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Modern Model Theory the Impact on Mathematics and Philosophy

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Modern Model Theory the Impact on Mathematics and Philosophy Modern Model Theory The impact on mathematics and philosophy John T. Baldwin Modern Model Theory Prelude: From The impact on mathematics and philosophy Logic to Model Theory Classification Theory John T. Baldwin Fundamental Structures Exploring Cantor's Paradise June 1, 2010 The role of Syntax Goals Modern Model Theory The impact on mathematics and philosophy John T. Baldwin Modern model theory has introduced many fundamental Prelude: From Logic to notions with philosophical significance that have masqueraded Model Theory as technical mathematical notions. Classification Theory I will describe some of the themes of this work. Fundamental Structures Exploring Cantor's Paradise The role of Syntax Outline Modern Model Theory The impact on mathematics and philosophy 1 Prelude: From Logic to Model Theory John T. Baldwin 2 Classification Theory Prelude: From Logic to Model Theory Classification 3 Fundamental Structures Theory Fundamental Structures 4 Exploring Cantor's Paradise Exploring Cantor's Paradise The role of 5 The role of Syntax Syntax What is model theory? Modern Model Theory The impact on mathematics and philosophy John T. Baldwin Definition for mathematicians Prelude: From Logic to A model theorist is a self-conscious mathematician Model Theory Classification who uses formal languages and semantics to prove Theory mathematical theorems. Fundamental Structures Exploring Cantor's Paradise The role of Syntax Prelude: From Logic to Model Theory Modern Model Theory The impact on mathematics and philosophy John T. Baldwin The decades of the 50's and the 60's saw a transition from Prelude: From model theory as one aspect of the study of (first order) logic to Logic to Model Theory the introduction of an independent subject of model theory. Classification Theory This talk will expound the philosophical and mathematical Fundamental impacts of that shift. Structures Exploring Cantor's Paradise The role of Syntax (A natural variant allows for variables and formulas with free variables.) Model-Theoretic Logics Modern Model Theory The impact on mathematics and A vocabulary τ is a list of function, constant and relation philosophy symbols. John T. Baldwin A τ-structure A is a set with an interpretation of each symbol Prelude: From in τ. Logic to Model Theory A logic L is a pair (L; j=L) such that Classification Theory L(τ) is a collection of τ-sentences and for each φ 2 L(τ), Fundamental A j=L φ Structures satisfies natural properties. Exploring Cantor's Paradise The role of Syntax Model-Theoretic Logics Modern Model Theory The impact on mathematics and A vocabulary τ is a list of function, constant and relation philosophy symbols. John T. Baldwin A τ-structure A is a set with an interpretation of each symbol Prelude: From in τ. Logic to Model Theory A logic L is a pair (L; j=L) such that Classification Theory L(τ) is a collection of τ-sentences and for each φ 2 L(τ), Fundamental A j=L φ Structures satisfies natural properties. Exploring Cantor's Paradise (A natural variant allows for variables and formulas with free The role of variables.) Syntax Properties of first order logic (1915-1955): Modern Model Theory The impact on mathematics and philosophy John T. Baldwin 1 Completeness and Compactness Prelude: From Logic to 2 Lowenheim-Skolem Model Theory Classification 3 syntactic characterization of preservation theorems Theory Fundamental 4 Interpolation Structures Exploring Cantor's Paradise The role of Syntax Theories Modern Model Theory The impact on mathematics and philosophy John T. Axiomatic theories arise from two distinct motivations. Baldwin Prelude: From 1 To understand a single significant structure such as Logic to Model Theory (N; +; ·) or (R; +; ·). Classification Theory 2 To find the common characteristics of a number of Fundamental structures; theories of the second sort include groups, Structures rings, fields etc. Exploring Cantor's Paradise The role of Syntax The Genesis of Modern Model Theory (1955-1970): Modern Model Theory The impact on mathematics A. Robinson. and philosophy Complete Theories. John T. North Holland, Amsterdam, 1956. Baldwin A. Tarski and R.L. Vaught. Prelude: From Logic to Arithmetical extensions of relational systems. Model Theory Compositio Mathematica, 13:81{102, 1956. Classification Theory A. Ehrenfeucht and A. Mostowski. Fundamental Structures Models of axiomatic theories admitting automophisms. Exploring Fund. Math., 43:50{68, 1956. Cantor's Paradise B. J´onsson. The role of Syntax Homogeneous universal relational systems. Mathematica Scandinavica, 8:137{142, 1960. Categoricity in Power: Modern Model Theory The impact on mathematics and philosophy John T. J.Los. Baldwin On the categoricity in power of elementary deductive Prelude: From systems and related problems. Logic to Model Theory Colloquium Mathematicum, 3:58{62, 1954. Classification Theory M. Morley. Fundamental Categoricity in power. Structures Transactions of the American Mathematical Society, Exploring Cantor's 114:514{538, 1965. Paradise The role of Syntax New Paradigm Modern Model Theory The impact on mathematics S. Shelah. and philosophy Stability, the f.c.p., and superstability; model theoretic John T. properties of formulas in first-order theories. Baldwin Annals of Mathematical Logic, 3:271{362, 1970. Prelude: From Logic to J.T. Baldwin and A.H. Lachlan. Model Theory Classification On strongly minimal sets. Theory Journal of Symbolic Logic, 36:79{96, 1971. Fundamental Structures B.I. Zil'ber. Exploring Cantor's Uncountably Categorical Theories. Paradise Translations of the American Mathematical Society. The role of Syntax American Mathematical Society, 1991. summary of earlier work. Recording the shift: Modern Model Theory The impact on mathematics and philosophy John T. Baldwin Prelude: From S. Shelah. Logic to Model Theory Classification Theory and the Number of Nonisomorphic Classification Models. Theory North-Holland, 1978. Fundamental Structures Exploring Cantor's Paradise The role of Syntax foundationS of mathematics Modern Model Theory The impact on mathematics and philosophy John T. Thesis: Baldwin Studying the model of different(complete first order) theories Prelude: From Logic to provides a framework for the understanding of the foundations Model Theory of specific areas of mathematics. Classification Theory This study cannot be carried out by interpreting the theory into Fundamental Structures an ¨uber theory such as ZFC; too much information is lost. Exploring Cantor's Categorical foundations may also be local in this sense. Paradise The role of Syntax Algebraic examples: complete theories Modern Model Theory The impact on Algebraic Geometry mathematics and philosophy Algebraic geometry is the study of definable subsets of John T. algebraically closed fields Baldwin Not quite: definable by positive formulas Prelude: From Logic to REMEDY: Zariski Geometries Model Theory Classification Chevalley-Tarski Theorem Theory Fundamental Chevalley: The projection of a constructible set is constructible. Structures Tarski: Acf admits elimination of quantifiers. Exploring Cantor's Paradise More precisely, this describes `Weil' style algebraic geometry. The role of Syntax Macintyre and Hrushovski (App. to computing the Galois grp....) have contrasting views on the need for `categorical foundations'. Algebraic consequences for complete theories Modern Model Theory The impact on mathematics and philosophy John T. 1 Artin-Schreier theorem (A. Robinson) Baldwin 2 Decidability and qe of the real field (Tarski) Prelude: From Logic to 3 Decidability and qe of the complex field (Tarski) Model Theory Classification 4 Decidability and model completeness of valued fields Theory (Ax-Kochen-Ershov) Fundamental Structures 5 quantifier elimination for p-adic fields (Macintyre) Exploring Cantor's 6 o-minimality of the real exponential field (Wilkie) Paradise The role of Syntax Recursive axiomatizations of particular areas (e.g. algebraic geometry) restores the decidability sought by Hilbert (which G¨odelshows is impossible for global foundations.) Why complete theories? Modern Model Theory The impact on mathematics and philosophy John T. Baldwin Studying elementary extensions is an important tool for Prelude: From studying definability in a particular structure. Logic to Model Theory Classification Theory Fundamental Structures Exploring Cantor's Paradise The role of Syntax Why complete theories? Modern Model Theory The impact on mathematics and philosophy John T. Baldwin Studying elementary extensions is an important tool for Prelude: From studying definability in a particular structure. Logic to Model Theory Recursive axiomatizations of particular areas (e.g. algebraic Classification Theory geometry) restores the decidability sought by Hilbert Fundamental (which G¨odelshows is impossible for global foundations.) Structures Exploring Cantor's Paradise The role of Syntax Classification Theory Modern Model Theory A pun on classify The impact on mathematics and 1 A class of models of a theory admit a structure theory philosophy (can be classified) if there is a function assigning set John T. Baldwin theoretic invariants to each model which determine the Prelude: From model up to isomorphism. Logic to Model Theory 2 The theories of a logic can be classified if there is a schema Classification determining which theories admit a structure theory. Theory Fundamental Structures Dicta: Shelah Exploring Cantor's A powerful tool for classification is to establish dichotomies: a Paradise property Φ such that Φ implies non-structure and :Φ The role of Syntax contributes to a structure result.
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