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Stable and Simple Theories (Lecture Notes)

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Stable and Simple Theories (Lecture Notes) Stable and simple theories (Lecture Notes) Enrique Casanovas Universidad de Barcelona September 5, 2007 2 Contents 1 Preliminaries 7 2 ϕ-types, stability and simplicity 9 3 ∆-types and the local rank D(π, ∆, k) 15 4 Forking 19 5 Independence 23 6 The local rank CB∆(π) 27 7 Heirs and coheirs 31 8 Stable forking 35 9 Lascar strong types 41 10 The independence theorem 45 11 Canonical bases 49 12 More on independence 55 13 Supersimple theories 63 14 More ranks 71 15 Hyperimaginaries 77 16 Forking for hyperimaginaries 83 17 Canonical bases revisited 91 3 4 CONTENTS 18 Elimination of hyperimaginaries 97 Preface These are lecture notes from a seminar imparted the academic years 2005-06 and 2006-07 at the Department of Logic of the University of Barcelona. They are incomplete, in process of revision and enlargement. In particular, the preliminaries contain only a few statements. Advanced knowledge of Model Theory is throughout assumed. The notes are based on the work of many modeltheorists. The names of John Baldwin, Byunghan Kim, Daniel Lascar, Anand Pillay, Bruno Poizat, Saharon Shelah, Frank Wagner and Martin Ziegler deserve special mention. I thank to the participants in the Seminar, Hans Adler, Silvia Barbina, Javier Moreno, Rodrigo Pel´aez, Juan Francisco Pons, and Joris Potier for their patience and their remarks. Of course, I am the only responsible for the mistakes or inaccuracies that might appear in the text. 5 6 CONTENTS Chapter 1 Preliminaries T is a complete theory of language L with infinite models and C is its monster model. A, B, C are subsets of C and a, b, c are sequences of elements of C. a ∈ A means that all the elements in the sequence a belong to A. We use x,y for single variables but also for sequences of variables. The existence of indiscernible sequences is usually established using Ramsey’s Theorem. It is convenient to introduce here a more powerful method based on Erd¨os-Rado Theorem. Proposition 1.1 If κ ≥ |T | is a cardinal number, λ = i(2κ)+ , |A| ≤ κ and (ai : i < λ) + is a sequence of sequences ai of fixed length α < κ , then there is an A-indiscernible sequence (bi : i < ω) such that for each n < ω there are i0 < . < in < λ such that b0, . , bn ≡A ai0 , . , ain . In most of the applications α is a natural number and therefore the cardinal number λ depends only on |T | and |A|. Corollary 1.2 1. If (ai : i < λ) is indiscernible over A, there is some model M ⊇ A such that (ai : i < λ) is indiscernible over M. eq 2. If (ai : i < λ) is indiscernible over A, then it is also indiscernible over acl (A). A canonical parameter of a definable relation R is an imaginary element c such that for all f ∈ Aut(C), f(R) = R if and only if f(c) = c. It is unique up to interdefinability and it can be constructed starting with some ϕ(x, y) such that ϕ(C, a) = R by c = a/E where E is the 0-definable equivalence relation given by E(b, d) ⇔ ϕ(C, b) = ϕ(C, d) The following result on definability and imaginaries will be useful: Proposition 1.3 The following are equivalent for any definable relation R: 1. R is definable over any model M ⊇ A. 2. R has only finitely many A-conjugates. 3. R is a union of equivalence classes of some A-definable finite (i.e. with finitely many classes) equivalence relation. 4. R is definable over acleq(A). 7 8 CHAPTER 1. PRELIMINARIES Chapter 2 ϕ-types, stability and simplicity Definition 2.1 Let ϕ(x, y) ∈ L.A ϕ-formula over A is a formula of the form ϕ(x, a) or ¬ϕ(x, a) with a ∈ A.A ϕ-type over A is a consistent set of ϕ-formulas over A.A ϕ-type p(x) over A is complete if for every a ∈ A either ϕ(x, a) ∈ p or ¬ϕ(x, a) ∈ p. The set of all complete ϕ-types over A is Sϕ(A). Definition 2.2 Let ϕ(x, y) ∈ L. A complete ϕ-type p(x) over A is definable over B if there is a formula ψ(y) ∈ L(B) such that for all a ∈ A, ϕ(x, a) ∈ p ⇔ |= ψ(a) If B is not mentioned we understand that A = B. A complete type p(x) ∈ S(A) is definable if all its restrictions p ϕ are definable. Lemma 2.3 Let p(x) ∈ Sϕ(M) be definable. Then for each A ⊇ M there is a unique q ∈ Sϕ(A) extending p which is definable over M. Proof: Let ψ(y) ∈ L(M) be a definition of p. It is easy to check that {ϕ(x, a): a ∈ A, |= ψ(a)} ∪ {¬ϕ(x, a): a ∈ A, |= ¬ψ(a)} is consistent and it is in fact a complete ϕ- type over A extending p. On the other hand, if q1, q2 ∈ Sϕ(A) are M-definable extensions of p with definitions ψ1(y), ψ2(y) ∈ L(M), then M |= ∀y(ψ1(y) ↔ ψ2(y)), which implies C |= ∀y(ψ1(y) ↔ ψ2(y)) and therefore q1 = q2. 2 Definition 2.4 Let λ be an infinite cardinal number. We say that ϕ is λ-stable or stable in λ if for any set A, |A| ≤ λ ⇒ |Sϕ(A)| ≤ λ It is said that ϕ is stable if it is stable in some λ. Otherwise ϕ is called unstable. Proposition 2.5 The following conditions are equivalent for ϕ = ϕ(x, y) ∈ L. 1. ϕ(x, y) is stable. 2. Γϕ(ω) is inconsistent, where for any ordinal α, Γ (α) = {ϕ(x , y )ν(i) : ν ∈ 2α, i < α} ϕ ν νi and where ϕ0 = ϕ and ϕ1 = ¬ϕ. 9 10 CHAPTER 2. ϕ-TYPES, STABILITY AND SIMPLICITY 3. For any set A, any type p(x) ∈ Sϕ(A) is definable. 4. ϕ(x, y) is λ-stable for all λ. Moreover in 3. one can add that p is definable by a formula of the form ψ(y) = ∃x1 . xn∃y1 . ymχ(y, x1, . , xn, y1, . ym) where χ is a conjunction of formulas of the form ϕ(xi, yj), ¬ϕ(xi, yj), ϕ(xi, y) and ϕ(xi, y)- formulas over A. Proof: 1. ⇒ 2. Assume Γϕ(ω) is consistent. Let λ be an infinite cardinal number and µ <µ let µ be the least cardinal number such that 2 > λ. Then 2 ≤ λ. Since Γϕ(µ) is also <µ µ consistent, there is a sequence (bν : ν ∈ 2 ) such that for every ν ∈ 2 the set of ϕ-formulas ν(i) p (x) = {ϕ(x, b ) : i < µ} is consistent. Since p (x) is inconsistent with p 0 (x) for ν νi ν ν 0 µ <µ ν 6= ν , it follows that there are 2 > λ complete ϕ-types over the set A = {bν : ν ∈ 2 } but |A| ≤ λ. This shows that ϕ(x, y) is not λ-stable. 2. ⇒ 3. Let p(x) ∈ Sϕ(A) and assume Γϕ(ω) is inconsistent. Then also Γϕ(ω) ∪ S ν∈2ω p(xν ) is inconsistent and by compactness there is a least natural number n for which [ Γϕ(n) ∪ p(xν ) ν∈2n is inconsistent. Again by compactness, there is a finite subset p0(x) ⊆ p(x) such that S Γϕ(n) ∪ ν∈2n p0(xν ) is inconsistent. Then n > 0 and one can check that for any a ∈ A, [ ϕ(x, a) ∈ p ⇔ Γϕ(n − 1) ∪ p0(xν ) ∪ {ϕ(xν , a)} is consistent ν∈2n−1 that is n−1 <n−1 ^ ^ ϕ(x, a) ∈ p ⇔ ∃(xν : ν ∈ 2 )∃(yη : η ∈ 2 )( Γϕ(n − 1) ∧ p0(xν ) ∧ ϕ(xν , a)) ν∈2n−1 which is a definition of p of the form indicated above. 3. ⇒ 4. Since there are at most λ many definitions of the described form over a set A with |A| ≤ λ, there are also λ many complete ϕ-types over A. This shows that ϕ(x, y) is λ-stable for any λ but uses the hypothesis 3. with the added information on the form of the definition. Without this information we can only guarantee that it is stable in any λ ≥ |T |. But this is enough since after all we have established that 1. implies 4. 2 Remark 2.6 If ϕ is stable any global ϕ-type p ∈ Sϕ(C) is definable. Proof: The proof of 2 ⇒ 3 given for Proposition 2.5 works also for p ∈ Sϕ(C). 2 Definition 2.7 ϕ = ϕ(x, y) ∈ L has the order property if for some ai, bi (i < ω) the following holds: |= ϕ(ai, bj) ⇔ i < j Remark 2.8 1. ϕ(x, y) has the order property if and only if for some ai, bi (i < ω), |= ϕ(ai, bj) ⇔ i ≤ j. 2. ϕ(x, y) has the order property if and only if ¬ϕ(x, y) has the order property. 11 3. In the definition of the order property one can change the index set ω and its order by any infinite linear ordering. Lemma 2.9 Assume ϕ = ϕ(x, y) does not have the order property. If p ∈ Sϕ(A) is finitely satisfiable in A (which is always true if A is a model), then p is definable by a positive boolean combination of formulas of the form ϕ(a, y) with a ∈ A. Proof: Let X1,...,Xn be a family of subsets of a set A.
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