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, |=L) such that Classification Theory L(τ) is a collection of τ-sentences and for each φ ∈ L(τ), Fundamental A |=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, |=L) such that Classification Theory L(τ) is a collection of τ-sentences and for each φ ∈ L(τ), Fundamental A |=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. dichotomies must have absolute/syntactic/internal form and an external semantic form (Dzjamonja) Classification Theory = Model Theory
Modern Model Theory The impact on mathematics and philosophy
John T. Baldwin In a review of Classification Theory Poizat wrote:
Prelude: From ‘Possibly, the editor found it too delicate to use a more Logic to Model Theory ambitious term: this could give an impression of arrogance, or Classification of a will to exclude people working in more traditional matters Theory
Fundamental (...) . Those of us who are free from that kind of scruple will Structures restore to this book the only name it deserves: Model theory.’ Exploring Cantor’s Paradise
The role of Syntax Properties of classes of theories (1970-present)
Modern Model Theory The stability hierarchy is defined by syntactical conditions on The impact on mathematics theories or (equivalently) by counting types over models of and philosophy different cardinalities. John T. Baldwin The Stability Hierarchy
Prelude: From Every complete first order theory falls into one of the following Logic to Model Theory 4 classes. Classification Theory 1 ω-stable Fundamental 2 superstable but not ω-stable Structures
Exploring 3 stable but not superstable Cantor’s Paradise 4 unstable The role of Syntax Two other classes, simple, and NIP (dependent) have been seen recently to have important structural consequences. The stability hierarchy: examples
Modern Model Theory ω-stable The impact on mathematics Algebraically closed fields (fixed characteristic), differentially and philosophy closed fields, complex compact manifolds John T. Baldwin strictly superstable Prelude: From Logic to ω ω Model Theory (Z, +), (2 , +) = (Z2 , Hi )i<ω,
Classification Theory strictly stable Fundamental Structures (Z, +)ω, separably closed fields, random graphs with edge prob Exploring −α Cantor’s n (α irrational Paradise
The role of Syntax unstable Arithmetic, Real closed fields, complex exponentiation, random graph More than Taxonomy
Modern Model Theory The impact on mathematics and philosophy This classification and the tools originally developed for the John T. Baldwin spectrum problem have had profound developments and
Prelude: From consequences across mathematics: Logic to Model Theory 1 Hrushovski’s work on Mordell-Lang Classification Theory 2 Study of differential fields and difference fields Fundamental Structures 3 compact complex varieties are ω-stable. Exploring 4 o-minimality– solution of a problem of Hardy Cantor’s Paradise
The role of Syntax Dimension Theory
Modern Model Theory The impact on mathematics and philosophy The Van der Waerden theory of dimension assigns a dimension John T. Baldwin to certain structures,
Prelude: From those which admit a combinatorial geometry. Logic to Model Theory Examples: Classification Theory Vector Spaces, Fundamental Structures transcendence degree of algebraically closed fields, Exploring Cantor’s rank of certain modules. Paradise
The role of Syntax (Families) of Combinatorial Geometries
Modern Model Theory The impact on mathematics and philosophy A stable theory admits a Notion of Independence that John T. Baldwin 1 is weaker in some ways than a combinatorial geometry; Prelude: From Logic to 2 but uniformly studies dimensions over various subsets; Model Theory
Classification 3 provides good dimension on sets definable by Theory weight one types; Fundamental Structures 4 provides a means for connecting the dimensions on various Exploring types. Cantor’s Paradise
The role of Syntax Consequences: Main Gap
Modern Model Theory The impact on mathematics and philosophy Using these ideas of dimension Shelah proved: John T. Baldwin Main Gap Prelude: From Logic to For every first order theory T , either Model Theory
Classification 1 Every model of T is decomposed into a tree of countable Theory models with uniform bound on the depth of the tree, or Fundamental Structures 2 The theory T has the maximal number of models in all Exploring Cantor’s uncountable cardinalities. Paradise
The role of Syntax What does this mean?
Modern Model Theory The impact on mathematics and philosophy For ‘classifiable theories’ every model is determined by its John T. countable submodels and a tree that serves as a skeleton. Baldwin Question Prelude: From Logic to Model Theory Is Shelah’s claim: Classification the existence of the maximal number of models signals Theory
Fundamental non-structure Structures correct? Exploring Cantor’s Paradise What are the criteria for answering such a question?
The role of Syntax Understanding Fundamental Structures
Modern Model Theory The impact on mathematics and philosophy John T. Real Exponentiation Baldwin
Prelude: From An ordered structure is o-minimal if every definable subset is a Logic to Model Theory Boolean combination of intervals.
Classification Theory Wilkie proved Th(<, +, ·, exp) is o-minimal.
Fundamental Structures The study of o-minimal theories, another syntactically defined
Exploring class, has substantial mathematical significance. Cantor’s Paradise
The role of Syntax Zilber’s Thesis
Modern Model Theory The impact on mathematics and philosophy
John T. Baldwin Fundamental structures are canonical Prelude: From Logic to Model Theory Fundamental mathematical structures can be characterized in
Classification an appropriate logic. Theory Conversely, characterizable structures are ‘fundamental’. Fundamental Structures
Exploring Cantor’s Paradise
The role of Syntax Canonical Structures are Fundamental
Modern Model Theory The impact on mathematics and philosophy Zilber Conjecture
John T. Baldwin Every strongly minimal first order theory is 1 Prelude: From disintegrated Logic to Model Theory 2 group-like
Classification 3 Theory field-like
Fundamental Structures Hrushovski refuted the precise conjecture; establishing a refined Exploring version of the underlying thesis is active work connecting model Cantor’s Paradise theory with complex analysis, algebraic geometry, and The role of non-commutative geometry. Syntax Logical Consequences from Mathematical Tools
Modern Model Theory The impact on mathematics Theorem: Zilber and philosophy There is no complete finitely axiomatizable first order theory T John T. that is categorical in every uncountable power. Baldwin
Prelude: From Logic to Ingredients Model Theory Classification 1 Deduce the existence of groups definable in the models of Theory
Fundamental categorical first order theories. Structures 2 Characterize those groups Exploring Cantor’s 3 Paradise Use the properties of those groups to analyze the fine
The role of structure of the model to show T has the finite model Syntax property. Understanding Fundamental Structures
Modern Model Theory The impact on mathematics and philosophy
John T. Baldwin Complex Exponentiation Prelude: From Logic to Model Theory Find an axiomatization for Th(C, +, ·, exp).
Classification Theory The conjectured axiomatization is in Lω1,ω(Q). The framework Fundamental is a special case of our next topic. Structures
Exploring Cantor’s Paradise
The role of Syntax Exploring Cantor’s Paradise
Modern Model Theory The impact on mathematics and philosophy John T. Shelah’s Attitude Baldwin “This work certainly reflects the authors preference to find Prelude: From Logic to something in the white part of the map, the ‘terra incognita’, Model Theory rather than to understand perfectly what we have reasonably Classification Theory understood to begin with ...” Fundamental Structures from preface to Classification Theory for Abstract Elementary Exploring Cantor’s Classes Paradise
The role of Syntax Exploring Cantor’s Paradise
Modern Model Theory The impact on mathematics and philosophy Much of core mathematics is coarse: John T. It studies either Baldwin 1 properties of particular structures of size at most the Prelude: From Logic to continuum Model Theory
Classification 2 or makes assertions that are totally cardinal independent. Theory E.g., if every element of a group has order two then the Fundamental Structures group is abelian.
Exploring Cantor’s Model theory, especially of infinitary logic recognizes distinct Paradise algebraic behavior at different cardinalities. The role of Syntax Other models matter
Modern Model Theory The impact on mathematics and philosophy
John T. Baldwin
Prelude: From A reason to study complete theories is that non-standard, in Logic to Model Theory particular, saturated models give information that is not
Classification available looking at one structure. Theory Fundamental The stability hierarchy illustrates the interaction of the behavior Structures of models in very different cardinalities. Exploring Cantor’s Paradise
The role of Syntax Big models matter
Modern Model Theory The impact on mathematics and philosophy
John T. Shelah’s dicta: Baldwin There may be noise in small cardinalities that hides eventual Prelude: From Logic to uniformity. Model Theory Classification + Theory 1 first order categoricity begins at |T | . Fundamental 2 superstability begins at (2|T |)+. Structures
Exploring 3 Infinitary categoricity may begin at . i Ls(K) + Cantor’s (2 ) Paradise
The role of Syntax Small models matter
Modern Model Theory The impact on mathematics and philosophy Sufficiently nice behavior on small cardinalities propagates to John T. all cardinals. Baldwin Categoricity can fail (for L ) at any point below ℵ . Prelude: From ω1,ω ω Logic to Model Theory Theorem: Shelah ZFC Classification Theory Excellence (n-dimensional amalgamation in ℵ0 for all n) implies Fundamental amalgamation in all cardinalities (and in particular the Structures existence of arbitrarily large models). Exploring Cantor’s Paradise ℵn ℵn+1 The role of Theorem: Shelah 2 < 2 Syntax For Lω1,ω, Categoricity below ℵω implies excellence. The role of Set Theory
Modern Model Theory The impact on mathematics and philosophy
John T. Baldwin In 1970 model theory seemed to be intimately involved with problems in axiomatic set theory. Prelude: From Logic to Model Theory The stability hierarchy established the absoluteness of most Classification important concepts and theorems in first order model theory. Theory Fundamental But for infinitary logic, weak extensions of ZFC seem to be Structures needed. Exploring Cantor’s Paradise
The role of Syntax The role of Syntax
Modern Model Theory The impact on mathematics and philosophy What is syntax?
John T. Baldwin vocabulary and structures
Prelude: From definable relations Logic to Model Theory proof Classification Theory
Fundamental David Pierce on Mathematicians vrs Model Theorists on Structures vocabulary Exploring Cantor’s pierce Paradise Similar examples appear in Emily Grosholz on Productive The role of Syntax Ambiguity. The Choice of Vocabulary
Modern Model Theory The impact on mathematics Model Theoretic Convenience and philosophy
John T. 1 Morley’s ‘quantifier elimination’ by force. Baldwin 2 Shelah’s transformation from sentences in Lω1,ω to atomic Prelude: From Logic to models of first order theories. Model Theory 3 C eq. Classification Theory
Fundamental Structures Mathematical Significance
Exploring Cantor’s Model theoretic algebra seeks a sufficiently powerful vocabulary Paradise but insists that the basic relations must be ‘meaningful’. The role of Syntax Frege’s concept analysis. Positive Applications of Syntax
Modern Model Theory The impact on mathematics Much of model theory concerns first order definable sets. and philosophy 1 The stability hierarchy is syntactic. John T. Baldwin 2 O-minimality allows the capture of topological properties
Prelude: From of the real numbers by first order logic. Logic to Model Theory 3 Work of Denef, Van den Dries et al essentially replaces Classification ‘blowing up of singularities’ by induction on quantifier Theory
Fundamental rank. Structures 4 Continued work on motivic integration Exploring Cantor’s 5 Bays, Gavrilovich, Zilber analysis of covers of abelian Paradise
The role of varieties. Syntax Detail on covers Model Theory without logic
Modern Model Theory The impact on mathematics Model theory is about classes of structures of a fixed similarity and philosophy type.
John T. Baldwin Must these classes be defined in a logic? 1 Zariski Structures Prelude: From Logic to Model Theory 2 Zilber’s theory of quasiminimal excellence. Classification 3 Shelah’s theory of abstract elementary classes (aka Theory accessible categories). Fundamental Structures
Exploring When do mathematical conditions on classes generate a logic? Cantor’s Paradise These approaches recall Tarski’s contrast of logical and The role of Syntax mathematical characterizations in the early 1950’s. Kennedy calls this move a species of ‘formalism freeness’. The role of Syntax
Modern Model Theory The impact on mathematics and What is a syntactical property? philosophy
John T. A first order theory is small if for each n, there are only Baldwin countably many n-types over the emptyset. Prelude: From Logic to Model Theory Vaught (1958) writes: Classification Theory “A little thought convinces one that a notion of ‘purely Fundamental syntactical condition’ wide enough to include ‘small theory’ Structures
Exploring would be so broad as to be pointless.” Cantor’s Paradise
The role of ‘small’ is an anachronism. Vaught defined the notion but didn’t Syntax name it. The technical developments of the 60’s and 70’s created specialized subareas of mathematical logic. We have argued that Modern Model Theory:
1 has profound mathematical consequences 2 provides both questions and tools for philosophical investigations. Next slide more topics.
The role of model theory?
Modern Model Theory The impact on mathematics The Conference manifesto reads and philosophy Model theory seems to have reached its zenith in the sixties
John T. and the seventies, when it was seen by many as virtually Baldwin identical to mathematical logic. Prelude: From Logic to Model Theory
Classification Theory
Fundamental Structures
Exploring Cantor’s Paradise
The role of Syntax The role of model theory?
Modern Model Theory The impact on mathematics The Conference manifesto reads and philosophy Model theory seems to have reached its zenith in the sixties
John T. and the seventies, when it was seen by many as virtually Baldwin identical to mathematical logic. Prelude: From Logic to Model Theory The technical developments of the 60’s and 70’s created Classification Theory specialized subareas of mathematical logic. We have argued
Fundamental that Modern Model Theory: Structures
Exploring Cantor’s 1 has profound mathematical consequences Paradise 2 provides both questions and tools for philosophical The role of Syntax investigations. Next slide more topics. Other Topics
Modern Model Theory The impact on mathematics and philosophy 1 Why is second order categoricity uninformative?
John T. 2 Baldwin When is categoricity in power the right notion of canonicity? Reals vrs complexes Prelude: From Logic to 3 transfer of ideas used to study uncountable structures and Model Theory vocabularies to the countable. Classification Theory 4 Conjectures in AEC: the diversity of infinity. Fundamental Structures 5 algebraic vrs combinatorial properties Exploring Cantor’s 6 Monster Models and Grothendieck universes. Paradise 7 The role of completions (existential, Morley, eq) Syntax Z-Covers of Algebraic Groups
Modern Model Theory The impact on mathematics and Definition A Z-cover of a commutative algebraic group (C) philosophy A
John T. is a short exact sequence Baldwin
Prelude: From exp Logic to 0 → Z N → V → (C) → 1. (1) Model Theory A
Classification Theory where V is a Q vector space and A is an algebraic group, Fundamental defined over k0 with the full structure imposed by (C, +, ·) and Structures so interdefinable with the field. Exploring Cantor’s Paradise N TA + Λ = Z denotes an Lω1,ω axiomatization of such a The role of structure. Syntax Categoricity Problem
Modern Model Theory The impact on mathematics and philosophy
John T. N Baldwin Is TA + Λ = Z categorical in uncountable powers?
Prelude: From paraphrasing Zilber: Logic to Model Theory Categoricity would mean the short exact sequence is a Classification reasonable ‘algebraic’ substitute for the classical Theory complex universal cover. Fundamental Structures Return Exploring Cantor’s Paradise
The role of Syntax Pierce on Spivak
Modern Spivak’s Calculus book asserts the following are equivalent Model Theory The impact on condition on . mathematics N and philosophy 1 induction (1 ∈ X and k ∈ X implies
John T. k + 1 ∈ X ) implies X = N. Baldwin 2 well-ordered Every non-empty subset has a Prelude: From least element. Logic to Model Theory 3 strong induction (1 ∈ X and for every m < k, Classification m ∈ X implies k ∈ X ) implies X = N. Theory 4 recursive definition For every unary f and Fundamental Structures element a there is a unique function h with Exploring h(1) = a and h(k + 1) = f (h(k)). Cantor’s Paradise But this doesn’t make sense: The role of Syntax 1 and 4 are properties of unary algebras 2 is a property of ordered sets (and doesn’t imply others) 3 is a property of ordered algebras. Return Other Topics
Modern Model Theory The impact on mathematics and philosophy 1 Why is second order categoricity uninformative? John T. Baldwin 2 When is categoricity in power the right notion of
Prelude: From categoricity? Logic to Model Theory 3 transfer of ideas used to study uncountable structures and Classification Theory vocabularies to the countable. Fundamental 4 Conjectures in AEC: the diversity of infinity. Structures
Exploring 5 algebraic vrs combinatorial Cantor’s Paradise 6 Monster Models and Grothendieck universes.
The role of Syntax