Model Theory

1. Max Euwe Model Theory Jouko Väänänen Model theory

2. Types • A an L-structure, XA, bA • The complete type of b in A over X is the set tpA(b/X)={(x,a) : A(b,a), aX} These are the formulas, or rather the sets defined by them Here b is a one element sequence A b Model theory

3. Example • tpG(b/X)={(x,a) : (G,E)(b,a), aX} • X = elements we ``know´´. • Type of b describes b: • ```Is not a neighbor of a´´ • `Has two neighbors´´ • ``Has a neighbor that is not a neighbor of a´´ • Etc, etc X a b G Model theory

4. Types • A complete type in A over X is any complete type of some sequence in an elementary extension of A • A type in A over X is any subset of a complete type in A over X. Complete type of b in A over X • complete type in A over X  typein A over X Model theory

5. Realized vs. omitted • A type is realized in A if it is contained in the complete type of some sequence of elements of A, otherwise omitted by A A A Model theory

6. Examples (Z,+,0,<) has an elementary extension with infinitely large elements, but it omits the type of an infinitely large element (R,+,×,0,1,<) has an elementary extension with ``infinitesimal´´ elements, but it omits the type of an ``infinitesimal´´. Model theory

7. Characterization of types • (x) is a typein A over X iff (x) is finitely realized in A • (x) is a complete typein A over X iff (x) is maximal w.r.t. being finitely realized in A. Model theory

8. Elementary amalgamation D   f g B C  b c Model theory

9. Taking isomorphic copy D’ of D D’  g B C b c Model theory

10. Special case: b=c lists the elements of an elementary substructure A We can throw C into the same structure as B keeping the common elementary substructure A fixed. D’  g C B A Model theory

11. Elementary amalgamation • Suppose (B,b)(C,c). • Claim: There are a model D and elementary embeddings f:BD and g:CD such that fb=gc. • Let us take a new constant symbol defor each e in B and a new constant symbol sefor each e in C. • Let T consist of all (de1,...,den) (”first kind”) where B (e1,...,en) and e1,...,enB, plus all (se1,...,sen) (”second kind”) where C (e1,...,en) and e1,...,enC, plusall dbi=sci (”third kind”) . Model theory

12. Claim: T is consistent • Suppose T0 is a finite subset of T. • Let (se1,...,sen) be the conjunction of the elements of T0 which are of the second kind (true in C). • We separate out those se1,...,sen where ei is from c. We get (sc1,...,scn,se’1,...,se’m). • Now Cx1... xm (sc1,...,scn,x1,...,xm). • Since (B,a)(C,c), we have Bx1... xm (da1,...,dan,x1,...,xm). • On the other hand, the sentences of the first kind can be satisfied in an expansion of B for trivial reasons. The sentences of the thrid kind are satisfied in a similar expansion of B by interpreting sci as ai. • So B can be expanded to a model of T0. Model theory

13. The amalgamation • Let D be a model of T. • If e is in B, let f(e)=(de)D. • If e is in C, let g(e)=(se)D. • B(e1,...,en)  D(de1,...,den)  D(f(e1),...,f(en)) • C(e1,...,en)  D(se1,...,sen)  D(g(e1),...,g(en)) • So f and g are elementary embeddings. Model theory

14. f(a)=g(c) • Ddai=sci • f(ai) = (dai)D = (sci)D =g(ci) • QED Model theory

15. Notes • We can take an isomorphic copy D’ of D and make f=id. • Alternatively we can take an isomorphic copy D’ of D and make g=id. Model theory

16. Application • For any structure A there is an elementary extension B such that every type over A with respect to A is relaized in B. • Proof: List the types pi, i<λ. Suppose Ai satisfies pi. Construct, by amalgamation, an elementary chain Bi, i<λ such that Bi satisfies pi. Model theory

17. B ... ... B3 A3 A2 B2 A1 B1 A0 B0 A Model theory