2/4. Simplicity Theory Frank Wagner's book Simple Theories. various articles. Nice Paper: From Stability to Simplicity / Kim & Pillay JSL? Stability = Theory of independence + multiplicity. types of models + how they are related Framework of FO model theory. Organization: Basic Definitions & Framework Simplicity & independence "Canonical" Constructions? - adding random predicates etc. "Generic" Constructions? Simple groups Lovely Pairs. Analogue of Monster Model in CAT. Defn: Let L be a signature, ∆ ⊆ L_0, i.e., set of formulas. Assume that ∆ is closed under positive boolean combinations. Then ∆ is a positive fragment of L. We fix a positive fragment ∆ or ∆. A "formula" always means a formula in ∆. Definition: An L-structure M is a (K is a cardinal) if it satisfies: K-universal domain Homogeneity: If A ⊆ M, |A| ≤ K, and f: A → M is a mapping st. ∀(x) ∈ ∆, ∀a ∈ A, M ⊨ φ(a) ⇒ M ⊨ φ(f(a)) [We say that f is a partial *-endomorphism of M.] Then ∃ E Aut(M) which extends f. 2/4 Simplicity Theory 2. Compactness: If Σ(x) is a set of ∆-formulas where x is a (possibly infinite) tuple of variables, then either Σ(x) is realized in U, or there is Σ₀ ⊆ Σ, Σ₀ finite, which is not realized in U. Fact: Assume M is a universal domain. a ∈ U, φ ∈ L and φ(a) ∧ ∃y ψ(a,y) ⊢ ∃y ψ(a,y) Then there is a formula ∀x ψ(x,c) ∈ tp(ā) st. ∀x ψ(x,c) → ∃y ψ(x,y) Proof: Set p(x) = {φ(x) : Σ ⊨ φ(a)}. Then p(x) is not realized in U. (by homogeneity. (Otherwise, we'd get c ∈ U st. p(c) ⊆ tp(a,b) so a₁: A → M is a partial endomorphism extending to an automorphism, f, and then U ⊨ φ(a) ⇒ U ⊨ φ(f(a)) is not realized in U) Remark: If ∆ is a universal domain, set ∆ = Ø and A = U. M is a universal domain wrt Ø. Proof: Homogeneity becomes easier. Compactness: by replacing each ∃y ψ(x,y) with ∀y ψ(x,y) new variables Therefore we allow ourselves the following additional assumption: Convention: For every φ(x,y) ∈ ∆, the formula ∃y ψ(x,y) is equivalent in ∆ to a partial ∆-type. We say that U is a universal domain, φ ∈ L, T = Th(U). Lemma: S_n(L,T) = {tp(ā)|ā ∈ Uⁿ} = all maximal sets of ∆-formulas. (x₁,...,xₙ) consistent with T. Proof: (1) Let p(x) = tp(ā) where ā ∈ Uⁿ. Since U ⊨ p(x) ∪ T, then p(x) is consistent with T. If b ⊨ p(x) ∪ T, (but b ≠ ā) then p(x) ∪ {x ≠ b} ⊆ p(x). So b ⊨ ∃y (x ≠ y ∧ ψ(x,y)) (by the fact) So p(x) ∪ {∃y (x ≠ y)} ∪ T is inconsistent. (2) Assume that p(x) is maximal (consistent with T. (by compactness) Since it is consistent with T, it is realized in U (by compactness) Say by ā ∈ U ⊨ p(x). Then p(x) ⊆ tp(ā), and tp(ā) is consistent with T. So by maximality p = tp(ā). From homogeneity, two tuples are the same type ⇔ they correspond by an automorphism of U. Therefore S_n(L,T) = Uⁿ/Aut(U). The logic topology on S_n(L,T): If ψ(x) ∈ ∆, then <ψ> ⊆ S_n(L,T) = {tp(ā): ψ(ā) ∈ p} Closed The closed sets are generated by the sets of this form.

18.996A Simplicity Theory: Lecture 1 Notes

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