What is a complete theory?
John T. Baldwin
Logical Considerations What is a complete theory? Covers of Semi-abelian varieties Mordell-Weil John T. Baldwin Theorem categoricity
January 2, 2009 Outline
What is a complete theory?
John T. Baldwin 1 Logical Considerations Logical Considerations
Covers of Semi-abelian 2 Covers of Semi-abelian varieties varieties
Mordell-Weil Theorem categoricity 3 Mordell-Weil Theorem
4 categoricity Themes
What is a complete theory?
John T. Baldwin
Logical Considerations
Covers of 1 Logic for Logic’s sake Semi-abelian varieties 2 Model theory as a tool for studying the methodology of Mordell-Weil mathematics. Theorem Each mathematical subject requires its own formalization. categoricity A basic notion
What is a complete theory?
John T. Baldwin
Logical Considerations A (consistent) theory T in a logic L is complete if for every Covers of L-sentence φ, Semi-abelian varieties T |= φ Mordell-Weil Theorem or categoricity T |= ¬φ. Complete First Order Theories
What is a complete theory?
John T. Baldwin
Logical Considerations 1 dense linear order (w/o endpoints) Covers of Semi-abelian varieties 2 algebraically closed fields (of fixed characteristic) Mordell-Weil 3 true arithmetic Theorem categoricity 4 real closed fields First order model theory
What is a complete theory?
John T. Baldwin
Logical The main tool of first order model theory is the classification of Considerations complete theories by stability-like notions. Covers of Semi-abelian varieties If complete theories have similar semi-syntactic theoretic
Mordell-Weil properties: ℵ1-categorical, ω-stable, o-minimal, strictly stable, Theorem then their class of models have similar algebraic properties: categoricity number of models, existence of dimension functions, interpretability of groups, existence of generic elements, The Standard Example
What is a complete theory?
John T. Baldwin
Logical Considerations
Covers of Semi-abelian varieties Th(M) for any M.
Mordell-Weil Theorem categoricity Thus, algebraic geometry is the model theory of (C, +, ·, 0, 1). This philosophy underlies Hrushovsky’s work on the geometric Mordell-Lang Conjecture.
Leitmotif
What is a complete theory?
John T. Baldwin
Logical Considerations Study a mathematical structure M by studying Th(M). Covers of Semi-abelian varieties
Mordell-Weil Theorem categoricity Leitmotif
What is a complete theory?
John T. Baldwin
Logical Considerations Study a mathematical structure M by studying Th(M). Covers of Semi-abelian varieties Thus, algebraic geometry is the model theory of (C, +, ·, 0, 1).
Mordell-Weil Theorem This philosophy underlies Hrushovsky’s work on the geometric categoricity Mordell-Lang Conjecture. Lω1,ω-completeness
What is a complete theory?
John T. Baldwin
Logical Considerations For ∆ a fragment of Lω1,ω, a ∆-theory T is complete if for Covers of every ∆-sentence φ, Semi-abelian varieties T |= φ Mordell-Weil Theorem or categoricity T |= ¬φ. L¨owenheim Skolem properties
What is a complete theory?
John T. Baldwin
Logical Considerations Downward: Every consistent countable set of Lω ,ω-sentences Covers of 1 Semi-abelian has a countable model. varieties Mordell-Weil No upward: There are sentences with maximal models in (that Theorem characterize) each ℵα and each α. categoricity i What is Th (R, +, ·, 0, 1)? Lω1,ω
What is the theory?
What is a complete theory?
John T. Baldwin
Logical Considerations Covers of What is Th (C, +, ·, 0, 1)? Semi-abelian Lω1,ω varieties
Mordell-Weil Theorem categoricity What is the theory?
What is a complete theory?
John T. Baldwin
Logical Considerations Covers of What is Th (C, +, ·, 0, 1)? Semi-abelian Lω1,ω varieties What is Th (R, +, ·, 0, 1)? Mordell-Weil Lω1,ω Theorem categoricity Vaught’s test
What is a complete theory?
John T. Baldwin
Logical Considerations Let T be a set of first order sentences with no finite models, in Covers of Semi-abelian a countable language. varieties
Mordell-Weil Theorem If T is κ-categorical for some κ ≥ ℵ0, categoricity then T is complete. Definition
An Lω1,ω-sentence φ is ∆-‘not so big’, if each model of φ is small (realizes only countably many complete ∆-types over the empty set).
Definition
An Lω1,ω-sentence φ is ∆-small if there is a set X countable of complete ∆-types over the empty set and each model realizes some subset of X .
‘small’ means ∆ = Lω1,ω
Small
What is a Let ∆ be a fragment of L that contains φ. complete ω1,ω theory?
John T. Definition Baldwin A τ-structure M is ∆-small for L∗ if M realizes only countably Logical many ∆-types (over the empty set). Considerations
Covers of Semi-abelian varieties
Mordell-Weil Theorem categoricity Definition
An Lω1,ω-sentence φ is ∆-small if there is a set X countable of complete ∆-types over the empty set and each model realizes some subset of X .
‘small’ means ∆ = Lω1,ω
Small
What is a Let ∆ be a fragment of L that contains φ. complete ω1,ω theory?
John T. Definition Baldwin A τ-structure M is ∆-small for L∗ if M realizes only countably Logical many ∆-types (over the empty set). Considerations
Covers of Semi-abelian Definition varieties
Mordell-Weil An Lω1,ω-sentence φ is ∆-‘not so big’, if each model of φ is Theorem small (realizes only countably many complete ∆-types over the categoricity empty set). Small
What is a Let ∆ be a fragment of L that contains φ. complete ω1,ω theory?
John T. Definition Baldwin A τ-structure M is ∆-small for L∗ if M realizes only countably Logical many ∆-types (over the empty set). Considerations
Covers of Semi-abelian Definition varieties
Mordell-Weil An Lω1,ω-sentence φ is ∆-‘not so big’, if each model of φ is Theorem small (realizes only countably many complete ∆-types over the categoricity empty set).
Definition
An Lω1,ω-sentence φ is ∆-small if there is a set X countable of complete ∆-types over the empty set and each model realizes some subset of X .
‘small’ means ∆ = Lω1,ω Small implies complet(able)
What is a complete theory?
John T. Baldwin
Logical Considerations If M is small then M satisfies a complete sentence. Covers of Semi-abelian varieties If φ is small then there is a complete sentence ψφ such that: Mordell-Weil φ ∧ ψφ have a countable model. Theorem categoricity So ψφ implies φ. The Lω1,ω-Vaught test
What is a complete theory?
John T. Baldwin Shelah If φ has an uncountable model M that is ∆-small for every Logical countable ∆ and φ is κ-categorical then φ is implied by a Considerations
Covers of complete sentence ψ with a model of cardinality κ. Semi-abelian ℵ1 varieties Keisler If φ has < 2 models of cardinality ℵ!, then for every Mordell-Weil countable ∆, φ is ∆ not so big. Theorem I.e. each model is ∆-small for every countable ∆. categoricity So we effectively have Vaught’s test. But only in ℵ1! And only for completability! Trivially, no. Take the disjunction of a ‘good’ sentence with one that has 2ℵ0 -countable models and no uncountable models.
Countable models I
What is a complete theory?
John T. Baldwin
Logical Considerations Must an ℵ -categorical sentence have only countably many Covers of 1 Semi-abelian countable models? varieties
Mordell-Weil Theorem categoricity Countable models I
What is a complete theory?
John T. Baldwin
Logical Considerations Must an ℵ -categorical sentence have only countably many Covers of 1 Semi-abelian countable models? varieties Mordell-Weil Trivially, no. Take the disjunction of a ‘good’ sentence with Theorem one that has 2ℵ0 -countable models and no uncountable models. categoricity Countable models II
What is a complete theory?
John T. Baldwin Is there a way to study the countable models of sufficiently nice Logical incomplete sentences? Considerations
Covers of Must an ℵ1-categorical sentence Semi-abelian varieties with the joint embedding property have only countably many
Mordell-Weil countable models? Theorem A direction: The Kesala-Hyttinen study of finitary abstract categoricity elementary classes. Another direction: Kierstead’s thesis using admissible model theory. Is categoricity in ℵ1 absolute?
Two specific research questions
What is a complete theory?
John T. Baldwin
Logical Considerations For φ a sentence in Lω1,ω: Covers of Semi-abelian varieties Does categoricity in κ > ℵ1 imply completeness
Mordell-Weil (completeability)? Theorem categoricity Two specific research questions
What is a complete theory?
John T. Baldwin
Logical Considerations For φ a sentence in Lω1,ω: Covers of Semi-abelian varieties Does categoricity in κ > ℵ1 imply completeness
Mordell-Weil (completeability)? Theorem categoricity Is categoricity in ℵ1 absolute? Model Theory and Mathematics
What is a complete theory?
John T. Baldwin Stability theory developed Logical Considerations 1 abstractly with the stability classification Covers of Semi-abelian 2 concretely by finding the stability class of important varieties mathematical theories and using the techniques of the Mordell-Weil Theorem abstract theory. categoricity The absoluteness of fundamental notions such as ℵ1-categoricity and stability liberated first order model theory from set theory. Infinitary Logic
What is a complete theory?
John T. Baldwin
Logical Considerations At the same time and largely unnoticed, Shelah developed the Covers of Semi-abelian fundamentals of stability theory for infinitary logic. varieties
Mordell-Weil It was not until Zilber’s exploration of complex exponentiation Theorem in the 1990’s that the significance of this work for mainstream categoricity mathematics was realized. More general questions
What is a complete theory?
John T. Baldwin A successful strategy for first order Logical Considerations Study the complete first order theory of the mathematical Covers of structure of interest. Semi-abelian varieties Mordell-Weil Infinitary logic Theorem categoricity What structures in what languages benefit from analysis in infinitary languages? What notion of complete is appropriate for such a study? Zilber: Yes!
A simple example
What is a complete theory?
John T. Baldwin
Logical ℵ Considerations Let (V , +) be a Q-vector space of cardinality 2 0 . ∗ Covers of Let h be a homomorphism from V to (C , ·) with kernel Z. Semi-abelian varieties Have I completely described a structure Mordell-Weil Theorem (V , +, h, C, +, ·)? categoricity A simple example
What is a complete theory?
John T. Baldwin
Logical ℵ Considerations Let (V , +) be a Q-vector space of cardinality 2 0 . ∗ Covers of Let h be a homomorphism from V to (C , ·) with kernel Z. Semi-abelian varieties Have I completely described a structure Mordell-Weil Theorem (V , +, h, C, +, ·)? categoricity Zilber: Yes! Acknowledgements
What is a complete theory?
John T. Baldwin
Logical Considerations
Covers of Semi-abelian Most of the ideas here are reworking and reorganizing Zilber’s varieties Ravello paper. Some proofs and some emphases are different. Mordell-Weil Theorem categoricity Covers of Algebraic Groups
What is a complete theory?
John T. Baldwin
Logical Definition A cover of a commutative algebraic group A(C) is a Considerations short exact sequence Covers of Semi-abelian varieties exp Mordell-Weil 0 → Z N → V → (C) → 1. (1) Theorem A categoricity where V is a Q vector space and A is an algebraic group, defined over k0 with the full structure imposed by (C, +, ·). Algebraic group
What is a complete theory?
John T. Baldwin
Logical For any algebraically closed field F and any algebraic group A Considerations there is also a set of formulas F defining an algebraically closed Covers of Semi-abelian field such that F(A(F )) and A(F ) are interdefinable. varieties If (V , A) is such a model with A = (F ), we identify F with Mordell-Weil A Theorem F(A). categoricity Thus, definable in the group is the same as definable in the underlying field. Axiomatizing Covers: first order
What is a complete theory?
John T. Baldwin Let A be a commutative algebraic group over an algebraically Logical closed field F . Considerations Let T be the first order theory asserting: Covers of A Semi-abelian varieties 1 (V , +, fq)q∈Q is a Q-vector space. Mordell-Weil Theorem 2 The complete first order theory of A(F ) in a language categoricity with a symbol for each k0-definable variety (where k0 is the field of definition of A). 3 exp is a group homomorphism from (V , +) to (A(F ), ·). Axiomatizing Covers: Lω1,ω
What is a complete theory?
John T. Baldwin
Logical Add to T Considerations A N Covers of Λ = Z asserting the kernel of exp is standard. Semi-abelian varieties
Mordell-Weil −1 N _ Theorem ((∃x ∈ (exp (1)) )(∀y)[exp(y) = 1 → Σi What is a complete theory? John T. Baldwin Logical For any : Considerations A Covers of N Semi-abelian TA + Λ = Z varieties Mordell-Weil 1 has arbitrarily large models Theorem categoricity 2 has the amalgamation property Associated Sequences What is a complete theory? John T. Baldwin Let K be a field and W ⊂ K r a variety defined over K. Logical 1 A sequence W associated with W over K is a family of Considerations 1/m Covers of varieties (defined over K) W such that Semi-abelian (W 1/mk )k = W 1/m, each W 1/m is a minimal K-variety. varieties Mordell-Weil 2 The sequence stabilizes with respect to p(x), an r-type Theorem over the empty set if there is exists an ` such that for categoricity every m, there is a unique K-definable variety V with V m = W 1/` and such that p(x) and hexp(x1/m`),... exp(xr /m`)i ∈ V is consistent. Pseudo-generating sequences What is a complete theory? John T. Baldwin Logical Considerations Let = (V , A) |= TA. hτ1, . . . , τN i ∈ V is a pseudogenerating Covers of V Semi-abelian tuple of Λ(V ) if for each m ∈ Z: varieties Mordell-Weil Theorem n1τ1 + ..., +nN τN ∈ mΛ iff gcd(n1,..., nN ) ∈ mZ. categoricity Expanded Language What is a complete theory? ∗ John T. L is the basic language. Form L by adding predicates: Baldwin We expand L to L∗ by adding the following formulas. Logical n Considerations 1 Ind (x) holds if x is a linearly independent n-tuple in V . ` Covers of 2 PG (x) holds if x is an `-tuple from Λ that satisfies for Semi-abelian varieties each m ∈ Z: Mordell-Weil Theorem n1τ1 + ..., +n`τ` ∈ mΛ iff gcd(n1,..., n`) ∈ mZ. categoricity W 3 Gen (x) holds if exp(x) satisfies the type of a generic point of the k-irreducible variety. −1 4 Rm(v) ↔ (∃y ∈ exp (1))[my = v]. Consequences I What is a complete theory? John T. Baldwin Logical N Considerations If TA + Λ(V ) = Z is small. Covers of Semi-abelian N ∗ varieties 1 TA + Λ(V ) = Z admits elimination of quantifiers in L . Mordell-Weil N Theorem 2 Every countable model of TA + Λ(V ) = Z + ‘infinite categoricity dimension’ is ω-homogeneous. Consequences II What is a complete theory? John T. Baldwin Logical Considerations Let r be the type of a pseudogenerating sequence. Covers of Semi-abelian N varieties If TA + Λ(V ) = Z is small. Mordell-Weil Theorem If k is finitely generated any sequence W associated with any categoricity W over k stabilizes with respect to r. Smallness and Completeness What is a complete theory? John T. Baldwin Logical Considerations Covers of Semi-abelian N varieties [JB] TA + Λ(V ) = Z has a finite number of completions. Mordell-Weil Theorem categoricity Aside: Characteristic p What is a complete theory? John T. Baldwin [Bays, Zilber] Consider Logical Considerations ∗ Covers of 0 → Z[1/p] → V → Fp → 0. Semi-abelian varieties ∗ Mordell-Weil where Z[1/p] is the localization at p and Fp is an infinite Theorem dimensional algebraically closed field of characteristic p. categoricity N ℵ TA + Λ(V ) = Z is not small. There are 2 0 completions - distinct minimal models. The theories must be analyzed separately; each is categorical. Choosing Roots What is a complete Definition theory? A multiplicatively closed divisible subgroup associated with John T. ∗ Baldwin a ∈ C , is a choice of a multiplicative subgroup isomorphic to Logical Q containing a . Considerations Covers of Definition Semi-abelian 1 1 varieties m Q m Q ∗ b1 ∈ b1 ,... b ∈ b ⊂ C , determine the isomorphism type of Mordell-Weil ` ` Theorem Q Q ∗ b1 ,... b` ⊂ C over F if given subgroups of the form categoricity Q Q ∗ c1 ,... c` ⊂ C and φm such that 1 1 1 1 m m m m φm : F (b1 ... b` ) → F (c1 ... c` ) is a field isomorphism it extends to Q Q Q Q φ∞ : F (b1 ,... b` ) → F (c1 ,... c` ). An Algebraic Condition What is a complete theory? John T. Baldwin Logical For = (C∗, ·): Considerations A Covers of F -Thumbtack Lemma Semi-abelian varieties ∗ Let F be a countable field. For any b1,... b` ⊂ C , there exists Mordell-Weil 1 1 Theorem m Q m Q ∗ an m such that b1 ∈ b1 ,... b` ∈ b` ⊂ C , determine the categoricity Q Q ∗ isomorphism type of b1 ,... b` ⊂ C over F . Context What is a complete theory? John T. Baldwin Logical Considerations ∗ Covers of For A = (C , ·), the thumbtack lemma is clearly stated and Semi-abelian varieties true. Mordell-Weil As A varies, the exact formulation is not clear (at least to me) Theorem and truth will vary with the choice of A. categoricity Proving smallness and more What is a complete theory? John T. Baldwin examining Zilber’s arguments Logical Considerations Smallness is equivalent to F -thumbtack for finitely generated Covers of Semi-abelian F . varieties ω-stability is equivalent to F -thumbtack for countable F . Mordell-Weil Theorem categoricity Zilber N The (full) Thumbtack Lemma is equivalent to TA + Λ = Z is excellent. Mordell-Weil Theorem What is a complete theory? John T. Baldwin Logical Considerations Covers of For a smooth elliptic curve, Semi-abelian A varieties If k is a finite algebraic extension Q, A(k) is a Mordell-Weil finitely generated abelian group. Theorem categoricity Smallness and Mordell-Weil What is a complete theory? John T. Baldwin Logical Considerations For any algebraic group : Covers of A Semi-abelian N varieties If TA + Λ(V ) = Z is small. Mordell-Weil Theorem If k is finitely generated over Q, Ator(k) is finite. categoricity smallness implies finite torsion: boundedness What is a complete theory? John T. Baldwin Definition Logical Considerations The algebraic group A is bounded if for every finitely Covers of Semi-abelian generated extension k of the field of definition k0 of A, there is varieties a d such that for every ` the Galois group of Gal(k˜, k) has only Mordell-Weil Theorem d-orbits on the set categoricity N ˜ ` X` = {ha1,..., aN i ∈ A` (k):(∃b)[a = exp(b/`) ∧ PG (b)] }. smallness implies bounded What is a complete theory? John T. Baldwin Lemma Logical If TA + Λ(V ) is small, the A is bounded. Considerations Covers of Proof. Every sequence over k associated with the type Semi-abelian ` varieties p = PG (x) stabilizes. Mordell-Weil Thus, there are only finitely many extensions of p to complete Theorem categoricity types over (V (K), A(K)) and by the homogeneity over the empty set we have a bound d on the number of orbits of pseudogenerating sets. But since each automorphism of V induces an automorphism of ˜ A`(k) for each `, we have the same bound in X`. smallness implies finite torsion What is a complete theory? Lemma John T. If is bounded, then for every finitely generated extension k of Baldwin A the field of definition k0, Ators(k) is finite. Logical Considerations Proof. We show that if φ(`) > d, there is no element of (k) Covers of A Semi-abelian that has order `. varieties Suppose a ∈ A(k) is a counterexample. Then a can be taken Mordell-Weil ˜ Theorem as the first element in an N-tuple a from A`(k) with categoricity a = exp(b/`) and PGN (b) For any m that is coprime to `, am also has order ` and can be extended to a sequence am, so that N am = exp(bm/`) with PG (bm). Thus the sequences am for m < ` and (m, `) = 1 represent distinct orbits in X` under Gal(k˜, k) (the first elements of the sequences are distinct elements of k). So if φ(`) > d, we have a contradiction. ω-stability What is a complete theory? John T. Baldwin Logical Definition Considerations φ is ω-stable if for every countable model of φ, there are only Covers of Semi-abelian countably many types over M that are realized in models of φ. varieties Mordell-Weil N Theorem If TA + Λ(V ) = Z is ω-stable. categoricity If k is countable any sequence W associated with any W over k stabilizes with respect to r. Quasiminimal Excellence What is a complete theory? John T. Baldwin A class (K, cl) is quasiminimal excellent if cl is a combinatorial Logical Considerations geometry which satisfies on each M ∈ K: Covers of Semi-abelian 1 there is a unique type of a basis, varieties 2 Mordell-Weil a technical homogeneity condition: Theorem ℵ0-homogeneity over ∅ and over models. categoricity 3 and the ‘excellence condition’ which follows. Conditions 1 and 2 are sufficient for ℵ1-categoricity. Necessary Notation What is a complete theory? John T. Baldwin In the following definition it is essential that ⊂ be understood Logical as proper subset. Considerations Covers of Definition Semi-abelian varieties − S Mordell-Weil 1 For any Y , cl (Y ) = X ⊂Y cl(X ). Theorem 2 We call C (the union of) an n-dimensional cl-independent categoricity system if C = cl−(Z) and Z is an independent set of cardinality n. Essence of Excellence What is a complete theory? John T. Baldwin Logical Considerations Covers of Semi-abelian There is a primary (unique prime) model over any finite varieties independent system. Mordell-Weil Theorem categoricity QM EXCELLENCE IMPLIES CATEGORICITY What is a complete theory? John T. Baldwin Logical QM Excellence implies by a direct limit argument: Considerations Covers of Lemma Semi-abelian varieties An isomorphism between independent X and Y extends to an Mordell-Weil Theorem isomorphism of cl(X ) and cl(Y ). categoricity This gives categoricity in all uncountable powers if the closure of finite sets is countable. Almost Quasiminimal Excellence What is a complete theory? John T. Let K be a class of L-structures which admit a function clM Baldwin mapping X ⊆ M to clM (X ) ⊆ M that satisfies the following Logical properties. Considerations Covers of 1 clM satisfies is a monotone idempotent closure operator Semi-abelian varieties with clM (X ) ∈ K that satisfies ‘excellence’ (But not Mordell-Weil exchange). Theorem 2 clM induces a quasiminimal excellent geometry on a categoricity distinguished sort U. 3 M = clM (U). 4 We call the class Almost Quasiminimal if the ‘excellence’ is dropped. Algebraic Formulations of Excellence What is a complete theory? John T. Baldwin Let S = {Fs : s ⊂ n} be an independent n-system of Logical algebraically closed fields contained in a suitable monster M. Considerations S Denote the subfield of M generated by ( Fs ) as k. Covers of s⊂n Semi-abelian varieties Canonical completions Mordell-Weil Theorem n Y categoricity A(k) = A ⊕ A(Fs ) s⊂n where An is a free Abelian group. Excellence What is a complete theory? John T. Baldwin ℵn ℵn+1 Logical The following are equivalent under VWGCH (2 < 2 ) Considerations Covers of 1 The cover of A is categorical in all uncountable κ. Semi-abelian varieties 2 The cover of A is categorical in all ℵn for n < ω. Mordell-Weil Theorem 3 The cover of A is almost quasiminimal excellent. categoricity 4 A satisfies the algebraic conditions ω-stability and homogeneity and has canonical completions. AQE and covers What is a complete theory? John T. Baldwin Logical Considerations Claim/Conjecture Covers of Semi-abelian varieties An almost quasiminimal class is ℵ1-categorical. Mordell-Weil Thus, an omega-stable cover is ℵ1-categorical. Theorem categoricity Are there A that are ω-stable but not excellent? AQE and covers What is a complete theory? John T. Baldwin Logical Considerations There are important mathematical topics that can only be Covers of Semi-abelian usefully formalized in infinitary logic. varieties Mordell-Weil There is a dynamic interplay between the study of such Theorem examples and the development of infinitary model theory. categoricity