CHAPTERVI SATUaATION OF ULTRAPRODUCTS VI. 0. Introduction The ultraproduct construction is one of the most important methods in model theory, and unlike the primary model, the ultraproduct does not depend on the specific theories we deal with. The ultraproduct n,,,M,/D, for D an ultrafilter, M, L-models, is the product when we identify any two elements of n,,,M, (i.e., functions from I)which me equal for “almost ” every i. The important properties of the ultraproduct me: (1) Log’s Theorem: a (first-order) sentence is satisfied by the ultra- product 8 it is satisfied by almost every M,. (2) The L,-reduct of the ultraproduct is the ultraproduct of the L,- reducts . It is also important to know: (3) The ultraproduct is A-compact if the ultrafilter is A-good and N,-incomplete (thus we can construct A-mturated models). (4) Two L-models are elementarily equivalent iff they have iso- morphic ultrapowers. So (4) gives an “algebraic ” characterization of elementary equivalence. It is also important that by ultraproducts, we can get an ccalgebraic’yproof of the compactness theorem. The ultraproduct is not central in this book (and outside this chapter, we shall use it only once or twice), and the first three sections of this chapter have nothing to do with stability. However we have developed here a classification of theories using stability and associated notions, different from the one Keisler [x67] gives (using his ordering a).We show that our classification is helpful in the analysis of Keisler’s order, thus giving more evidence of the naturalness of the classification. We also deal with the categoricity of pseudo-elementary classes, and saturation of ultraproducts. In Section 1 we give the basic properties of ultraproducts and re- duced products and regular filters. In Section 2 we deal mainly with A-good filters. This property is 32 1 defined combinatorically, but its main property is: an ultrdter D is A-good iff for every family of A-saturated L-models M,,i E I, M,/D is A-compaot iff for every A-saturated atomic Boolean algebra My Mr/Dis A-saturated. If we require D to be N,-incomplete aa well, we .can omit the condition that the models am A-eaturated. The Boolean algebra comes quite naturally, but we can replace them by dense order. This is done aa follows: suppose M1/Dis A+-compacf for every dense order M.For any model N,we expand it by encoding the set of finite sequences of formulae (with parameters) which are consistent, and some natural relations and functions and get Nl. Note they form naturally a tree and by it we can “spk” on sequences of formula in N’,/D which are not nedy“standard”. We look in N’,/D,and any 1-type p in W/D,1p1 < A, and define by induction on i s lpl elements a, of the tree, increasing with i, and “consistent ” with p (in the inner sense we demand this for every finite subset of p separately and this is expres- sible): and letting p, = {v,: i < IpI}, we want that 9, will “appear” in the “sequence” a,, the element “realizing” alpl is as required. The translation of the tree to the dense (linear) order is by intervals. Now from this we can deduce for singular A that A-goodness implies A + -goodness. The importance of constructing ultrafilters is clear, and this is the subjeot of Section 3. Some of those problems are connected to problems on large oardinale, and consistency results and are outside our scope (e.g., N,-complete ultrafilters, non-regular ultrafilters). It is easy to construct regular ultrafilters as they sped on the existence of a family of sets, and this is done in 1.3(4). Good ultrafilters posed a more difficult problem ae their definition says: for every function into D there is a function into D . .. Keisler [K 641, constructs a good ultrafilter on A when A+ = 2” in A+ steps; in each step we have a uni- form filter generated by A subsets of A. When 2” > A+ ,this seems to fail, but Kunen [Ku 721 suggests an alternative proof he takes a family of 2” independent functions from A onto A; and construct the ultrafilter in 2* steps, after the ith step, we have to delete s 8, + li I of them only. Now it is natural to ask where there is a (regular) p-good ultrdter over h whioh is not p+-good. But p s A+, and by 2.10 p is regular. For p a successor we use the product of ultrafilters, whose most important property is NIX*/Dl x D, = (Mr/Dl)J/Dl,i.e., if we take ultrapower twice, it is like taking it once for the product. By this we get h+good, not A + +-good ultrafilters over p 2 A. We use independent functions to construct ultrafilters with various properties; e.g., m. 0 01 INTRODUaTION 323 (1) For p 2 A, h regular, over p them is a regular ultrafilter which is A-good but not h + -good. (2) If 2” 2 h = A%, then there is a regular ultrafilter D over p and At < w (t E I) suoh that ntsInt/D = h (the conditions h = Po s 2Y are neoeeeary (for an infinite A), see Exercise 2.10). (3) If 2” 2 A, p 2 K; h > K regular, then there is a regular ultrafilter D over p suoh that h = lcf(K, D) =d& mink: in {a E #ID: #/D t a < a for every a < K} there is a set unbounded from below of cardinality x}. We prove also that Thd is essentially “simpler” than the theory of linear order. In the exeroises, we indicate why any two elementarily equivalent models have isomorphic ultmproducts, and other results. In Section 4 we define Keisler’s order and prove some theorems on it. We oould change the definition slightly, e.g., by omitting the require- ment that D is an ultrafilter over A, and &ill get the same results. We also show that some ultraproducts are not A-compact. Section 6 is the heavy section, utilizing the results from I1 on d-n- indiscernible sets and their dimemione, to determine how saturated are ultraproducts. We find which elementary olasses (of countable type) contain categorical pseudo-elementary claases. E.g. , we prove THEOREM 0.1: If bf W a model of T, T countable and without the f.c.p., D an K,-implete UUraJilter Over I, then M1/DiS ~~/D-eaturated. THEOREM 0.2: If T ie countable, euperdable and without the f.c.p., then theie a theory TI,T E TI,lTIl = 2u0, 8Wh that any uncountable model of T, haa eaturated L( T)-red&. Other theorems (6.1, VIII, 2.1) show this result is the best possible. Summing up our results we get the following picture for Keisler’s order on countable theories. Let H- = {T:T countable, without the f.c.p.}, Kscp= (5”: T#K,,,,, is countable and stable with the f.c.p.}, = {T:T countable, unstable K~~(T)< OO}, Kcdt= {T:T countable, Kodt(T)= GO, T without the strict order property}, H,, = {T:T countable and with the strict order property}. THEOREM 0.3: (1) If T,,TI are both Hmin[Kaop][Hmu]then they are @‘-equi~alent,i.e., TI@ Ta @ Tl. In Section 6 we find how compaot rn ultralimite. PROBLEM 0.1: It would be very desirable to prove that (1A) T,, Ta E Ktndimplies TI, Ta are @-equivalent, (1B) T,, T, E Kdt (or we should aek &o whether K,,(T) = 00, K,,,(T)= 00) implies T,, Ta rn @- equhalent. This will complete the model-theoretio sham of investigating Keisler’s order for countable theories. For this it seems reasonable to try to find for T E KiDda theory parallel to 11, Section 2 for stable theories. PROBLEM 0.2: It would be desirable to replace in 0.3(2) the “con- sistent with ,’ by “provable form ,’. This is a problem in constructing ultrafilters. Another problem on ultrafilters is 6.1. PROBLEM 0.3: On Keisler’s order for uncountable T see Shelah [Sh 721. We conjecture that for every T there are a countable T, and simple Ta (Definition 6.4) such that T’@ T o [T’@ T, and T’@ Ta]. PROBLEM 0.4: We conjecture: if T is not @-minimal then T is not @,,,-minimal, and if M1/Dis (21’I)+-compact, M is A-compact, then M’/D is A-compact (see 6.7). In this chapter D will always denote a filter (or ultrafilter) over I (or sometimes J)(For the definition and some theorems see Section 1 of the Appendix.) VI.1. Reduced praduots and regular filters DEWINITION1.1 : Let D be ti filter over I and for each iE I let Mt be an L-model. (1) We define an equivalence relation zD (or z for short) over ni,1 IJft I* f z g iff {i E I:f(i)= g(i)} E D. (2) We define the L-model n,,M,/D (the reduced product of the Mi’s): OH. VI, $13 RIDDUOED PRODUUTS AND REUULAR FILTER8 325 (A) Its universe is {f/k : f E n,,, [Mil}. (B) If R is en n-place predicate symbol in L then RM = {(f1/#, . .,fn/z):{i €1:(fl(i), . .,f,,(i))E RMd}E D}. (C) If P is an n-place function symbol in L then PM(fl,.