & Viktor V. Verbovskiy - 2024 - Annals of Pure and Applied Logic 175 (2):103376.
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  29. Almost strongly minimal theories. II.John T. Baldwin - 1972 - Journal of Symbolic Logic 37 (4):657-660.
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  30.  51
    Diverse classes.John T. Baldwin - 1989 - Journal of Symbolic Logic 54 (3):875-893.
    Let $\mathbf{I}(\mu,K)$ denote the number of nonisomorphic models of power $\mu$ and $\mathbf{IE}(\mu,K)$ the number of nonmutually embeddable models. We define in this paper the notion of a diverse class and use it to prove a number of results. The major result is Theorem B: For any diverse class $K$ and $\mu$ greater than the cardinality of the language of $K$, $\mathbf{IE}(\mu,K) \geq \min(2^\mu,\beth_2).$ From it we deduce both an old result of Shelah, Theorem C: If $T$ is countable and (...)
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  31.  40
    Expansions of geometries.John T. Baldwin - 2003 - Journal of Symbolic Logic 68 (3):803-827.
    For $n < \omega$ , expand the structure (n, S, I, F) (with S the successor relation, I, F as the initial and final element) by forming graphs with edge probability n-α for irrational α, with $0 < \alpha < 1$ . The sentences in the expanded language, which have limit probability 1, form a complete and stable theory.
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  32.  28
    Some EC ∑ Classes of Rings.John T. Baldwin - 1978 - Mathematical Logic Quarterly 24 (31-36):489-492.
  33.  18
    Maximal models up to the first measurable in ZFC.John T. Baldwin & Saharon Shelah - 2023 - Journal of Mathematical Logic 24 (1).
    Theorem: There is a complete sentence [Formula: see text] of [Formula: see text] such that [Formula: see text] has maximal models in a set of cardinals [Formula: see text] that is cofinal in the first measurable [Formula: see text] while [Formula: see text] has no maximal models in any [Formula: see text].
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  34.  30
    Categoricity and generalized model completeness.G. Ahlbrandt & John T. Baldwin - 1988 - Archive for Mathematical Logic 27 (1):1-4.
  35.  51
    A model theoretic approach to malcev conditions.John T. Baldwin & Joel Berman - 1977 - Journal of Symbolic Logic 42 (2):277-288.
    A varietyV satisfies a strong Malcev condition ∃f1,…, ∃fnθ where θ is a conjunction of equations in the function variablesf1, …,fnand the individual variablesx1, …,xm, if there are polynomial symbolsp1, …,pnin the language ofVsuch that ∀x1, …,xmθ is a law ofV. Thus a strong Malcev condition involves restricted second order quantification of a strange sort. The quantification is restricted to functions which are “polynomially definable”. This notion was introduced by Malcev [6] who used it to describe those varieties all of (...)
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  36. A selection of papers presented at the" Stability in Model Theory III" conference.John T. Baldwin & Annalisa Marcja - 1993 - Annals of Pure and Applied Logic 62 (2).
  37. E-mail: marat@ niimm. kazan. su.John T. Baldwin & Masanori Itai - 1995 - Bulletin of Symbolic Logic 1 (1).
     
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  38.  15
    Images in Mathematics.John T. Baldwin - 2021 - Theoria 87 (4):913-936.
    Mathematical images occur in lectures, books, notes and posters, and on the internet. We extend Kennedy's proposal for classifying these images. In doing so we distinguish three uses of images in mathematics: iconic images; incidental images; and integral images. An iconic image is one that so captures the essence of a concept or proof that it serves for a community of mathematicians as a motto or a meme for an area or a result. A system such as Euclid's can combine (...)
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  39.  23
    John W. Rosenthal. A new proof of a theorem of Shelah. The journal of symbolic logic, vol. 37 , pp. 133–134.John T. Baldwin - 1973 - Journal of Symbolic Logic 38 (4):649.
  40.  16
    Preface.John T. Baldwin - 1993 - Annals of Pure and Applied Logic 62 (2):81.
  41.  15
    Preface.John T. Baldwin & Annalisa Marcja - 1989 - Annals of Pure and Applied Logic 45 (2):103.
  42.  36
    Problems in Set Theory, Mathematical Logic, and the Theory of Algorithms.John T. Baldwin - 2004 - Bulletin of Symbolic Logic 10 (2):222-223.
  43.  78
    Some contributions to definability theory for languages with generalized quantifiers.John T. Baldwin & Douglas E. Miller - 1982 - Journal of Symbolic Logic 47 (3):572-586.
  44.  38
    Saharon Shelah. There are just four second-order quantifiers. Israel journal of mathematics, vol. 15 , pp. 282–300.John T. Baldwin - 1986 - Journal of Symbolic Logic 51 (1):234.
  45.  21
    The metamathematics of random graphs.John T. Baldwin - 2006 - Annals of Pure and Applied Logic 143 (1-3):20-28.
    We explain and summarize the use of logic to provide a uniform perspective for studying limit laws on finite probability spaces. This work connects developments in stability theory, finite model theory, abstract model theory, and probability. We conclude by linking this context with work on the Urysohn space.
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  46.  47
    The spectrum of resplendency.John T. Baldwin - 1990 - Journal of Symbolic Logic 55 (2):626-636.
    Let T be a complete countable first order theory and λ an uncountable cardinal. Theorem 1. If T is not superstable, T has 2 λ resplendent models of power λ. Theorem 2. If T is strictly superstable, then T has at least $\min(2^\lambda,\beth_2)$ resplendent models of power λ. Theorem 3. If T is not superstable or is small and strictly superstable, then every resplendent homogeneous model of T is saturated. Theorem 4 (with Knight). For each μ ∈ ω ∪ {ω, (...)
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  47.  36
    (1 other version)Uncountable categoricity of local abstract elementary classes with amalgamation.John T. Baldwin & Olivier Lessmann - 2006 - Annals of Pure and Applied Logic 143 (1-3):29-42.
    We give a complete and elementary proof of the following upward categoricity theorem: let be a local abstract elementary class with amalgamation and joint embedding, arbitrarily large models, and countable Löwenheim–Skolem number. If is categorical in 1 then is categorical in every uncountable cardinal. In particular, this provides a new proof of the upward part of Morley’s theorem in first order logic without any use of prime models or heavy stability theoretic machinery.
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  48.  52
    Local Homogeneity.Bektur Baizhanov & John T. Baldwin - 2004 - Journal of Symbolic Logic 69 (4):1243 - 1260.
    We study the expansion of stable structures by adding predicates for arbitrary subsets. Generalizing work of Poizat-Bouscaren on the one hand and Baldwin-Benedikt-Casanovas-Ziegler on the other we provide a sufficient condition (Theorem 4.7) for such an expansion to be stable. This generalization weakens the original definitions in two ways: dealing with arbitrary subsets rather than just submodels and removing the 'small' or 'belles paires' hypothesis. We use this generalization to characterize in terms of pairs, the 'triviality' of the geometry on (...)
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  49.  34
    Subsets of Superstable Structures Are Weakly Benign.Bektur Baizhanov, John T. Baldwin & Saharon Shelah - 2005 - Journal of Symbolic Logic 70 (1):142 - 150.
  50.  58
    Meeting of the association for symbolic logic: Biloxi, 1979.Daniel Halpern, William Tait & John T. Baldwin - 1981 - Journal of Symbolic Logic 46 (1):191-198.
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