65 found
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  1. 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.  9
    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.  41
    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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  4.  7
    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.  16
    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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  6.  10
    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.
  7.  22
    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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  8.  7
    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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  9. Classification of Δ-Invariant Amalgamation Classes.Roman D. Aref'ev, John T. Baldwin & Marco Mazzucco - 1999 - Journal of Symbolic Logic 64 (4):1743-1750.
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  10.  49
    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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  11. Almost Strongly Minimal Theories. II.John T. Baldwin - 1972 - Journal of Symbolic Logic 37 (4):657-660.
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  12.  3
    Almost Galois Ω-Stable Classes.John T. Baldwin, Paul B. Larson & Saharon Shelah - 2015 - Journal of Symbolic Logic 80 (3):763-784.
  13.  3
    How Big Should the Monster Model Be? [REVIEW]John T. Baldwin - 2015 - In Andrés Villaveces, Roman Kossak, Juha Kontinen & Åsa Hirvonen (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics. De Gruyter. pp. 31-50.
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  14.  17
    Completeness and Categoricity : 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 as virtuous: the property must have significant mathematical consequences for the theory. 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 this virtue to other complete theories. The interaction of (...)
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  15. Almost Strongly Minimal Theories. I.John T. Baldwin - 1972 - Journal of Symbolic Logic 37 (3):487-493.
  16.  2
    Images in Mathematics.John T. Baldwin - forthcoming - Theoria.
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  17.  9
    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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  18.  19
    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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  19.  52
    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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  20.  6
    First-Order Theories of Abstract Dependence Relations.John T. Baldwin - 1984 - Annals of Pure and Applied Logic 26 (3):215-243.
  21.  9
    As an Abstract Elementary Class.John T. Baldwin, Paul C. Eklof & Jan Trlifaj - 2007 - Annals of Pure and Applied Logic 149 (1):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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  22.  27
    Model Companions of $T_{\Rm Aut}$ 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 . For any theory T, form a new theory $T_{\rm Aut}$ 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 $T_{\rm Aut}$ has a model companion. (...)
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  23.  19
    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.
  24.  47
    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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  25.  18
    Jane Bridge. Beginning Model Theory. The Completeness Theorem and Some Consequences. Oxford Logic Guides. Clarendon Press, Oxford1977, Viii + 143 Pp. [REVIEW]John T. Baldwin - 1979 - Journal of Symbolic Logic 44 (2):283.
  26.  15
    DOP and FCP in Generic Structures.John T. Baldwin & Saharon Shelah - 1998 - Journal of Symbolic Logic 63 (2):427-438.
  27.  12
    A Hanf Number for Saturation and Omission: The Superstable Case.John T. Baldwin & Saharon Shelah - 2014 - Mathematical Logic Quarterly 60 (6):437-443.
  28.  15
    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.
  29.  24
    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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  30.  17
    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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  31.  6
    Trivial Pursuit: Remarks on the Main Gap.John T. Baldwin & Leo Harrington - 1987 - Annals of Pure and Applied Logic 34 (3):209-230.
  32.  12
    The Metamathematics of Random Graphs.John T. Baldwin - 2006 - Annals of Pure and Applied Logic 143 (1):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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  33.  19
    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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  34.  22
    Transfering Saturation, the Finite Cover Property, and Stability.John T. Baldwin, Rami Grossberg & Saharon Shelah - 1999 - Journal of Symbolic Logic 64 (2):678-684.
    $\underline{\text{Saturation is} (\mu, \kappa)-\text{transferable in} T}$ if and only if there is an expansion T 1 of T with ∣ T 1 ∣ = ∣ T ∣ such that if M is a μ-saturated model of T 1 and ∣ M ∣ ≥ κ then the reduct M ∣ L(T) is κ-saturated. We characterize theories which are superstable without f.c.p., or without f.c.p. as, respectively those where saturation is (ℵ 0 , λ)- transferable or (κ (T), λ)-transferable for all λ. (...)
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  35.  5
    Stability Theory and Algebra.John T. Baldwin - 1979 - Journal of Symbolic Logic 44 (4):599-608.
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  36.  5
    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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  37.  17
    Some EC∑ Classes of Rings.John T. Baldwin - 1978 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 24 (31-36):489-492.
  38.  19
    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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  39.  4
    Upward Categoricity of Very Tame Abstract Elementary Classes with Amalgamation.John T. Baldwin & Olivier Lessmann - 2006 - Annals of Pure and Applied Logic 143 (1-3):29-42.
  40.  33
    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.
  41.  30
    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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  42.  15
    Problems in Set Theory, Mathematical Logic, and the Theory of Algorithms.John T. Baldwin - 2004 - Bulletin of Symbolic Logic 10 (2):222-223.
  43.  26
    The Vaught Conjecture: Do Uncountable Models Count?John T. Baldwin - 2007 - Notre Dame Journal of Formal Logic 48 (1):79-92.
    We give a model theoretic proof, replacing admissible set theory by the Lopez-Escobar theorem, of Makkai's theorem: Every counterexample to Vaught's Conjecture has an uncountable model which realizes only countably many ℒ$_{ω₁,ω}$-types. The following result is new. Theorem: If a first-order theory is a counterexample to the Vaught Conjecture then it has 2\sp ℵ₁ models of cardinality ℵ₁.
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  44.  32
    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.
  45.  15
    Categoricity and Generalized Model Completeness.G. Ahlbrandt & John T. Baldwin - 1988 - Archive for Mathematical Logic 27 (1):1-4.
  46.  10
    Reviews. Selected Papers of Abraham Robinson. Volume 1. Model Theory and Algebra. Edited and with an Introduction by H. J. Keisler. Yale University Press, New Haven and London 1979, Xxxvii + 694 Pp. George B. Selioman. Biography of Abraham Robinson, Pp. Xiii–Xxxii. H. J. Keisler. Introduction, Pp. Xxxiii–Xxxvii. Abraham Robinson. On the Application of Symbolic Logic to Algebra, Pp. 3–11. A Reprint of XVIII 182. Abraham Robinson. Recent Developments in Model Theory, Pp. 12–31. A Reprint of XL 269. Abraham Robinson. On the Construction of Models, Pp. 32–42. A Reprint of XL 506. Abraham Robinson, Metamathematical Problems, Pp. 43–59. , Pp. 500–516.) Abraham Robinson. Model Theory as a Framework for Algebra, Pp. 60–83. Abraham Robinson. A Result on Consistency and its Application to the Theory of Definition, Pp. 87–98. A Reprint of XXV 174. Abraham Robinson. Ordered Structures and Related Concepts, Pp. 99–104. A Reprint of XXV 170. [REVIEW]John T. Baldwin - 1982 - Journal of Symbolic Logic 47 (1):197-203.
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  47.  12
    Uncountable Categoricity of Local Abstract Elementary Classes with Amalgamation.John T. Baldwin & Olivier Lessmann - 2006 - Annals of Pure and Applied Logic 143 (1):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.  3
    Some EC∑ Classes of Rings.John T. Baldwin - 1978 - Mathematical Logic Quarterly 24 (31‐36):489-492.
  49.  14
    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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  50.  17
    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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