Model Theoretic Perspectives on the Philosophy of Mathematics John T. Baldwin∗ Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago January 11, 2010

PRELIMINARY VERSION, DEC 2009 And some infinites are larger than other infinites and some are smaller. Robert Grosseteste 13th century [Fre54] We approach the ‘practice based philosophy of logic’ by examining the practice in one specific area of logic, model theory, over the last century. From this we try to draw lessons not for the philosophy of logic but for the philosophy of mathematics. We argue in fact that the philosophical impact of the developments in mathematical logic during the last half of the twentieth century were obscured by their mathematical depth and by the intertwining with mathematics. That is, that concepts which are normally regarded by both mathematicians and philosophers as ‘simply mathematics’ have philosophical importance. We make two claims. First is that the mere fact that logical methods have had mathematical impact is important for any investigation of mathematical methodology. Twentieth century logic introduced techniques that were important not just for the problems they were originally designed to solve (arising out of Hilbert’s program) but across broad areas of mathematics. But, from a philosophical standpoint, there is a further impact. These methods actually provide tools for the analysis of mathematical methodology. The longtime standard definition of logic is ”the analysis of methods of reasoning”. This does not describe the perspective of a contemporary model theorist. A model theorist is a self-conscious mathematician. A model theorist uses various formal languages and semantics to prove mathematical theorems. But there is an inherently metamathematical aspect. The very notion of model theory involves seeking common patterns across distinct areas of mathematical investigation. One of our goals below is to make precise this notion of ‘distinct area’. We view the philosophy of mathematics as a broad inquiry into and critical analysis of the conceptual foundations of actual mathematics work1 . This investigation also include a study of the basic methodologies and proof techniques of the subject 2 . The foundationalist goal of justifying mathematics is a part of this study. But the study we envision cannot be carried out by interpreting the theory into an u ¨ber theory such as ZFC; too much information is lost. The coding does not reflect the ethos of the particular subject area of mathematics. The intuition behind fundamental ideas such as homomorphism or manifold disappears when looking at a complicated definition of the notion in a language whose only symbol is . Tools must be developed for the analysis and comparison of distinct areas of mathematics in a way that maintains meaning; a simple truth ∗ We give special thanks to the Mittag-Leffler Institute where we were able to rethink and focus the ideas of this talk. Baldwin was partially supported by NSF-0500841. The paper builds on a presentation at Notre Dame in Fall of 2008. 1 This is a paraphrase of part of Dutilh-Novaes presentation. 2 For a broad investigation the philosophy of mathematics including a study of leading contemporary mathematicians (e.g. Grothendieck, Langlands, Shelah, Zilber) see [Zal09]

1

preserving transformation into statements of set theory is inherently inadequate. The traditional foundationalist approach sacrifices explanation on the altar of justification. The discussion below has both a sociological and a philosophical aspect. Sociologically, in the remainder of this introduction, we describe recent examples to illustrate the practice of the model theoretic species of logician. Philosophically, in the two main sections of the paper we propose some tools for studying the methodology of mathematics. We aim to sketch a program for using model theoretic concepts for a) formalizing a notion of ‘area of mathematics’ and b) analyzing basic concepts of mathematics. In Section 1) we sketch the history of model theory in the twentieth century and in particular the development of the notion of a complete theory. We argue for the notion of a first order complete theory as a useful unit of analysis for describing ‘an area of mathematics’. We conclude the historical discussion with an introduction to the sophisticated model theoretic methods developed in the last 40 years. In Section 2) we discuss how these methods can provide insight into the way fundamental notions are specified in different areas of mathematics. We discuss in detail the analysis of one particular mathematical notion, dimension, using model theoretic notions. In both cases, our main point is that model theoretic tools can be brought to bear. We are simply giving introductory sketches of illustrations of that thesis. For brevity, most of the emphasis is on first order logic. But important extensions to infinitary logic and even ‘syntax deprived’ model theory will appear later in the paper. In the sociological mode, we now list the titles of papers from the Mid-Atlantic Model Theory conference held in the Fall of 2008 at Rutgers. Our summary of this conference focuses on two currents of ‘main-stream’ model theory represented at this conference. It does not encompass a number of other areas of model theory such as models of arithmetic, finite model theory, model theory in computer science, higher order and other extensions of first order logic, and universal algebra. 1. Model theory and non-archimedean geometry 2. The valuation inequality for complex analytic structure 3. Cherlin’s Conjecture and Generix’s Adventures in Groupland 4. ω-stable semi-Abelian varieties 5. O-minimal triangulation respecting a standard part map 6. Some modest attempts at defining the notions of groups and fields of dimension one, and establishing their algebraic properties 7. Dependent theories: limit model existence and recounting the number of types 8. The non-elementary model theory of analytic Zariski structures 9. Difference fields, model theory and applications 10. Model Theory of the Adeles The two currents of model theory that I want to contrast focus, broadly speaking, a) on the use of model theory in various parts of mathematics and b) on the development of an independent subject area of ‘model theory’. In the early 70’s these seemed wildly divergent subjects. But now, at least seven of the papers above, even those focused on algebraic notions such as non-archimedean geometry, semi-abelian varieties, or difference fields integrate the fundamental concepts introduced in the pure theory. For example, paper 3) concerns a conjecture of Cherlin which uses model theoretic concepts to lift the program of the classification of finite simple groups to the classification of simple groups of finite Morley rank. Even the statement of the problem is posed in model theoretic terms (that we discuss below). But this terminology provides a way to organize topics that 2

are already in the mathematical air. The investigation involves significant techniques from model theory, finite group theory, and algebraic groups. Even the relatively few papers at this conference that were ‘pure’ developed concepts central to current research in e.g. the model theory of valued fields3 .

1

Historical Survey of model theory

The integral connections of model theory with modern mathematics as described in the introduction are often (and often correctly) seen as a falling away from philosophical concerns. But as we’ll see below, many of these interactions do stem from concerns about explanation and coherence of mathematical ideas that have a philosophical basis. The divorce is from narrow concern with the formal justification of results. And there are natural philosophical issues that arise from more technical results. As noted below there are vast differences between the role of ℵ0 and any uncountable cardinal in the study of categoricity. What is so different about countability? After a survey of the history of model theory I expound the use of model theoretic concepts as a tool for such an analysis of the foundations of mathematics. We review this history from a standpoint similar to this paper but with an emphasis on the mathematical applications in [Bal0x]. We distinguish three types of analysis in first order model theory: 1. Properties of first order logic (1930-1965) 2. Properties of complete theories (1950-present) 3. Properties of classes of theories (1970-present)

1.1

Properties of first order logic (1930-1965)

The essence of model theory is a clear distinction between syntax and semantics. Sentences in a formal language for a vocabulary τ are true or false in structures for τ . While the full formal treatment of this notion first appears in [Tar35], the basic idea is essential for G´ odel’s completeness theorem [G¨od29] to even make sense. While the completeness theorem plays a fundamental role in first order model theory, a formal proof system is not essential to formulating many of the crucial concepts. The prehistory of model theory include the work before 1950 L¨owenheim, Skolem, G¨odel, Malcev, and Tarski. They isolated the fundamental properties of first order logic such as completeness, compactness, and the Lowenheim-Skolem-Tarski theorem. The prehistorical aspect is illustrated by references in logic courses to the ‘Lowenheim-Skolem-Tarski theorem’ and its proof by Malcev and G¨odel. The term model theory was popularised in the early 1950’s, especially by Tarski and Robinson. Work in that decade provided syntactic characterization of preservation properties. E.g., The models of a first order theory are closed under unions of chains if and only the theory is axiomatized by ‘for all, there exist’ sentences. But what we might now call ‘syntactic’ and ‘semantic’ formulations are described as more of a contrast between ‘logical’ and ‘mathematical’. In [Tar54], Tarski writes ‘universal classes can be characterized in a purely mathematical terms’. The compactness theorem is given both ‘logical’ proofs from the completeness theorem and ‘mathematical’ proofs 3 Shelah’s concept of theories without the independence property (nip or dependent depending on the author) were expounded in the least-applied talk. Hrushovski’s paper ‘Stable groups and approximate group theory’[Hru09], which uses the model theoretic analysis of these theories as a tool for the study of groups, was the subject of semester-long seminars at UCLA, Berkeley, Urbana, and Leeds in the Fall of 2009. Fields medalist Terrence Tao discusses the progress of the UCLA seminar in the blog at http: //terrytao.wordpress.com/.

3

via ultraproducts. Tarski and Vaught [TV56] define the notion of elementary extension and prove the union of elementary chain is an elementary extension of each member of the chain; this both refines the original theory and helps to develop the correct category for model theory. Further general properties of first order logic developed in the 50’s included interpolation theorems and the Robinson Consistency theorem. Much model theoretic work in the 60’s and 70’s extended these kinds of notions to logic with infinite conjunctions or with generalized quantifiers of various sorts. But we want to focus on a crucial idea that crystalized in the 1950’s: a complete theory. Before proceeding to complete theories we discuss a different notion with the same name: completeness of a logic. By a logic, we mean as in [BF85] a syntactical notion of a collection of sentences L(τ ) for a vocabulary τ and a satisfaction relation |=L between sentences φ ∈ L(τ ) and τ -structures M . The logic is complete if there is some proof system `L of L such that: `L φ if and only if for every M |=L φ. A theory T is a consistent set of sentences in a logic L. (We will consider first order, second order, Lω1 ,ω and Lω1 ,ω (Q).) Our discussion of prehistoric times is not complete without mentioning the American Postulate Theorists [AR02a, AR02b]. Already in 1902, Huntington introduced the notion of an axiom system having exactly one model. By 1904 [Veb04], this notion had been christened ‘categoricity’ and Veblen proves the categoricity of a set of (second order) axioms for geometry. Following the terminology of [AR02a], which is reasonably standard, we say. Definition 1 1. A theory T is semantically L-complete if for each L-sentence φ and any pair of models M, N of T , M |=L φ if and only if N |=L φ. 2. A theory T is deductively (or syntactically) L-complete if for each L-sentence φ either T `L φ or T `L ¬φ If L satisfies the (extended) completeness theorem then these notions are equivalent. Again as reported in [AR02a], Fraenkel [Fra28] had distinguished these notions without establishing that they are really distinct. In [Ken], Kennedy discusses the significance of the first paragraph of G¨odel’s thesis. She points out this distinction becomes clear only with G¨ odel’s proof of the completeness theorem. Kennedy further notes that G¨odel argues that categoricity and an effective proof theory implies syntactic completeness. Thus G¨odel foreshadows the incompleteness theorem in his argument that a proof is needed for completeness (contrary to the view that ‘consistency implies existence’ is tautological). There is a categorical axiomatization of the real numbers with arithmetic in second order logic; this yields semantic, but not syntactic completeness of the second order theory. Vaught’s proof [Vau54] of the Los-Vaught test (a first order theory with no finite models that is categorical in some infinite power is complete) writes the argument in modern terms4 : Categoricity plus upward and downward L¨owenheim-Skolem implies semantic completeness; syntactic completeness follows by G¨odel. What now seem obvious compactness arguments for the existence of non-standard models were clearly not in the air in 1930 [Ken, Vau86]. Note that for any structure M , Th(M ) = {φ : M |= φ} is a semantically complete theory for every logic L is under consideration. This method of obtaining complete theories is fundamental. 4 There

is no indication of a connection with the G¨ odels argument cited above.

4

1.2

Properties of complete theories (1950-present)

The mathematical significance of the fundamental notion of a first order complete theory was stressed by Abraham Robinson [Rob56]. He provides a number of mathematically interesting examples of complete first order theories and shows common model theoretic characteristics involving the form of the axiomatization or quantifier elimination for a number of them. Axiomatic theories arise from two distinct motivations. One is to understand a single significant structure such as (N, +, ·) or (R, +, ·). The other is to find the common characteristics of a number of structures; theories of the second sort include groups, rings, fields etc. There are a number of second order theories of the first sort that are categorical. Both of these motivations aim at studying fundamental properties which determine all properties of a structure or a group of structures. But the axiomatizations have quite different impact. The (usually) second order axioms characterizing a single important structure delineate exactly what makes that structure unique. These axioms illuminate a key feature of the structure: the reals are the unique complete ordered field with a countable dense subset. But this light is shed on the particular structure. Bourbaki represents a triumph of axiomatization for the second reason. Large parts of mathematics were organized into coherent topics by providing informative axiomatizations. Let us consider the relation with categoricity. To avoid trivialities, we deal only with infinite models. T is categorical if it has exactly one model (up to isomorphism). T is categorical in power κ if it has exactly one model in cardinality κ. Note that under these definitions, every categorical first order theory is semantically complete. Further every theory in a logic which admits the upward and downward L¨owenheim-Skolem theorem for theories that is categorical in some infinite cardinality is semantically complete. First order logic is the only one of our examples that satisfies this condition. Semantic (and indeed syntactic completeness) can be deduced from ℵ1 -categoricity for sentences of Lω1 ,ω [She83a, She83b, Bal09]. At present the ℵ1 plays an essential role in the proof. Most people have an intuition for only a few infinite structures: arithmetic on the natural numbers, the rationals, and perhaps on the reals. Most mathematicians extend this to the complex numbers and then to a deeper understanding of various structures depending on their own specialization: (SL2 (

PRELIMINARY VERSION, DEC 2009 And some infinites are larger than other infinites and some are smaller. Robert Grosseteste 13th century [Fre54] We approach the ‘practice based philosophy of logic’ by examining the practice in one specific area of logic, model theory, over the last century. From this we try to draw lessons not for the philosophy of logic but for the philosophy of mathematics. We argue in fact that the philosophical impact of the developments in mathematical logic during the last half of the twentieth century were obscured by their mathematical depth and by the intertwining with mathematics. That is, that concepts which are normally regarded by both mathematicians and philosophers as ‘simply mathematics’ have philosophical importance. We make two claims. First is that the mere fact that logical methods have had mathematical impact is important for any investigation of mathematical methodology. Twentieth century logic introduced techniques that were important not just for the problems they were originally designed to solve (arising out of Hilbert’s program) but across broad areas of mathematics. But, from a philosophical standpoint, there is a further impact. These methods actually provide tools for the analysis of mathematical methodology. The longtime standard definition of logic is ”the analysis of methods of reasoning”. This does not describe the perspective of a contemporary model theorist. A model theorist is a self-conscious mathematician. A model theorist uses various formal languages and semantics to prove mathematical theorems. But there is an inherently metamathematical aspect. The very notion of model theory involves seeking common patterns across distinct areas of mathematical investigation. One of our goals below is to make precise this notion of ‘distinct area’. We view the philosophy of mathematics as a broad inquiry into and critical analysis of the conceptual foundations of actual mathematics work1 . This investigation also include a study of the basic methodologies and proof techniques of the subject 2 . The foundationalist goal of justifying mathematics is a part of this study. But the study we envision cannot be carried out by interpreting the theory into an u ¨ber theory such as ZFC; too much information is lost. The coding does not reflect the ethos of the particular subject area of mathematics. The intuition behind fundamental ideas such as homomorphism or manifold disappears when looking at a complicated definition of the notion in a language whose only symbol is . Tools must be developed for the analysis and comparison of distinct areas of mathematics in a way that maintains meaning; a simple truth ∗ We give special thanks to the Mittag-Leffler Institute where we were able to rethink and focus the ideas of this talk. Baldwin was partially supported by NSF-0500841. The paper builds on a presentation at Notre Dame in Fall of 2008. 1 This is a paraphrase of part of Dutilh-Novaes presentation. 2 For a broad investigation the philosophy of mathematics including a study of leading contemporary mathematicians (e.g. Grothendieck, Langlands, Shelah, Zilber) see [Zal09]

1

preserving transformation into statements of set theory is inherently inadequate. The traditional foundationalist approach sacrifices explanation on the altar of justification. The discussion below has both a sociological and a philosophical aspect. Sociologically, in the remainder of this introduction, we describe recent examples to illustrate the practice of the model theoretic species of logician. Philosophically, in the two main sections of the paper we propose some tools for studying the methodology of mathematics. We aim to sketch a program for using model theoretic concepts for a) formalizing a notion of ‘area of mathematics’ and b) analyzing basic concepts of mathematics. In Section 1) we sketch the history of model theory in the twentieth century and in particular the development of the notion of a complete theory. We argue for the notion of a first order complete theory as a useful unit of analysis for describing ‘an area of mathematics’. We conclude the historical discussion with an introduction to the sophisticated model theoretic methods developed in the last 40 years. In Section 2) we discuss how these methods can provide insight into the way fundamental notions are specified in different areas of mathematics. We discuss in detail the analysis of one particular mathematical notion, dimension, using model theoretic notions. In both cases, our main point is that model theoretic tools can be brought to bear. We are simply giving introductory sketches of illustrations of that thesis. For brevity, most of the emphasis is on first order logic. But important extensions to infinitary logic and even ‘syntax deprived’ model theory will appear later in the paper. In the sociological mode, we now list the titles of papers from the Mid-Atlantic Model Theory conference held in the Fall of 2008 at Rutgers. Our summary of this conference focuses on two currents of ‘main-stream’ model theory represented at this conference. It does not encompass a number of other areas of model theory such as models of arithmetic, finite model theory, model theory in computer science, higher order and other extensions of first order logic, and universal algebra. 1. Model theory and non-archimedean geometry 2. The valuation inequality for complex analytic structure 3. Cherlin’s Conjecture and Generix’s Adventures in Groupland 4. ω-stable semi-Abelian varieties 5. O-minimal triangulation respecting a standard part map 6. Some modest attempts at defining the notions of groups and fields of dimension one, and establishing their algebraic properties 7. Dependent theories: limit model existence and recounting the number of types 8. The non-elementary model theory of analytic Zariski structures 9. Difference fields, model theory and applications 10. Model Theory of the Adeles The two currents of model theory that I want to contrast focus, broadly speaking, a) on the use of model theory in various parts of mathematics and b) on the development of an independent subject area of ‘model theory’. In the early 70’s these seemed wildly divergent subjects. But now, at least seven of the papers above, even those focused on algebraic notions such as non-archimedean geometry, semi-abelian varieties, or difference fields integrate the fundamental concepts introduced in the pure theory. For example, paper 3) concerns a conjecture of Cherlin which uses model theoretic concepts to lift the program of the classification of finite simple groups to the classification of simple groups of finite Morley rank. Even the statement of the problem is posed in model theoretic terms (that we discuss below). But this terminology provides a way to organize topics that 2

are already in the mathematical air. The investigation involves significant techniques from model theory, finite group theory, and algebraic groups. Even the relatively few papers at this conference that were ‘pure’ developed concepts central to current research in e.g. the model theory of valued fields3 .

1

Historical Survey of model theory

The integral connections of model theory with modern mathematics as described in the introduction are often (and often correctly) seen as a falling away from philosophical concerns. But as we’ll see below, many of these interactions do stem from concerns about explanation and coherence of mathematical ideas that have a philosophical basis. The divorce is from narrow concern with the formal justification of results. And there are natural philosophical issues that arise from more technical results. As noted below there are vast differences between the role of ℵ0 and any uncountable cardinal in the study of categoricity. What is so different about countability? After a survey of the history of model theory I expound the use of model theoretic concepts as a tool for such an analysis of the foundations of mathematics. We review this history from a standpoint similar to this paper but with an emphasis on the mathematical applications in [Bal0x]. We distinguish three types of analysis in first order model theory: 1. Properties of first order logic (1930-1965) 2. Properties of complete theories (1950-present) 3. Properties of classes of theories (1970-present)

1.1

Properties of first order logic (1930-1965)

The essence of model theory is a clear distinction between syntax and semantics. Sentences in a formal language for a vocabulary τ are true or false in structures for τ . While the full formal treatment of this notion first appears in [Tar35], the basic idea is essential for G´ odel’s completeness theorem [G¨od29] to even make sense. While the completeness theorem plays a fundamental role in first order model theory, a formal proof system is not essential to formulating many of the crucial concepts. The prehistory of model theory include the work before 1950 L¨owenheim, Skolem, G¨odel, Malcev, and Tarski. They isolated the fundamental properties of first order logic such as completeness, compactness, and the Lowenheim-Skolem-Tarski theorem. The prehistorical aspect is illustrated by references in logic courses to the ‘Lowenheim-Skolem-Tarski theorem’ and its proof by Malcev and G¨odel. The term model theory was popularised in the early 1950’s, especially by Tarski and Robinson. Work in that decade provided syntactic characterization of preservation properties. E.g., The models of a first order theory are closed under unions of chains if and only the theory is axiomatized by ‘for all, there exist’ sentences. But what we might now call ‘syntactic’ and ‘semantic’ formulations are described as more of a contrast between ‘logical’ and ‘mathematical’. In [Tar54], Tarski writes ‘universal classes can be characterized in a purely mathematical terms’. The compactness theorem is given both ‘logical’ proofs from the completeness theorem and ‘mathematical’ proofs 3 Shelah’s concept of theories without the independence property (nip or dependent depending on the author) were expounded in the least-applied talk. Hrushovski’s paper ‘Stable groups and approximate group theory’[Hru09], which uses the model theoretic analysis of these theories as a tool for the study of groups, was the subject of semester-long seminars at UCLA, Berkeley, Urbana, and Leeds in the Fall of 2009. Fields medalist Terrence Tao discusses the progress of the UCLA seminar in the blog at http: //terrytao.wordpress.com/.

3

via ultraproducts. Tarski and Vaught [TV56] define the notion of elementary extension and prove the union of elementary chain is an elementary extension of each member of the chain; this both refines the original theory and helps to develop the correct category for model theory. Further general properties of first order logic developed in the 50’s included interpolation theorems and the Robinson Consistency theorem. Much model theoretic work in the 60’s and 70’s extended these kinds of notions to logic with infinite conjunctions or with generalized quantifiers of various sorts. But we want to focus on a crucial idea that crystalized in the 1950’s: a complete theory. Before proceeding to complete theories we discuss a different notion with the same name: completeness of a logic. By a logic, we mean as in [BF85] a syntactical notion of a collection of sentences L(τ ) for a vocabulary τ and a satisfaction relation |=L between sentences φ ∈ L(τ ) and τ -structures M . The logic is complete if there is some proof system `L of L such that: `L φ if and only if for every M |=L φ. A theory T is a consistent set of sentences in a logic L. (We will consider first order, second order, Lω1 ,ω and Lω1 ,ω (Q).) Our discussion of prehistoric times is not complete without mentioning the American Postulate Theorists [AR02a, AR02b]. Already in 1902, Huntington introduced the notion of an axiom system having exactly one model. By 1904 [Veb04], this notion had been christened ‘categoricity’ and Veblen proves the categoricity of a set of (second order) axioms for geometry. Following the terminology of [AR02a], which is reasonably standard, we say. Definition 1 1. A theory T is semantically L-complete if for each L-sentence φ and any pair of models M, N of T , M |=L φ if and only if N |=L φ. 2. A theory T is deductively (or syntactically) L-complete if for each L-sentence φ either T `L φ or T `L ¬φ If L satisfies the (extended) completeness theorem then these notions are equivalent. Again as reported in [AR02a], Fraenkel [Fra28] had distinguished these notions without establishing that they are really distinct. In [Ken], Kennedy discusses the significance of the first paragraph of G¨odel’s thesis. She points out this distinction becomes clear only with G¨ odel’s proof of the completeness theorem. Kennedy further notes that G¨odel argues that categoricity and an effective proof theory implies syntactic completeness. Thus G¨odel foreshadows the incompleteness theorem in his argument that a proof is needed for completeness (contrary to the view that ‘consistency implies existence’ is tautological). There is a categorical axiomatization of the real numbers with arithmetic in second order logic; this yields semantic, but not syntactic completeness of the second order theory. Vaught’s proof [Vau54] of the Los-Vaught test (a first order theory with no finite models that is categorical in some infinite power is complete) writes the argument in modern terms4 : Categoricity plus upward and downward L¨owenheim-Skolem implies semantic completeness; syntactic completeness follows by G¨odel. What now seem obvious compactness arguments for the existence of non-standard models were clearly not in the air in 1930 [Ken, Vau86]. Note that for any structure M , Th(M ) = {φ : M |= φ} is a semantically complete theory for every logic L is under consideration. This method of obtaining complete theories is fundamental. 4 There

is no indication of a connection with the G¨ odels argument cited above.

4

1.2

Properties of complete theories (1950-present)

The mathematical significance of the fundamental notion of a first order complete theory was stressed by Abraham Robinson [Rob56]. He provides a number of mathematically interesting examples of complete first order theories and shows common model theoretic characteristics involving the form of the axiomatization or quantifier elimination for a number of them. Axiomatic theories arise from two distinct motivations. One is to understand a single significant structure such as (N, +, ·) or (R, +, ·). The other is to find the common characteristics of a number of structures; theories of the second sort include groups, rings, fields etc. There are a number of second order theories of the first sort that are categorical. Both of these motivations aim at studying fundamental properties which determine all properties of a structure or a group of structures. But the axiomatizations have quite different impact. The (usually) second order axioms characterizing a single important structure delineate exactly what makes that structure unique. These axioms illuminate a key feature of the structure: the reals are the unique complete ordered field with a countable dense subset. But this light is shed on the particular structure. Bourbaki represents a triumph of axiomatization for the second reason. Large parts of mathematics were organized into coherent topics by providing informative axiomatizations. Let us consider the relation with categoricity. To avoid trivialities, we deal only with infinite models. T is categorical if it has exactly one model (up to isomorphism). T is categorical in power κ if it has exactly one model in cardinality κ. Note that under these definitions, every categorical first order theory is semantically complete. Further every theory in a logic which admits the upward and downward L¨owenheim-Skolem theorem for theories that is categorical in some infinite cardinality is semantically complete. First order logic is the only one of our examples that satisfies this condition. Semantic (and indeed syntactic completeness) can be deduced from ℵ1 -categoricity for sentences of Lω1 ,ω [She83a, She83b, Bal09]. At present the ℵ1 plays an essential role in the proof. Most people have an intuition for only a few infinite structures: arithmetic on the natural numbers, the rationals, and perhaps on the reals. Most mathematicians extend this to the complex numbers and then to a deeper understanding of various structures depending on their own specialization: (SL2 (