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,. . . ,fn) = f where for every i E I,f(i)= PMi(f,(i),. . .,fn(i)) and PM(f1/#,. . . ,fn/z) = PM(f,,.. .,fn)/#. (3) If D is an ultrafilter, the redud product is called an ultra- product. If M, = N for every i E I, then we write N1/Dfor nt,g M,/D and call it a power instead of a product. Remurk. As D is a filter, z really is an equivalence relation andf, z gl for 1 5 1 s n implies {i: (f,(i),. . .,f,,(i)) E R'c} E D o {i: (gl(i), . . ., gn(i )) E RMg}E D adPM(fly * * * ,fn) z P"(g1, * * - 8 gn) 80 ntsl Mt/D is a well-defined L-model. Notice that for a term T, 7(f1,. . . ,fn)(i)3: 4fl(i), - ' * Y fn(i)I* Notation. (1) If M, is an L-model for i E I, P # L, N = nrSIMt/D and (a,,P,) is an L u {P}-model for every i E I, then in an abuse of nota- tion (N,PN) = n,,, (X;, P,)/D. (2) We do not strictly distinguish between a E In,,, M,/DI and a representative of a in n,,,lM,l. We write (a[i]:i E I) for the repre- sentative of a. Also (in this chapter) if ii = (aoy..., a,,-l) E In,, M,/DIythen 7i[iI = (ao[i],. .. , a,". (3) We identify a and Remark. In Definition 2.1(1) we om take n = 0, so every atomio formula is a basic Horn formula. Also, every negation of an atomic formula is a baeio Horn formula. THEOREM 1.2: If p(l) i8 u Hmforncukc, M = n,,MJD, iz E 1M1 and {i E I:Mi C tp,P[i]]}E D, thiK b p[iz]. Proof. By induction on 'p. DERTNITION 1.3: (1) The family {Xi:i c A} of subsets of I is regular if for w E A, nx, z 0 1~1< N,. f6Y (2) The above-mentioned family regularizes D (which is a filter over I)if it is regular and X,E D for i < A. (3) The filter D over I is A-regular if some {X,:i < A} regularizes it. (4) D is regular if it ia 111-regular. (6) D is A-incomplete if there are X,E D( i < a c A), nlSaX, = 0. (6) D is A-complete if for X,E D (i c a c A), X,E D. LEMMA 1.3: (1) If D is A-regular, p 5 A, then D ii3 p-regular. (2) Tk$&r D i8 N,-btiX~m~Zeteiff D i8 K,-reg&r. (3) D is not 111 +-regular. (4) Over every infinite I there is a IIl-regular filter, hence a regular ultrafilter. Proof. (1) Immediate. (2) Immediate. (3) If {X,: i < 111 +} c D is regular, each X,,being a member of D, is non-empty, so choosej, E X,.Clearly for somej*, I{i c 111 + :j, = j*}l = 111 + but n {X,:j, = j*} 2 {j*} # 0, contradiction. (4) Clearly it suffices to find a set J, I JI 3: II] over which there is a regular filter. So let: J = &,(I) = {w: w E I, lw] c No}, for WEJ, Oa. VI, 0 13 REDUOED PRODUaTB AND REGULAR FILTERS 327 X, = {w: WEJ,w c w} c J. The filter D generated by {X,: WEJ) (see Definition 1.1 and Theorem 1.1(1), (2) of the Appendix) is reg- ularized by {X{f,:i E 4. The second phrase follows by 1.1(4) of the Appendix. Proof of the Cene88 Th-mrem (I, 1.1): From 1.1 and 1.3 we get a proof of the Compactness Theorem. Let T be a set of sentences, such that every finite t c T has a model Mt. Let J = BN0(T)and let D be the filter over J defined in the proof of 1.3(4), and D* an ultrafilter over J,extending D (existsby 1.1( 4) of the Appendix). Let M 2: n,, M,/D*, thenforevery#ET,{wEJ:M, C#} 2 {wEJ:#Ew}= X,,,ED E D*. Hence, by hg’s Theorem, M C #, so M is a model of T. DEFINITION1.4: A type is atomic if all its formulas are atomic. THEOREM 1.4: (1) Let D be afilter over I. Then the following condition% are quid&. (A) D i8 A+*. (B) For ewery L and every family Mt (i E I)of L-models, every atomic type p aver 0 in each Mi, 1p1 5 A, i8 reat%zed in nisrMJD. (C) For every M every atonzic type g over in W/D,lpl 5 A, is realized in M’/D (notice: p is in M iff it is in MIID). (D) Condition (C) holds for M = (rSNo(A), r). (2) If in (1) D is an ultrafilter we can replace “atomic type” by “type” in the umditifma. Proof. (1) (A) * (B) Let N = nfsrMi/D, {X,: a < A} regularize D, p = {v,(Z): a c A} be the type. For my finite w E A let a,[<] E 1M,1 realize {v,(Z): a E w}. Now as {X,: a c A} is regular, for every i E I, w(i) = {a < A: i e X,} is finite. Define E IN1 by 8[i] = GwiJi]. Hence for a < A {i E I: Mi C rp,p[i]}2 {i E I:a E w(i)} = {i E I:i E X,} = X, E D. Hence N C tpa[i3], so a realizes p. (B)* (C) Use (B) for Mi = (Mya)oslarl, for every i E I. (Clearly p is a type in M.) (C) * (D) Holds by their definition. (D) * (A) Let N = M1/Dand let p = {{a} E 2: a < A}, so clearly p is finitely satisfiable in My(hence in N). Suppose a E IN1 realizes p, and let for a < h X, = {i E I:M C {a} E a[i]}. 328 SATURATION OR’ ULTRAPRODUCTS [a.m, 8 1 Then X, E D aa N C {a} c a, and for w c A, nUeu,Xu= {i~l:w c a[i]}, so clearly it is empty iff w is irhite. (2) The same proof using LOPSTheorem. DEFINITION1.5: M is (A, d, m)-compact if every (A,m)-typep over 1M1, in M, 1p1 < A, is realized in M. For d the set of atomic formulaa of L(M), we write at. Similarly the sets of quantifier free formulas, con- junctions of atomic formulaa, and formulaa of quantifier depth sn are denoted by qf, cnat, and qd,, respectively; ifm = 1 we omit it. DEFINITION1.6: (1) For models M,, Ml and an ordinal a, (L(M,) n L(Ml) 2 L) we define a game Gt(Ho,Ml) between the players I and I1 aa follows: At the @h move player I chooses I E {0,1} and afi E lMil, and then player I1 chooses a;-i E ~Ml-i~. The play ends after a moves and then player I1 wins if for every atomic formula cp(z,,. . . ,2,) E L and ordinals p(O), . . ., @(n) < a, Jfo C ~[a&),- - - 9 $(,)I * C ~[aj(o),* - * 9 a;,,)], and player 1 wins otherwise. (2) A strategy (of a player) is a sequence of functions fB @ < a) which “tells” him what to do (fBfor the /?th move) depending only on the previous choices in the play. A winning strategy is a strategy such that in any play in which the player chooses according to it, he wins. A player wine in the game if he has a winning strategy. (3) We omit L if L = L(M,) = L(Ml). LEMMA 1.5: (1) In the game Gz(Mo,Ml) nd bothpZaymu win. (2) The relation “player I1 wine in Gz(M,N)” is an equivalence re- lation among models. (3) If player 11 [I] wine in Gi(MoyM,), L(1) E L, /3 5 a [a 5 8, L E L(l)] then he m’ne in G&l,(M,, Ml). (4) If M,, M, are ~sonwrph~c,then player I1 Wine in Ua(M,, Ml). (5) If M,, M, are elementarily equivalent, and A-eaturated, then player I1 wine in G”M,, MI). Proof. (1) Suppose both players win, then in some play, both use their winning strategy, but one of them must lose this play, contradiction. (2) Combine the strategies to prove transitivity. Symmetry is trivial, and reflexivity follows by (4). (3) Trivial. OH. m,8 11 REDUCED PRODUCTS AND REUULAR BILTEFtS 329 (4) If 8: IM,l+ lMll is the isomorphism then player I1 chooses a;-1 so that g(a;) = a;. (5) Player I1 has to choose at-' such that tp,({az: a zs; p), 0, 22,) = tP*({d: a PI, 0, Ml). Proof. Immediate. THEOREM 1.7: The L*-mud.ele M,, M,are elementarily equivalent iff for every jinite L E L*, n < w, player II wine in G;(M,, Ml). Proof. Let us define by induction on n, the sets F;,,,(L), F;,JL) of formulas ~(z,,, . ., z,,,-,)E L. (i) F;,,,(L) is the set of atomic formulas y(s,, . . . , z,,,-,)E L. (ii) If F;,,,(L) is defined, F;,,,(L) is the set of formulas of the form A {cp(Z)u(@'6r): cp(@ E F;,,,,(L)} for F c F;,,,(L). (iii) If F;,,,+,(L) is defined e+l,m(L)= {(~~rn)F(~o,* - - 9 2,): dzo, * * * 9 z,) E %n+l(L)I* Clearly any sentence in L is equivalent to a disjunction of some of the sentences in F;,, for every sufficiently large n (in fact for its quantifier depth), and this holds for any formula. If M,, M, are elementarily equivalent, L c L* finite, n < W, a winning strategy for player I1 in Gft(M,, M,)is to preserve the satis- faction of (*I M, C ?[at,. . ., a!-,] iff M, C cp[ua, . . ., a:-,] for every F(zO, * * -9 zk-l) E F;-k.k(L)- For k = 0 this holds as M,,M, me elementarily equivalent; and by the definition of F;-k,k(L) it is easy to play so that (*) is preserved. For the other direction, it suffices to prove that if ut, . . . , a!-,, a;, . . . , a;- have been chosen in a play in which player I1 uses a winning strategy for Gft(M,, M,) then (*) holds (because then for any 330 SATURATION OF ULT'RDRODUOTS [CH. VI, 5 1 $ E (J I'l,o(L), L c_ L*, L finite, Mo C $ e Ml C $, and aa any sentence $ of L* is a sentence of some finite sublenguage, bl,, M,are elementarily equivalent). The proof is may, by induction on n - k. THEOREM 1.8: If &I$',Mi are elementarily equivalent L-modeze for i E I, ILI S A, D a A-regeclarfilter over I, a < A+, thenplayer I1 wiw in Ga(Mo,M1), wlrere No = no Mf'/D, M1= n16xMi/D. (80 Mo, M1 are elementarily equivalent.) Proof. The number of formulaa 'p(~~<~,,. . ., z4(,,)), E L,p(1) < a,is 5 A, 80 let {'pj(~sco,n,. . ., z,,~,,~,.,): j < A} be an enumeration of the set of thew formulaa (possibly with repetitions). Let {Xj:j < A} E D regularize D, and for t E I, let w(t) = {j: t E Xj},(which is finite), let hO.t), - - - Zv(Mt).t)} = u {%#(o.n,.. .} where y(0, t) < - - - < y(n(t),t). j€ur(t) Let Lt be the minimal sublanguage of L such that vjE & for j E w(t). So by the previous theorem, player I1 wins in P&(t)+l(Mf,Mi)for every t. Now we shall describe the winning strategy of player I1 in GC(M0,W): when he haa to choose E lM1-il he chooses each aj-I[t] separately. If /? $ {y(k,t): k I n(t)},he chooses a;-I[t] arbitrarily. If fl = y(k, t)he imagines he is playing qt)+'(Yp,Mi), that a~(,,,,[t], , . ., a:(&-l,o[t], ak- la[t] and a:(k,t,[t]have been chosen, and he chooses ai,;;.',,[t]by his winning strategy in GZt)+l(Mf,M;). It is eaay to check that this is a winning strategy for Gg(M0,Ml). CONCLUSION 1.9: If MyN are elementarily equivalent, D ie a A-regular Jilter over I then Mx/Dis (A, A, m)-wmpad iff Nx/Di8 (A, A, m)-wmpa&. LEMMA 1.10: If for each t E I player I1 win8 in Ga(Mf,Mt), D a jElter over I, then player I1 wins in Ga(n,, Mf/D, n,,,Mt/D). Proof. In view of the proofs of 1.8, 1.9, we leave it to the reader. CONCLUSION 1.11: If M,, N, are elementarily equivalent and h- saturated, D a Ner, then M = n,,, Mt/Die (A, A, m)-wmpact ifl N = n,,,Nt/D i8 (A, 4 m)*pa&. Proof. By 1.6(6) player I1 wins in Gh(Mt,N,). By 1.10 player I1 wins in GA(M,N). Henoe, if A > No,by l.6(4) M is (h,A,m)-compact iff fl is (A, A, m)-compact. If h = KO,every model is (A, A, m)-compact; so we finish. CH. m,8 11 REDUCED PRODUCTS AND REGULAR FILTERS 33 1 Notation. If A,, i E I is a family of non-empty sets and D is a filter over I we define nt,, A,/D = (flz :f E nierA,} where z is the equivalence relation from Definition 1.1. If A, is a ctlrdind A, for every i E I, we write nter &/Dfor IrIteI WDI and nier A, for Inw 4-$0 llrI:er WDII = ntsr ll~:ll/D. If Dis afilter over I, J c I then D 1 J = {X n J:XED}so when I - J$D, D tJisnottrivid. EXERCIBE 1.2: (1) If D is a filter over I, J E D then nt, M,/D z nteJMt/D J (the isomorphism maps (art]:t E I)/Dto (a[t]:t EJ)/D); 80 Titer &/D = nteJ &/D IJ. (2) If J c I, I - J$D, D a p-regulrtr filter over I then D J ie p-regular. EXERCIBE 1.3: (1) Show that if (M,P) = nteI (Mt,PJ/D then IPI = nt,r IPtIID. (2) Show that n,,, A,/D < nlolA, ; and {t : A, < p,}ED implies nter &/D s nter pt/D. (3) Show that if QE1A,/D > 0 then n,,, A,/D 2 n,,, A,/D, r J, for any filter D, 1 D,J E I. THEOREM 1.12: (1) If D is a p-reguZurjZter over I, & (t E I)(injnite) cardinale, and h = nter &/D > 0, x = mink: {t E I: & s x} E D} then h 1 x". (2) If in (1) p = IIl then h = f. (3) Moreover tkre are natural numbers nt mlr that nt,l nt/D 2 2", and if D is reghr ntsr nt/D = 21'1. Proqf. (1) By Exercise 1.3(3) we can assume D is an ultrafilter (by extending the original filter to an ultrafilter containing {t: & > x,} for each x1 < x (by 1.1(2), (3) of the Appendix). Case I: x > p. By Exercise 1.2(1) we can wume that for every t, p 5 & s x. Now we want to define models Mt (t E I)so that llMt 11 = At 332 SATURATION OF ULTRAPRODUUTB rm. w, 8 1 and f pairwise contradictory types will be realized in nt,, dd,/D. Let L consist of the one-place predicates Pa (i < x, a < p). We define Mt such that for each a, for i < & the Pi(Mt)are pairwise disjoint, for & 5 i < x Pk(iWt)= lM,l and for every finite w E p and function f:w+ XM,I= (3x)[/\,,,P{(")(x)]. It is quite easy to construct such a model : lMtl = {cy: w c p, IwI < No,f: w + & a function}, Pk(Mt) = {c/" E lMtl: a E w,f (a)= i or i 2 &}. By the definition of x, for every i < j < x, a < p {t E I:M: I=-1(3%)(pk(%) A Pi(%))} 1 {t E I: li 1, lj I < &} E D, hence M = MJD C 7(3s)[Pk(x)A Pi(z)].Let for each r) E 'x, p,, = {e[al(z):a < p}, 80 the p,,'~are pairwise contradictory; 60 it suffices to prove they are realized in M. But clearly each p,, is finitely satisfiable in each Mt, 80 by 1.4 it is realized in M. Cme 11: x 5 p. By Exercise 1.3(2) it follows from 1.12(3). (2) h 2 xfi by (1) and h Ixu as we can assume (again by Exercise 1/41)) A, 5 x for every t, so by Exercise 1.3(2), &/D5 n:pr h: S x~r~= f . (3) We use a method similar to the one used in (1). Let L consist of the one place predicates Pf, (1 = 0,1, a < p), {X,:j < p} regularize D, and w(t) = {j:t E X,} which is finite. We define Mt such that llMtll = 2Iw(:)l, and denote this number by nt. We define it such that forj E w(t), P:, Pf are complementary and for any finite w E p Mt I= A Po,(%) A A Pi(%)). crew crew(:) - w For this let ]M*l = {cw: w G w(t)}, Pf,(M,)= {cw: 1 = 0 and u E w or I = 1 and a E w(t) - w or cc $ w(t)}. Let M = nfs,Mt/D,then P:(M), Pi(M)are a partition of IMI, and for every w E p, p, = {Pf,(x):a < p and a E w e 1 = 0) is realized in M, 80 llMll 2 2", but 11M11 = ntel llMtll/D = n:EIn,/D. When D is regular we get equality as in (2). CONCLUSION 1.13: If D is a regular filter over I, t7m hr/D = hir!. OH. VI, 8 21 FILTERS AND COMPACTNESS OF PRODUCTS 333 EXERCI8E 1.4: Suppose the filter D is p-regular for every p < A, and Xis singular. Prove that D is A-regular. [Hint: If {Xf : i < p} regularizes D for each p < A, K = cf A < A, A = 2 p(a), p(a) < A, then {Xf@)n Xi: a < K, i < p(a)} a EXERCISE 1.5: Suppose A is a regular cardinal, prove there is a filter D which is p-regulw for each p < A, but is not A-regular. [Hint: D will be generated by {Xf : i < p < A}, for each p, {Xf : i < p} is regular, but for every a < w, and distinct pn < h (n < a), and finite sets w, c {i: i < p,} nnsanlEtu. Xtm # 0. Use 1.4 of the Appendix to prow this is possible.] QUESTION 1.6: In Exercise 1.6, if K 2 AWo clearly we can find such a D over K. Can we always find such a D over A? (For A a successor it is trivial.) EXERCISE 1.7: Prove that in 1.4(1) (and (2)) to the four equivalent conditions we can add: (6) If F is a set of I A atomic formulas (in 5 A variables) which is finitely satisfiable in each Mt, then F is satisfiable in n,,, MJD. (6) In (6) instead of atomic formulas we can allow Horn formulas. EXERCIBE 1.8: Prove that for any ultrafilter D end cardinal A, D is A-incomplete iff D is not A-complete. EXERCIAE 1.9: If M = (A, R,, . . .), M1/D= N = (A', Ri, . . .) and A E lMll, then N; = (MI,A, R,, . . .),/D = (M{/D,A', R;, . . .). (Itis convenient to use this for MI = (H(K),E, M), where H(K)is the family of sets hereditarily of power < K, M E B(K)). VI.2. Goad Biters and compactness of reduced products DEFINITION2.1: (1) A filter D (over I) is A-good if for every p < A, every f:A&) + D which is monotonic [i.e., w c u f(u) c f(w)]has a refinement g : flH0(p)--t D [i.e., g(w) E f(w)]which is multiplicative [i.e., g(w u u) = g(w)n ~(u)]. (2) D is good if it is 111 +-good. Remark. (1) Clearly we can add “g is monotonic”, because multiplicity impliea monotonicity . (2) We can delete the ctssumption “f is m~n~tonic”becauee for any f:8&) + D, let f * be defined by f *(w) = nUswf (u).AS l{u: u E w}l = 21wl c KO,and D, being a filter, is closed under finite intersections, f *(w) E D, and any multiplicative refinement off *, is a multiplicative refinement of fi and f * is monotonic. CW2.1 : Euery j&r is Et,-good. Prmf. Let f:Bb(w) --+ D. Define g(w) = n {f (u):mas u s max w}. As for emh w E~~(w)max w < u,and the number of u c {n:n 5 max w} is finite, clearly g(w)E D. The function g is multiplicative aa max(u U 20) = max{max v, max w}. Proof. (1) * (2) Let N = nter Yt/D,p be an (at,m)-type over IN1 in N, 1131 < Asp = {v,(Z; a,): a < p < A} (v, atomic). Let for w f (w) = {t E I: Mt t (33) A,o,,, tpa(P; BJt])}. As p is finitely satisfiable in N,some 8, E IN1 realims {cp,@; aa):a E w}, so X& = {t: Mt t cpa[Zw[t], a,$)]} E D, (for a E w) hence naewX;w E D, but clearly it is a subset of f(w), henm f(w) E D. As D is A-good, there is a multiplicative function g :El,&) + D refining f. Let for t E I, ~(t)= {a < p: t E g({a})}. NOW,for every finite w E w(t), a E w t E g({a}), hence by the multiplicativity of g, t E g(w); and aa g refinesf, t E f (w). Hence Mt t (32) A,,, q~,(3,7i,[t]),and &B this holds for any finite w c w(t), pt = {v,(Z; Za[t]): a ~w(t)}is finitely satisfiable in Ht. As Mt is h-compact, there is 8[t] E lMtl which rsalizes pt, and let 8 = (. . . ,8[t], . . .)ter/D.Let us prove that 8 realizes p; for every a {t: Mt C v,@[t], 7i,[t]]} 2 {t: a E w(t)} = {t: t E g({a})} = g({a}) E D hence N t v,[& a,]. OH. w, 8 21 FILTERS AND OOMPAOTNESS OF PRODUCTS 335 (2) + (3) Trivial. (3) + (1) Let p < h andf : LYNo(p)3 D be monotonic. For each t E I we define a&] E ]Mi for a < p such that: (*) for every finite w rzLYN0(p), bl c (3z)( A z c a&] A P(z) e t Ef (w). aEw ) [If we want to define a,[t] for finitely many a's only, we can easily choose them in M,,.By the h-saturation of M we can choose all a,[t], a < PI. So let a, = (. . . , a,[t], . . .)ter/D and p = {z E a,: a < p} u {P(z)}. If p c p is finite, then for some finite w c p p E {z c a,:aEw}U {P(z)},and, by (*) {t: M t (3z)[/\aew 2 %[tI A P(z)I}= {t: t Ef(W)} = f(w)E D. Hence M t (3%)A p, so p is snitely satisfiable in N = Mf/D and by (3) it is realized in it, so let c E IN1 realize p. Define g(w) = {t €1: Aaswc[t] c a&] A P(c[t])},and it is easy to check that g: LYwo(p)+ D is multiplicative and refines f. THEOREM 2.3: Let D be a$lter over I, h a cardinal >KO then tk follm'ng conditions are equivdent : (1) D is &good and N,-implete. (2) Por every family of L-models Mt (t E I), ntpfMJD is (A, at, m)- co?wt. (3) N = Mi/D ie (p+,at)-compact for every p < h (M,,--asde$nedin 2.2( 3)). Proof. (2) =r (3) Trivial. (3) * (1) The set {{a} c z: a < p} is a set of formulas over lM,,l, and it is snitely satisfiable in N,,,hence some c E IN1 realizes it. Let for a CLAIM 2.4: If D ie pf-goOd and N,-incomplete then D is p-regular. Proof. As D is X,-incomplete, there are X, E D for n < w, X, = 0. Let f :S,(p) + D be defined by f (w)= Xlwl,and let g : BN0(p)+ D be multiplicative refinement off. Then for a < p, g({a}) E D,and for any infinite w E p, if t E naewg({a}), then for each n < w, choose w(n) E w, lw(n)l = 12, and then t E g({a}) = g(w(n))c.X,, hence t E n,,<,X, = 0, so necessarily naewg({a}) = 0, hence {g({a}): a < p} regularizes D. THEOREM 2.5: Let D be a A-good, K,-incomplete Jilter over I, D, a Jilter over I, D E D,. Let ILI < A, L = L(MP) = L(Mi) for t E I and M' = rite, MflD,. If NP,Mi are elementarily equivalent for every t E I, then player I1 wine in GA(M0,M1). Remark. Compare with 1.8, 1.10. We assume more and prove more. See Exercise 2.3. Proof. We use the notation of Definition 1.6. Players 11's strategy is to play so that for every formula ~(q,. . ., z,) E L, and a, > - . - > a, {t E I:MP C ~(a:~[t],. . ., aEn[t])} = {t E I:Mi C ~(at,[t],. . ., ain[t])}mod D and therefore mod D, since D E D, (where X = Y mod D means I - (X - Y)- (Y - X)ED). Suppose a:, a: have been chosen for i < a < A, and player I has chosen 1 E (0, 11, af E IM'I, and player I1 has to choose at-'. By the symmetry we can assume 1 = 0. Let @ be the set of formulas Q = Q(Z, ql,.. . , xi*), Q EL, i,, . . ., in < a. Let @ = {Q~:fi < p = I@]}; clearly p < A, hence by 2.4 D is p-regular, so some family {XB: fi < p} regularizes D. Let h(t) = @ < p: MP c ~~(aE[tl,aPo&I, . . .I>. For w E SHo(p)we define f (W)= {t E I:fi E W * t E x,9, and Mi b (3%) 'p,9(X; ai,,8))[t],. . . )}. E€wnh(t)A By the induction assumption {t E 1: M: b (3%) A 'p8(Z; a;l,8)[t1,. . . Eewnh(t) )> = {t E I:Mi k (32) A ~~(2;ai,,,,[t], . . . t?ewnh(t) CH. m,8 21 FILTERS AND COMPACTNESS OF PRODUCTS 337 hence it is eaay to check that f (w)E D. So some g : S,,(p) + D is multiplicative and refines f. Notice that @: t E g({/3})} c (Is: t EX,} which is finite. So for each t E I, define a:[t] E lM1l so that it realizes (~(2;a:(1,8,[t], . . .): /3 E g(t)} and a: = (. .., a;[t],. . .),,I/D. For the induction hypothesis, for each Q, we get only an inclusion ; by applying it also for -Q, we get equality mod D. THEOREM 2.6: The following ditione on theJilter D over I, and the cardinal A > No are equivalent: (1) D iS A-good. (2) For every A-saturated nmkl M of Tora[the theory of deme linear order I with no Jinite or last element], MI/D is (A, at)-mpact (where M = (pq, I)). (3) For every A-saturated dlM of Tor,, and for every set A s IMI/DI which is linearly ordered by s , every atomic 1-type p over A in MIID, IpI < A, is realized in MIID. Proof. The implication (1)=. (2) follows by 2.2 and (2) =- (3) is trivial; so assume (3), and we shall prove (1). CLAIM 2.7: Assume (3) from 2.6, and let N = (IN[,I) be a A-saturated rooted tree (i.e., I is a partial order, and for every a E IN I, {b E IN I : b I a} is linearly ordered, and 0 I a for every a E IN/);8uch that each a E IN1 hag 11 NII = 11 NI( immediate successors, and for every a, b E IN I there is is a nmxiwl c, c I a and c I b. Then (1) If N'/D C cf I c, for i < j < a, where u < A, then for some c, NI/D C c, I c for i < a. (2) If "/D ca1, I 41, A Ch, I CXa, A CXa, I cL,fori(l) < $1) < u,i(2) < j(2) < p, where u,p < A, then for some c, NI/D C c: I c I cs for i < a,j < p. Proof. We first note that if A is a linearly ordered subset of IM'/DI, M a model of Tor&,then if every (at, 1)-type over A in M1/Dis realized in MIID, then for every m, 1 I m < w, every (cnat, m)-type over A in MI/D is realized in MIID. Now let us choose for each a E IN I ; a A-saturated linear order (2) u < by 1 = 0, (3) b < u,E = 1, (4) let c be the maximal element for which c s a,c s b;c < a, c < by and a' cCb' where a', b' ES'~,a' 5 a, b' s b. It is easy to check that N, = (A*, s*)is B dense hem order, with firat and last elements and it is A-saturated. Hence it follows 888iIy by our hypothesis that if A E IN',/DI is linemly ordered by s*, then every atomic (n + 1)-type p over A in Ni/D, 1p1 < A, n < o is realized in N',/D. Now define a function P. (1) For a E IN],F(a) = (Po(a),Fl(a)) where Po(a)= (a, 0), Fl(a)= (a, 1) so Po(a),Fl(a) E lNll and for a E INr/DI, ~(a)= << . . ...~o(a[t~) - - .>tJD, < - Pi WI)- * - >tdD)E IWDI- (2) For atomic formulas 'p = p(z; y), define m'p) = P('p)(%B,, = P(z = Y) = 1% = Yo A 21 = YiI, m 5 9) = [zo s Yo A Yo 5 Y1 A Y1 5 511, F(y 5 2) = [Yo S 5 A 20 5 21 A z1 5 YJ. (3) For an atomic formula 'p(z; a) define Wz;4) = P('p)(zo:O,z1; p(a. (4) For a set p of atomic formulas 'p(z; a) let P(p)= {F('p@;4): 'pk 4 EP}. Now it is eaay to check that for every such type p in N, p is realized in N iff F(p)is realized in N,. Hence by 2.8 below for each atomic type p in Nr/D,p is realized in Nr/Diff P(p)is realized in N',/D. For each p as in Z.Q(A) or (B), F(p) is an atomic 2-type over some linearly ordered subset of JN',/DJin N',/D, so F(p)is realized in ",/D, hence p is realized in Nr/D,hence we get our conclusion. i.e., P(F~(Z', = #I($, 61) (I = 1,2) implies 91 = pa * #1 = $2. (3) For every p E Dom F, p ia redid in M ifl P(p) = {P[p(Zl, a)]:y(Z1; a) Ep} ie redid in N. Then if p = {pt(Z1;at):i < a} is a set of atormi0 fotrnulae in Mr/D, undfmwyi < a{tEI:pt(P;~i[t])~DomE3~Dt~piar~idin Mr/Diff P(p) = {P[yf(Z1;Q]: i < a} ia redid in Nr/D. Prmf. Suppose E realizes ~(EElMI/Dl); for every t EI let w(t) = {i < a: M C pt(8[t]; a#]) and pi(%';a#]) E Dom E") E D and define P[t] E IN1 so that it realizes {P[pl(Z1; qt])]: i E w(t)}. It is easy to oheok that Z1 realizes F(p).The other direotion is eaqtoo. Continuation of the pmf of 2.6. Let M be any h-saturated model, IL(M)l s IlMll, and we shall prove that iKz/Dis (A, at)-compaot. This is suffioient by 2.2. Let r={(po(2,a~),...,pm-l(2,a~-l)): m < o,at~lJflr(~r~L(Jf), pf atomio, and Af C (32) A pt@,af)}. iem Clearly II'J = J1MJ1,so there is a one-to-one function g from ]MIonto I'. Let us define some new relations and functions over lMl (aeauming that they 4 L(N)). (0) c* will be g'l(( )). (1) s : a s b o g(a) is an initial segment of g(b). (2) Fl: P(a)is an element of M realizing g(a). (3) Pe (for eaoh conjunotion p(z; @)of atomio formulas): PJb, a) o -oh oonjunot of p(z;a) appears in g(b). (4) Qe (for eaoh oonjunotion p(z;ji) of atomic formulas): Q,(b, a) o cp(x;a)is consistent withg(b),i.e., g(b)^ (+o(x;d'),.. . , pnpl(x;a"-') ~r where p(z; a) = At<,,#{(2; #). (6) FZ (for eaoh p(z; 1) &B in (4): PZ(b, a) = c ifQ,(b, a) and g(c) = g(b)^(#o(s; a'), . . ., #,,-l(s; a"-')) or +,(b, a) and c = b. 340 SA"RA!FIOW OR' ULTRAPRODUOTS [m. 5 2 (notice that in (6) if M C (34942;a), then M C (3z)[Af., qr(z;at) A dx;a)l). Let us define N = (Myc*, s , PlyP,, &,, P:, P:),, and let N1be a A-saturated elementary extension of N, [INl11 = [INl11 , and Ml be the L(M)-reduct of N,. Clearly Ml is A-saturated, hence by 1.11 M*/Dis (A, at)-compaet iff M, = Mi/Dis (A, at)-compact, so it suffices to prove that Mi/D is (A, at)-compact. Clearly Ma is the L(M)-reduct of N, = N',/D. It is easy to check that (lN1l,5%) satisfies the requirement of claim 2.7, hence also the conclusion of that claim. Let p = {tpf(z;af): i < a. < A} be a cnat-type in M, which is w.1.o.g. closed under conjunctions, and we will prove that it is realized in M,. We define by induction on i Ia. elements cf E such that: (A) j < i * N, C C$ 5 cf. (B) For every fl < ao, Na C &(pa(cfs3fi)- (C) If i = j + 1, N, C P,,(Cf,a,). Case I: i = 0. Let co = c* (the root of the tree). Conditions (A), (C) are vacuous, as for condition (B), for every /3 N C (Vij)(Vz)[rps(z, 8)+ &,&*, jj)] and this is a Horn sentence, so N, also satisfies it. As p is a type in Ma, N, C (32)9~&, Zfi) hence Na C QCp4(c*,a#). Case 11: i + 1. Let cf+l = P:,(cfyZf). For condition (A) notice that (i) N t (Vzyz)(x 5 y A y ls; z +z 5 z), (ii) N I= (VzP)(z s P:JXY P)), and both are Horn sentences, hence Na satisfies them. By (ii) N, C cf s c,+~,and by (i) and the induction hypothesis j < i * N, Ccj 5 c:+1. As for Condition (B), for each fl < a. there is y = y(fl) < a. such that p,(z; PI = 'pv(z;Pi, Sa) = vfi(z;Pi) A vi(% 9,) and 8, = afi-4, (remem- ber p is closed under conjunctions). Clearly N C (VZ, zl,gl, ~,)[&,,(z; jj1, P,) A 21 = P:Jz9 Pa) +&cpp(zly 8113 and as this is a Horn sentence Na satisfies it. As N, C &,&; a, si,) A cf+1 = qc,Y a clearly N2 c &,a[c:+l,%I. OH. VI, 8 21 FILTERS AND COMPACTNESS OF PRODUCTS 34 1 As for Condition (C),N C (Vg, zl,z)[&,,(z, 8) A z, = P;Jz, 8)+ P,Jzl, g)] so by condition (B) clearly N, C P,,(ct+,,at). Case 111: i = 6 is a limit ordinal. By Claim 2.7( 1) and Condition (A) for j < 8, clearly there is do E lN,l such that j < 6 =r N, C c, 4 do. So do satisfies Condition (A) on c6, but not necessarily Condition (B). We now define by induction of j 4 a. elements d, E lN,l such that (*) jl < j, 4 ao, y < 6 * N, C cy 4 dj2 A d5* I We have already defined do; for limit j, the existence of d, follows by part (2) of Claim 2.7 (using (*) and Condition (A)). For j -+- 1 let a?,+, = Pz,(d,, a,), and we shall prove that (*) holds. Notice that N C (vW~)(q,(z9a, 2) and this is a Horn sentence, hence N, satisfies it, hence N, C d,,, Id,, hence (as &'a is transitive) j, < j + 1 5 N, C d,+, Id,l. On the other hand, N C (~8NVZl9Z,"l I 2, A &a& 8)--f 21 I F:,(z,, a11 and this is a Horn sentence, hence N, satisfies it. As for y < 6 N, C cy 5 d, (by the induction hypothesis on j)and N, C &,,(cy, 8,) (by the induction hypothesis on i = 6) clearly N, C cy I P&(d,, a,), but P:,(d,, 8,) = d,+,. So we prove (*) forj + 1. So we finish defining d,,j I ao,and let c6 = dao. Condition (A) holds by (*), Condition (C) is vacuous and Condition (B) holds because for eachp < ao, N c (vg)(vzl,z)r&,,(z, 8) /t 21 5 z +QQJq(Z1r ?a1 and this is a Horn sentence, hence N, satisfies it, and N, I= duo s dB+,. Now N, C &e,,(dB +is 3,) because N C (vz, 8,z)[~g(z, 8) +Q,p,(P:,(z, ji), 811; this is a Horn sentence, and as p is a type in M,, for some b E ]N,l N, t ag[b, @gI hence N, I=&,p(p:,(d,, %), ad. So we finish the definition of ct, i Iao. We shall prove that P1(cao) realizes p and thus finish the proof. Now for each /I < a. N C (Vg, zl,z2)[Pe,(z1, 8) A z1 S 22 + P,,(z,, a)] and N C (Vz)(Vjj)[P&, 8)+ tpB(P1(z),g)] and they are Horn sentences hence N, satisfies it, and N, C Pmp(cB+,,Gg) A cg+, I c,,, by Condition (C). Hence N, C rpg[P1(c,,), BB] for every /3 < ao. THEOREM 2.9: Let h > No, D a Jilter over I. Then the following conditions are equivalent. 342 SATURATION OF ULTRAPRODUCTS [CH. 8 2 (1) D is A-good and K,-inmmplete. (2) For every model H of Tor,, M'/D is (A, at)-mpact. (3) NLlD is (A, at)-compact, where N, = <"'p, <), <--whit order, for euch p < A. Proqf. (1) + (2), (1) =+- (3) Hold by 2.3. (2) * (1) By 2.6 it sdces to prove that D is 8,-incomplete. As (&, 5 )I/Dis (A, at)-compact (&-therationals) for some c E I(&, s)I/D~, n 5 c for each n < w, and let X, = {t E I:n < ~[t]}.Clearly X, E D, (I X, = 0. n (3) * (1) The proof is similar to that of 2.6, hence we leave it to the reader. CONCLUSION 2.10: If A i.9 a singular Cardinal and thefilter D is A-good, thD i8 A+-good. Proqf. We use 2.6(3), so let A€ be a A+-saturatedmodel of Tor,, A s IN1 = IM'/DI be linearly ordered by s , and p be an atomic 1-type over A in N, 1p1 s A, and we shall prove that p is realized in N, thus finishing. If for some a, (z = a) EP, a realizes p (because p is finitely satisfiable in N).SO let p = {U 5 Z: u E A,} U {X < a:u E As}. As A, c_ A, A, is linearly ordered by 5 , so it hm a cofinal sequence {a,:i < pl}, pl a regular cardinal (so (Vu~A,)(3i< p,)(a s a,) and a, E Al). As p1 is regular, p1 < A. Similarly there are a' E Aa (i < pa) such that (Va~A,)(3i< pa)(ai s a), pa regular. As M satisfies the Horn sentence saying 5 is transitive, an element realizes p if it realizes p1 = {a, s x: i < pl} u {z 4 a,: i c pa}. As lpll c A, and D is A-good, by 2.2 N realizes plyso we finish. Rernurk. See Exercise 2.7. Up to now we have dealt with filters and atomic compactness. By the following lemma, we do not need to deal separately with ultrafilters and compactness. LEMMA 2.11 : Let D be an ultrafilter over I, A > Noa cardinal. (1) For every MyMI/D is (A, &)-cumpact iff for every A€, W/Dis A--pact. OH. 5 21 FILTERS AND UOMPAUl"Es8 OW PRODUaTS 343 (2) For evey L, and LdlsMty n:,, MJD is (A, ati)-wmpwt iff for every L and L-nwaeb Mt, ntsI Mt/D $8 h-wm?mct. (3) A8 (I) [2] W for AdWtZ&& M[MJ Proof. In all parts the if part is trivial. For the other direction, for each model iK define M* aa follows: LW*)= {%a: 9m E L(M)}Y (%3) w Places) 1M*1 = 1M1, R$*) = {a E IN!:M k tp[a]}. By Logs Theorem it is easy to check that (ntsIM,/D)* = nt,I Mr/D hence Mr/Dis A-compact iff AZ*'/D is (A, at)-compact and nsI Mt/D is A-compact ii€ nt61M:/D is (A, at)-compact, and iK is A-saturated iff M* is A-saturated. Hence the "only if" part follows easily. THEOREM 2.12: 8uppo8e 2" = A+, IIl = A, D a good, K,incumplete, UltraCjElter over I; MyN are elementarily equidnt nzodeb of mraidity Proof. By I, 1.11 every two elementarily equivalent A+-satursted models of power A+ are isomorphic. By Log's Theorem Mr/D,Nr/D am elementdy equivalent. So, by the symmetry, it suffices to prove that Mr/D haa cardinality s h + and is A + -saturated. Now llMr/Dll < ~~M~~~r~< (A+)A < 2"'" = A+ and by 2.3 and 2.11 Mr/Dis A+-compact. As JL(M)Is A, HI/D is A+- saturated, so we finish. THEOREM 2.13: If D & a good ultrafler over I, nternt/D 2 KO,then nteI nJD = 2Irl. Proof. If D is 8,-complete, clearly for some n {t: n, = n} E D hence nt/D = n; hence D is XI-incomplete (by Exercise 1.8) so by 2.4 D is regular. For each n let M, be the following model: M,, = (n + 2", P,, Bn>, where P,,,B,, am one-place and two place relations respectively, P,,= (0, .. . , n- l} and R, is such that for every w E P, for some i,n < i < n+2", and for every k < n,kEwoR,(k,i). Let for t~1,m(t) = [log, ntl - 1, 80 ll~m(:)ll5 %, but for every n {t: llJfln(:)ll 5 .} E 344 8A"RATION OF ULTRAPRODUCTS [OH. VI, 5 2 {t: nt s 2"+l} $ D. Hence M = (A, P, R) = nteI M,,,,,)/D is an in- finite model and is 111 +-saturated. So clearly JPI > IIJ (otherwise {P(z)A x # a:a~P}is omitted by M) and ]A1 2 21'1 (for let C E P, ICI = 111, then for everyB s C, p, = {R(a;x)if(aEB): ~EC} is realized by some b,eA, so IAl 2 21'1 = 21'1. We can conclude and the other inequality is Exercise 1.3(2). EXERCISE 2.1: Show that to the list of equivalent conditions in 2.3 we can add: (4) If p is an atomic 1-type in M = niEIMf/D of cardinality EXERClSE 2.2: Show that in 2.5 instead of assuming q,Mi are elementarily equivalent for every t E I it suffices to assume nter M!/D and rite, MilD are elementarily equivalent. EXERCISE 2.3: Show that in 2.6 we can omit "Dis 8,-incomplete" but must then demand that for each t E I, M;, M: are A-saturated and elementarily equivalent. EXERCISE 2.4: Show that to the list of equivalent conditions in 2.9 we can add. (4) (i) N:/D is (A, at)-compact for some p. (ii) D is p-regular for each p < A. EXERCILYE 2.5: Let E be a family of subsets of I closed under finite intersection, and let E generate the filter D. Every monotonic f :SNo(p) -+ E has a refinement g :SNo(p) + E which is multiplicative iff D is p+-good. EXERCISE 2.6: (1) Every A-complete filter is A+-good. (Hint: See 2.1.) (2) If D is p-complete for each p < A, A singular, then D is A+- complete. EXERCISE 2.7: If A is regular 5 Ill, then over I there is a non-A+- complete filter D which is A-complete. Moreover D is p-good iff p S, A+ (Hint: D is generated by {X,:i < A}, X, decreasing, 0,X, = 0.) CH. M,5 31 CONSTRUOTINQ ULTRAFILTERS 345 EXERCISE 2.8: Suppose the filter D over I is p+-incomplete and A+-good. Show there am X,E D (i < A) such that the intersection of any p X1)s is empty. EXERCISE 2.9: In Definition 2.1(1), when A = p$, it suffices to take I.1 = Po. EXERCISE 2.10: Suppose D is an 24,-incomplete filter over I, and A = ntErnt/D2 24,. Prove that Po= A. [Hint: Let M = (w, <, +, x ,P). where P(x) = [GI,N = Mr/D,and n* = (. . ., nt,. . .)/D.and for a E INI, 1.1 = l{b E INI: N t a < b}l. Then In*] = A, 1.1 2 KO =- 1.1 = IP(a)l,and let nt = P(n*),nf+, = P(nf),and for every sequenoe = (b,,; n < w), bk < nf, let 1); = (n,*bo+ n:b, + -.. + nzbk < x < ntob0+ - - . + nfb, + nz: k < w} clearly p; is realized by some ai;, thus providing nkInf I = AHo distinct elements of {b: b < n*}.] EXERCISE 2.11: In Theorem 2.12 instead “D good” we can assume only “there is a good K,-incomplete filter D, G D.” EXERCISE 2.12: Show that no 24,-incomplete filter over I is 1I1+ +- good. VI.3. Constructing ultrafilters We shall show that over every cardinality there is a good ultrafilter. We define the product of ultrafilters, and find how regular and good the product is. Then we assume 2Wo > N, and MA (Martin’s axiom) to prove the existence of an ultrafilter D over w, such that for every N,- saturated model M of Find,Ma/D is N,-saturated. THEOREM 3.1 : Over any A there ia a good K,-inunnplete ultrajlter. Proof, We first give a definition and prove some claims. Meanwhile let D denote a filter over A, and 59’ denote a family of functions from A onto A. DEFINITION3.1 : 59’ is called independent mod D if for every n < w and distinct go, . . ., qn-, E 59’ and every jo,. . . ,j,-, < h {a < A: go(a) = jo,. . ., g,,-l(a) = jn-,}# 0 mod D (and D is non-trivial). CLAIM 3.2: Let Do be the@er over A generded by {A}. Ththe ia a famay 8 of eardidity 2" which b id- mod Do. Procf. This is a reetatement of 1.6 of the Appendix for a pdculm c&88. CLAIM 3.3: Let Q be hui?epemod D, and 8 E A. Then for 81nne jinite W C 0,5%- 8' ia id- mod Dl or mod Day wheDI[Da] ia thejElter generated by D U {IS)[DU {A - IS)]. Proof. If 8 is not independent mod D,, then for some n < w there are distinct go, . . ., g,,- ,E Q and there are joy.. .,j,,-, < A such that: W, = {a < A:go(a) =joy...,g,,-,(a) =j,,-,}= 0 mod D,, hence W,c A - 8 mod D (notice that n = 0 mems D, is trivial). Let 5%' = 8- {go, .. . , g,, - and assume 8' is not independent mod Day 80 there are m < w and ju,. . .,jm-l< A and distinct go,. . ., gm-' E Q' such that Wa = {a < A: go(.) = jO,. . ., #'"'''(a) = jm-'}= 0 mod Day hence Wa c 8 mod D. So W,n Wa = 0 mod D, contradicting the independence of 8 mod D. CLAIM 3.4: 8-e Q is independent mod D, g E Q, 9' = B-{g}, and f :BNo(A)+ D W momtonic. Then there is a fler D, D c D, and a mdiplicdve function f' ;tlN0(A) --t D' refining f mhthat Q' W id- pended mod D'. Procf. Let {w,: a < A} be an enumeration of Lgn,,(A), let f'(w) = @: g@) = a, 8 E f (w,), w C w,} and D the filter generated by D u v({i}):i < A}. Clearly D' is non-trivial and f' is multiplicative, into D', and a refinement off. Let us prove 3' is independent mod D, so let jo,. . ., j,,- < A, and go, . . . , gn-, E Q', the glYsbeing distinct and W = (8: go(8) = jo, . . .Y !&I-,@)= it-,}. We must prove W # 0 mod D.For this it suilhes to prove that if w E&~(A), W nf'(w) # 0 mod D. Let w = w,, 80 so we finish, &B 8 is independent mod D and f (w)E D. OH. VI, 8 31 CONSTaUCTINa ULTRAFILTERS 347 Proof of 3.1. Let (8,: a < 2”, a > 0 even} be an enumeratiiga of (8:8 G A}, and dfor: a < 2A, a odd} an enumeration of the functions f : &,(A) +- {S:8 G A}, each appearing 2A times. Let Do be the filter over A generated by {A}, and Sobe independent mod Do and of cardin- ality 2A. Now we define by induction on a D,, g, such that: (i) ’3, is independent mod D,, (5) 1’3 - 5 KO + lal, (iii) for /3 < a, DB G D,, ’38 2 ’3,, (iv) if a is a limit ordinal D, = U,<, D,, ’3, = n8<,g,, (v) if a is even, a > 0, 8, E D,+, or A - 8, E D,+l, (vi) if a is odd, f, : &,(A) + D,, fa monotonic then there is a multi- plicative refinement f: of fa, f: : &,(A) + I),+,. The induction is easy: for a = 0 we have defined, for a limit see (iv) (using iii), for a + 1, a even, a > 0 use Claim 3.3 and for a + 1, a odd use Claim 3.4, and for a + 1, a = 0 let D, = {{a:n < g(a) < w}: n < w}, ’3, = ’3,,-{g], where g E go. Now D,A is an ultrafilter by (v). Iff : &,(A) + DaA, then for some a < 2A,f :&,(A) +- D,, and for some /3, a < /3 < 2”, f,3 = f, /3 is odd, so (when f is monotonic) there is a multiplicative refinement f’ off, f’ : &,(A) + DB+1 E D+ SO D,A is a A+ -good ultraater over h. AS D, DaA, D,A is N,-incomplete. CONCLUSION 3.6: If My N are elementarily equivalent models of T, of cardinality 5 A+, ITI 5 A, 2A = A+, then for some ultrajlter D over A, MA/Dg NA/D. Proof. Immediate by 2.12 and 3.1. DEFINITION3.2: If D, is a filter over I, (1 = 1, 2) then D1 x D, is the family of sets 8 E I, z I, such that: {i E I,: {j E I,: (j, i)€8) E Dl} E D,. LEMMA 3.6: (1) D1 x D, is ajlter over I, x I,. (2) If D, is an ultrafilter over I, (I = 1, 2) then D, x Da ie an dtra- jlter over I, x I,. (3) I%i,j>er1 Jfi,j/Di x Da ILrP(nrer, Jfi,j/Di)/Da. (4) Mrixra/Dlx D, 2 (Mrl/DJa/Da. Proof. Immediate. 348 SATURATION OF ULTRAPRODUCTS [OH. VI, 8 3 LEMMA 3.7: (1) If D, OT Da i8 h-regular then D, x Da is h-regular. (2) D, x D, i8 h-gd ifl D, iS hyood; PO& that D, i8 81- incomplete. Prmf. (1) Immediate by 1.4(1)(A), (C) and 3.6(4). (2) Immediate by 2.3 and 3.6(4). If D, is h-good, for any h-saturated model M, Mri ra/Dl x D, (M'l/Dl)'a/D, (by 3.6) is (A, at)-compact (by 2.3). Hence by 2.3 D, x D, is h-good. If D1 x Da is h-good we get the result similarly. CONCLUSION 3.8: POTevery h Ip, there is a regular ultrafilter D Over p Whkh is h+-good but not A+ +-good. If A+ < 2", D i8 not A+ +-good but n,<,n,/D 2 Noimpliea n, Prmf. Let D, be a regular ultrafilter over p, and D, a A+-good K,- incomplete ultrafilter over h (exists by 3.1) D, is not A+ +-good by 2.4 and 1.3(3). So D, x D, is an ultrafilter over p x h which is h+-good but not A+ +-good. As IX x pI = p, we finish. The second part follows by 2.13. LEMMA 3.9 (MA): There i8 an ultrajlter D Over w such that: If IP"nI I No,and M = n,< Ir) M,/D and p is an m-type in M of cardinality < and P(xo)A - A P(x,,,-,)~p, then p i8 realized in M. Remark. On Martin's Axiom the rertder can consult, e.g., Jech [Je 741. It is consistent with ZFC + 2N0 = K, for any regular Eta, if ZFC is consistent and it implies h < 2No + 2" = ~Ho.As we use it only rarely, we do not elaborate. Proof. It is sufficientto prove the lemma just for models M, such that llJfnII < 2'0, IUMn)J < 2'0. So let {S,: a < a odd} be an enumeration of {S: S E w} and {(p,, (Mt: n < w)): a < 2N0, a even} be an enumeration of all pairs (p,(M,: n < w)) where L = L(M,) has cardinality < 2N0, IIM,II < 2N0, and p is a set of cardinality < 2No of ~(3,7i)(I@) = m, E E n,, (4) for a even if for every tpl(Z, a,), . . ., tpn(P, a,,) €pa, {i < W: Zt b (%)(p(Zo) A * *. A p(Zm-l) A tpl(Zs %[iI) A * + * A Ts(z, an[iI))} E [#a] then for some 6 E nieOMf, 6 = (bo, . . . ,b, - I), for every tpl(Z; 8,) E pa {i < w: Mf k P(bo[i]) A * * * A P(b,-,[i]) A p1(6[i], 81[i])} E [Ba+l]. Clearly if we succeed then [,?#,NO] is the ultrafilter we want: and there is no problem for a = 0 or a limit, or a odd (by 1.1 of the Appendix). So aasume a is even and pa,M: (i < w) satisfy the hypothesis of (4) (the other cases are trivial). Let us define a set V of “forcing conditions” a forcing condition is a finite set w of equations xl[i] = a where I < m, i < w, a E PI;such that q[i] = a E w and zr[i] = a’ E w implies a = a’. V is a partially ordered by inclusion, and it is countable, hence there are no N, pairwise incompatible conditions. For eachq(E;G) = {Aq: q C pa,q finite} and finite interseation S of members of Ea let w,tpm a)) = {w E V:for some i E 8 and bo, . . . , b, - E P(Mf) Mp c cp[bo,. . ., bm-,, tqi]], {ZO[i] = bo,. . ., X,-l[i] = b,-,} E u}. Clearly each V(8,tp(Z; a)) is dense in V (by our hypothesis from (4)) and their number is <2% Hence by MA there is a generic V* E V, i.e., every two members of V* have a common upper bound, and V* n V(S,tp@; a)) # 0 for every tp(Z; a) EI);, S a finite intersection of members of Ea.Let bl E ni E,,, = Ea U{{i < o:Mf C tp(6[i], a[i])}: cp(Z; a) €pa}. [Ea+,]is a proper Glter as V* intersects each V(S,~(f, a)) end the con- clusion of (4) clearly holds. THEOREM 3.10 (MA): (1) There is a regular ultrafilter D Over w, 8uch that if M is a A-snturated &el of Ti,,(see 11, 4.8), A 5 2H0, then Mm/D ie A-saturated too. (2) If h < 2’0, there is a regular ultraJilter D over h which is not N,- good such that for any model M of Tind,MAID is A+-saturated. 350 SATURATION OF ULTRAPBODUOTS [OH. VI, 0 3 Proof. (1) Let D be the ultrafilter from 3.9. If p is any 1-type over P/D,1pI = p < A, let p = {q&; a=):a < p}, let A, = U {aa[i]:a < p} by 1.6 of the Appendix, and the elimination of quantifiera of Tina,and the p+-eaturation of bl, there is BI E 1611, lBil = KO,such tha&any non-algebraic finite type over A, is realized by an element hmBI. Let us expand M to Mi = (M,P'a), Pa= BI and apply 3.9. (2) Let D, be a good K,-incornplete ultrafilter over A (exista by 3.1)) so D, is regular by 2.4, and Da the ultrafilter from (l),then D = D, x D, satisfies our demands (Dl x Do is regular by 3.7(1) and MAX"/D= (MA/D,)"/D,,iKA/Dl is A+-eaturatedby 2.3 and 2.11 as D, is A+-good and K,-incomplete. So by (1) (MA/D1)"/D,is A+-saturated too). Let 9 denote a family of functions g : A 3 A, g onto A, D a filter over A. Exercises 3.2-3.4 are from [Sh 71~1. DEFINITION3.3: (g,,g2, D) is K-independent if whenever j, < A (5 < to< K) g' E ga (1 < n < o) and f,, f' E g1 me distinct (5 < to, 2 c n) then {a c A: fr(a)= j, for t < to,f'(a) = g'(a) for 1 < n} # 0 mod D. EXERCISE 3.1 :There is a gl of cardinality 2Asuch that 8, Do) is K-independent, Do = {A}, provided that A = A<". EXERCISE 3.2: If (g,,a,D) is K-independent, D = [El (the filter generated by E), Yta is a family of functions from A into a, a < K, then there is '3; G g,, Igl - yP;( s 111 + Igsl i-K, such that ('3;, ga, D) is K-independent. EXERCISE 3.3: If (g,,0, D) is K-independent, S c h then for some 9;c g, Ig, - gil < K and (g,,0, D') is K-independent, where D' = [D u {s)] or D' = [D u {A - A!?}]. EXERCISE 3.4,: Assume (gl,0, D) is K-independent, D = [El. Assume also that Mi (i < A) is an L-model, lPMiI s x < K, aB,,,E n~<~Mi forB < Po < 2", 1 s rn s n(B).AssurneqBELand{qs(z,YB,,,. . ., Y~,,,(~)):/3 c Po) is closed under conjunctions, and for every B < B0 OH. VI, 8 33 OONSTRVOTWO WLTRAR'ILTIRS 35 1 EXERCILYE 3.5: (1) There is an ultrafilter D over A suoh that, if h = A<",then: (i) If M,N are elementarily equivalent and of cardinality CK, then MA/Dz NA/D. (5)If IlMll < K, 2Y s 2Athen MA/Dis p+-saturated. ($) If IpMtl s KO < K, 2' s 2A,p a l-tme in n{<~H{/D, P(X0) lpl s p then p is realized in nfeAM,/D. (iv) If 2" s P, 2%= A, M a model of Thd of mrdinality sA, then M"/D is p+-satuahd. (v) If IlHilJ5 x < K, MI, M' are elementarily equivalent where M' = ntxAM!/D then M', M' isomorphio (Z = 1,2) provided that IL(M1)1 s A. Remark. Notice the following difference between the proof of Exeroise 3.6 aketohed by the preceeding exercises, and the proof of 3.1. In the proof of 3.1 the funations in g are used as partitions of A, whereaa in the proof of Exeroise 3.6(1) they are used as elements in the ultra- produote. EXERCI8E 3.5: (2) In Exeroise 3.6(1) (and Exercise 3.1-4) instead of atesting with the filter Do = {A}, we can stwt with any K-complete filter D, over h provided the oonolusion of Exeroise 3.1 holds. (In Exeroises 3.2 and 3.4 [D, u El replace [El, and in Exeroise 3.6 D, c D.] Of oourse, we c)an start also with [D, u El, provided that IEI < 2", and the ooncluaion of Exeroise 3.1 holds. If there me A, c A, A, # mod D,, A, n A, = 0 (for i < j c A) then the c#>nclusionof Exeroise 3.1 holds. EXERCIHE 3.6: Let f be a 2-place function from 8%(p)to subset of I, let D be a p + -good filter over I, and X, (i < p) be subsets of I. Suppose (1) For my 8, t Ergwo(P), x, n n(I - x,)c f(8, t)mod D. te8 tct (2) If 8, t €8&), 8 A t # 8 thenf(8, t) = 8. (3) f(0,O) = 1. 352 SATURAlTON OF ULTRAPRODUOTS [a.m, 8 3 (4) If8,t, WE&&), 8nt= 8, 8 UtC U, then f(8, t) u (f(81, ti): 8 C 81 C U,t E ti c U, 81 u = U}. Then there am subsets Y1(i < p) of I such that: (A) Y1= X1mod D for each i c p. (B) For every 8, t E &&), n y1n n(I - yl) E f(8, t). 168 1e: On Exercises 3.6 and 3.7 see [Sh 72~1. EXERCIBE 3.7: Prove that the following conditions on the filter D over I and the cardinality A > No are equivalent. (1) For every set of L-models Mt (t E I)n,, M,/D is A-compact. (2) D is K,-incomplete, A-good and B(D)is A-saturated, where B(D) is the Boolean algebra of the subsets of I mod D. (Hint: Use the Fefer- man-Vaught theorem; see e.g. [CK 731 and Exercise 3.6.) EXERCI8E 3.8: In Exercise 3.7 we can replace A-compact and A- saturated by (A,d)-compact for d the set of Z,,(or n,)formulas (A C,-formula is a formula of the form #(#) = (3Zl)(VZ2). . .‘p(Zl, . . ., Z,,, g), ‘p quantifier free; a lI,,-formula is the negation of a C,-formula). EXERCI8E 3.9: Prove that B(D)is isomorphic to Mi/D where M, is the Boolean algebra with two elements. EXERCI8E 3.10: Show that in general, in 2.3(3) we cannot replace “for any p < A” by “for some p < A”. (Hint: Use Exercise 3.5.) EXERCIHE 3.11: (1) Show that for any A-good filter D, B(D)is (A, at)- compact (See Exercise 3.12). (2) Suppose M, is a (A+, at)-compact Boolean algebra, of cardinality A+, and 2A= A+. Prove that there is an K,-incomplete good filter D over A such that B(D)is isomorphic to M,. (3) We can suppose M, is a compact Boolean algebra of cardinality 2A, and (Vp < ZA))(2J’s 2”), and get the same conclusion. Remark. In (3) the existence of M, implies (Vp < 2”(2” s 2A)by VIII, 4.7. Similarly in (2), it implies 2“ = A+. [Proof: (2), (3). Similar to the proof of 3.1. In the ath step we have a filter D, = [E,], lEal s A + lal, a sub- algebra N, of No, llNIIll s 1.1 + H,, an embedding Ha of N, to the OONSTRVOTWO WLTRAR'ILTIRS Boolean algebra of the subsets of I such that 0 # a E lMol implies &(a) # 0 mod D,, and a family 9, of functions from h to A, such that for every distinct g,EY, and j, < h (I < n < w) and a€ Val,a # 0 {i< h:go(i) =jo,...,g,,-l(i) =j,,,}nH(a) # OmodDaandlS- s h + 1.1. (Clearly Nu,Ha, E, increaae with a, and S, deareaees with it, eaoh of them oontinuously. In the end D = D,n, H = H2~.) We have steps of thkinds: Kind I: We want to include b in the domain of H. Choose g E 9,, and let Nu+,be generated by Nu and b, = {i: g(i) = 0}, S,+, = S, - (s) and B,+, = E, u (I - Ha+,@,)n H,+,(a):a E Nu,a n b, = 0 and b, is b or its complement}. Kind 11: We want to refine the monotonic funofion f : S&) + D,. We do it exactly aa in 3.1. Kind 111: We want to decide the fate of S c h (that is, we want that for some a E No+,,8 = Ha+,@)mod Let here A denote a function: A = {(gf,jf): i < io} where io < A+ + 1.1 1.1 + and gf E S are distinct. For such A let P be the filter generated by Eh9 ~h = B, u {&+(a> = jf:a < A}: i < io} and Sh = g, - {gf: i < io}. Cme (i): For some a E Nuand A, H&) = S mod Dh. We then dehe: Nu+,= Na,H,+l = H,,9,+l = gh,E,+, = Eh(andofcoureeD,+, = [E, + ,I). It is eaay to check our demands hold. Cme (ii): Not (i), but there are a, b EN,, A such that H,(a) E 8 c H,(b) mod Dh, and b - a is an atom of Nu(i.e., (Vc E N,)(b - a E c v (b - a) n c = 0)). We then define Nu+,= Nu,So+, = gh, I#,+, = Eh u {A - (IS - Ha@))}.The only way in which our demands may fail is that for some A,, A E A,, and c EN,, c # 0 such that H,(c) G S - Ha@)mod P'. But then Ha@)n Ha@) = 0 mod D', hence c n a = 0; but also H,(c) E HJb) mod P',hence c c b; so as b - a is an atom of Nu,c = b - a. Now it follows that Ha@) = Ha@)U H,(c) = S mod P',so caae (i) holds, contradiction. Cue (iii):Not (ii) or (ii)but for some a E Nuand A, H(a) E S mod Dh, and for every b EN^, h' : h E hl, H,(b) c S mod Dh' implies b E a (in N,, of course). We then choose S,+,, B,+, as in 0&88 (ii). The only way in which our demands may fail is that for some CEN~, c # 0 and hl, h E hl, the following holds: H(c) s 8 - HJa) mod P', but then again c n a = 0, and H(c u a) c S mod P',so c u a c a, henoe c = 0, contrediotion. Cme (iv): Not (i),(ii) or (iii), but for some b E Nu, h: S c H(b)mod P, andic h1,c~Na,8cH(c)modP'implieebcc. The pmf is similar to that of c&88 (iii); this time Ea+, = Eh U {A - (H(b) - S)). Cme (v): None of the previous ones. Choose h such that if c E Nu, c f 0, and 8, E {S, h - 8),B c I,, and Ha@)n S1 = 0 mod Dh' then Ha@)n 8, = 8 mod P.Now letp = (8 c b: b EN,,S c Ha@)mod P} u {a E x: a E Nu, &(a) E S mod P}.It is easy to check p is a type in Nu, and s llNall + No s A, p is atomic, and N is (A+,&)- compect, some c E M,,rdizes p. Dehe Nu+, aa the subalgebra of No generatedbyIN,I u {c},Ha+,eXtendHa,Ha+,(c)= 8,andE,+, = Eh. We leave the reader the checking.] EXERCISE 3.12: Show that the Boolean algebra Mo is (A,&)- compact ifffor every a,b < A, a,,bt E Nosatisfying No C a,,,, s a,,, s bj(,) s a$(,)for i(1) < i(2) < a,j(1) < j(2) < there is c E Mo, Mo k a, I; c s bjfori < a,j < P. EXERCISE 3.13: Suppoee M,,is a Boolean algebra of cardin&ty 2". Then for some regular, goad filter D over A, Mo is elementarily embed- dable into B(D),and B(D)is p-saturated, where p = mink: 2' > 2"). PROBLEN 3.14: For any A, characterize the possible B(D) for D a filter over I. EXERCISE 3.15: Prove that for any n and A, there is a filter D over h such that B(D) is (A+, C,,)-compact but not (A+, l7,,)-comprtct, and vice vem. (Hint for Exercises 3.13 and 3.15: The proof is like Exercise 3.11, but Nu (a < 2A) is not predetermined.) EXERCIBE 3.16: Define a h-good filter in a Boolean algebra just as in Definition 2.1. Prove the parallel of Exercise 3.6. Show that if B, axe Boolean algebras, D, a filter in B,, Ir : 8, --+ 8, a homomorphism onto h'l(1) = D,, and D,, D, are h-good, then the filter h-l(D,) is &good. [Hint: Let g : S,&) + h-l(DS),p < No;so hg :S&) --+ D2, so it ha OONSTRVOTWO WLTRAR'ILTIRS a multiplioative refinement g1 :B&) + D, and choose X, E B,, h(X,) = g,({i}). Hence X, E g(e) mod D,. Define g1 :B,&) --+ D, bY g1(8) = u { 1 - (nX, - g(t)) : t .}, S' :~~;EOW -+ D, t8t a multiplimtive refinement of gl, and g*: 8,&) --t h-l(D,) defined by p(8) = n,., X, n fl(8) is aa regarded.] EXERCISE 3.17: Supp~eep s x = x<= < 2n, and ahow thare is a regdar, K-pod filter over p generated by a frtmily of sx sets. (Hint: Use 3.1, and show that any regular, good ultrafilter over p haa such subfilh, using Exeroise 2.6.) EXERCIBE 3.18: Suppose E is a family of subsets of A, IEI s A dosed under finite intersections, 0 $ E, and D is a A-regular filter over I. (1) There is a function B : [El --+ D suoh that: (i) for A, B E [El,B(A n B) = G(A)n G(B),A c B +G(A) c U(B), (ii) if A, €[El (i < S), nA, = 0 Q(A,) = 0. l EXERCIBE 3.19: If D is A-regular, A<= = A, then there is a K-good, A-regular filter D, s D. DEWINITION3.4: D is a, uniform filter if all members have the same Oardindity (it is IIl when D is a filter over I). EXERCIBE 3.20: (1) A ultrafilter D over A is dorm ii€ E D where @ = {A E A: IA - 41 < A} which is auniform filter. (2) If D is a uniform ultrafilter D over A, of[(A, < )"D] > A. (3) If D is a regular ultrafilter over A, 6, limit, then r EXERCIBE 3.21: Iffor some ultrafilter D over A, Do c D, Do a filter and o~[(K,< )"/Dl = p, then there rn functionsfr : A --f K, (i < p) suoh that for no g : A-K for every i < p (K, c)~/D,Cfr < g. 356 SATUI&ILTION OF ULTRAPRODUUTS [a.VI, B 3 EXERCIBE 3.22: If Do is a filter over A and there are functionf, : A 3 K (i < p) such that for every g : A -+ K for some i < p (K, < )"Do C g < f,, then for every ultrafilter D over A, extending Do cf[(K, < )"D] s p. If also i < a < p implies (K, <)"Do Cf, < fa, then cf[(K, <)"/Dl = p. Remark. The family of ft's as in Exercise 3.22 are called a (Do,p)-scale for K. On independence results conoerning scales see, e.g. [He 741. The existence of an 2b-s~aleis weaker than MA. The following exercises am an improvement of Theorem 3.10. EXERCISE 3.23: Suppose D is A-regular M = nt,rMt/D, A c ]MI, IAI S A then there are finite Pt c lMtl such that A s PM where W, PM)= IItar (Mt, pt)/D. EXERCIBE 3.24: Suppose Dan ultrafilter over I, llMtll s A, A c 1M1, nter Jft/D. (1) If IAI < cf[(h, <)'/D],then for some P, s lMtl, lPtl < A, and A c PM where (MyPM) = n,,I (Mt,P,)/D. (2) If A = Ha+,, IAI < cf[(N,+,, <)'/Dl for 0 s 2 s n, then for some P, c lM,l, lPtl < K, and A E P', where (Jf,PM) = n(Mt, PJD. t€t Remurk. Now a combination of Exercise 3.6 and 3.24 gives results like 3.10 (for the existence of such filters the existence of scales is sufficient, see Exercise 3.22). EXERCISE 3.25: Suppose A = 2" < 2"+, and some p+-complete filter D, over A satisfies Exercise 3.1 and a (Dl,A+ +)-scale for p+ exists. Then (1) over A there is a regular and good ultrafilter D, such that for every A+ +-saturated model of Thd, MA/Dis A+ +-saturated; (2) over A+ there is a regular not good ultrafilter D, such that for every model M of Tina,M"+/D is A+ +-saturated. EXERCISE 3.26: (1) If A = KO and MA holds, M, N are elementarily equivalent IL(M)I 5 A, 11M11 + IlNll s A then for some regular ultra- filter D over A, MAID = NX/D. (2) If, e.g., IlMll + llNll + IL(M)I s p+, M, N elementarily equiv- alent and the assumption of Exercise 3.26 holds, then for some regular ultrafilter D over h MA/D= NA/Dprovided that A+ + = 2". OONSTRVOTWO WLTRAR'ILTIRS CONJECTURE 3.26: (1) In Exercise 3.26(1) we can eliminate MA. (2) It is consistent with ZFC, that for A = KO for every regular ultrafilter D over A+, of@+, <)"+/D= A++, cf(h, <)"+ID> A++ (thus Exerciee 3.6(1) is essentially best possible). We may repltloe A, A+ by A+, 2". DEFINITION3.6: For an ultrafilter D (over I) and reguIar cadinal K; 1Cf(K, D) is the smallest cardinality A such that: there is a subset of {a E d/D:d/D C a < a for each a < K} which is unbounded from below, and has cardinality A. Remarke. (1) We do not distinguish between K and the model (K, <) (till the end of this section). (2) Clwly lCf(K, D) is an Wteregular cardinal or 1. (3) Till the end of thh section, an ultrafilter is over I if not mentioned otherwise. CLAIM3.11:(1)IfK > III~~@?',IOf(K, D) = 1;inf&hf(K, D) = 1 iff D ie cf ~damndinglycomplete (8ee De$nition 6.1). (2) d/Dk not (K + lcf(K, D))+-wt. (3) If D ie a A-good N,-inco))zlpleteultrajilter and K < A, then lcf(K, D) 2 A. Procf. Immediate. THEOREM 3.12: For every p = pNo 5 2", and regular K, No < K s p there is a regular ultrajEZter D over A such that lcf(Ko,D) = K and THEOREM 3.13: (1)Suppose No = A0 < A, < - * * < A,, = A+ each 4 i8 regular and hI+1 s pl s 2", pl regular (for 1 < n). Thenfor 801126 regular A1-good ultra$lter D over A, lcf(K, D) = pl whenever A, 5 K < A1+,. In fact D i8 not A?-good. (2) 80 in particular for every h and regular p s A, thre i8 a regular ultraMer over A which ia p-good but not p+-good. THEOREM 3.14: Stqpme in the previowr thorem that A, = K, and k < w, xl= xp (1 $ k) are given xo < x1 <*a- < xk.= 2A and xf (i < 2, 1 < k or i = 0, 1 = k) are regular mrdid >No, X! s xi, XI 5 XI. Let Jl = (a E oA/D: 1.1 = xl} 358 where 1.1 = na[i]/D= l{b E d/D:oA/D C b < a}], f DEFINITION 3.6: (1) In Definitions 3.1 and 3.3 we allow the functions in 9, g1resp., not be neclessarily onto A, and then demmdj, E Range@,), jc E RtmgeVt) reSP* (2) For a family 9 with domain A let (FI for finite interseotion): FI(9) = {h: h a funotion, Dom h c 9, lDom hl < No and h(f)E Rengedf)}. For hEFI(9) let A, = {i < A: f ~Domh implies f(i) = h(f)}. FI,(Y) = {A,,: h E FI(9)). CLAIM 3.15: (1) All relevant krnmaa remain true ~drthe new Dejinition 3.6: 3.2, 3.3, 3.4, Exerci8es 3.1,2,3 and 4. E.g., 3.2 becomee if IJ,l s A for i < 2”, then there ia a fmily (fi: i < 2”}, fc fr.m A onto Jt, independent mod{A}. (2) If 9 is i- mod D, for a < 6,D, (a < 8) an ilacrerreing 87Of $&3’8 Over I th9 6 ~dpendentmod ua<6D,. (3) If D i8 a@er over A, 9 idpendent mod D, then there is a dd jEJter D* 2 D, modulo which 9 is id- (i.e., there is no D1 # D*, and D12 D* such ticat Q itt idpendent mod Dl). (4) 9 is idpendent mod D iff A,, # 0 mod D for every h E FI(9); crnd I1 -C 1, iff Ah, E AhI. (We &Card the weIRange f I = 1.) DEFTNITION 3.7: (1) In a Boolean algebra, a partifion is a maximal set of pairwise disjoint non-zero elements. An element is baaed on a partition W,if b E W impties b c a or b n a = 0; the element is supported by a set W of elements if it is baeed on some partition W1E W (notice if W is a partition, a is baaed on W 8 W supports a). A set W is dense iffor every non-zero a there is in W b c a. (2) If D is a flhr over I, instead of speaking on B(D), AID, . . ., we speak on {A: A s I), A,. . . (and then everything is mod D.) (3) CC(B),for a Boolean algebra B, is the minimal regular cardinal A 2 Nosuch that every partition of B hae omdimlity CLAIM 3.16: In a Bodean algebra (1) aiebaaedtm Wiffl -aiebaeedon W, (2) if q ie baaed on Wl (Z = 1,2) then a1 n a,, a, u a,, a, - a, are Basedon{cnd:ctsW1,cEc Wa,cn&# O}wh&hieapadition, (3) for any eet W of ehnente olosed under finite intereectiorce 8uch that everyaiz WbeJonstoapwtition E W,theeetofelenzenteitswpporbiea d+brcz, adid& aU ekmmh of W, (4)if a,b are supported by Wand c E W =+ [c E a o c c b] then a = b. CLAIM 3.17: Buppoae D ie a maxidjEuer Over I modulo which 9 ie i-. (1) FI,(S)ie hemod D; hence euch ehnent ie twppted by it. (2) Fw each f E a, U-l(j):j E Rangef} is a pddtbn mod D. (3) If An E FI($), Dom hn(n < w) are pairtoke cEi.$Xnt, A E I, and A n A,, = 0 mod D then A = 0 mod D. (4) Led 9 be the diejoint union of Yr,, 9a, A E I ie eu- by FI8(g1),h E FI(EB), A = h1 U ha, h1E FI(gJ. If A, c A mod D then A,, c A mod D. (6) CC(B(D)) = KOiff for dyfinitely many f E B ie IRange(f)l > 1, and for 7u) one IRange(f)j = No. Othertoke CC(D) ie the fir8t regular A > No ezlol, that f E 9 =s IRange(f)l < A. Moreover, if p L h ie regular, A, # Omod D for i < p, then there ie 8sp, 181 = p euch t7mt for n < w, and dietilcct i(Z)€8, nl<,, A,(!, # 0 mod D. (6) IB(D)I s 191tre+ 2'"~hK = CC(B(D)). Prmf.(l)OtherwieeforsomeA E I,A # €imodDandA, $ AmodD for eaoh h EFI(~)hence Ah A (I - A) # 0mod D. Hence 9 is independent mod[Du{I - A}], oontradioting the maximality of D (aa A # 0 mod D). (2) Clearly the f-l(j)'s are pairwiee disjoint mod D. Suppose A n f-'(j)= 0 mod D for eaoh j E Range(f),but A # 0 mod D. By (1) for some h E FI(O) A, s A mod D, and olearly for some h,, h E hl E FI(Q),f E Dom hl. So also A,, c A mod D; but let h,(f) = jo,so A n A,, E A nf-l(j0)= 0 mod D, mntradiotion. (3) Suppoee A # 8 mod D heme by (1) for some Is E FI(g),Ah E A mod D; for all but hitely many n's, Dom h n Dom hn = 0, so h U hn E FI(9). NOW Ahuh,, # 0 mod D and Aruh, E A mod D, and A n &,ha E A n A,, = 0 mod D mntrdotion. (4) Let A be based on the partition {Ahl: g < g(O)}, h, E FI(9,). Now, AhEAmodD, hence AnAh,P0modD~8,,n8,=0rnodD 360 SATURATION OF ULTBdPtlODUCTS [m. 8 3 hence (m g1 u 9a is independent mod D) A n Ahr = 0 mod D =- Ah1 n A,, = 0 mod D hence (otherwise Ah1 - A contradicts the maxi- mality of the partition) Ah1 E A mad D. (6) Notice tht if f,,E are distinct, [Rangef,,l > 1, j: # j: E Range f,,,and h,, is defied by h,,(j',) = jpfor 1 < n and h,,(f,,) = j:, then h,, E FI(9) and A,JD (n < o)are KOpairwise disjoint non-zero elements of B(D). By 3.17(2) f E 9 3 ]Rangef I < CC(B(D)),hence the first phrase follows, and the second follows from the third. So let p L h be regular, A, # 0 mod D for i < p. By 3.17(1) there are h,EFI(Q) such tht A,,, c A, mod D. By its definition h > No, hence p > 8, and we msume p is regular; and Dom hi is finite. So by 1.4 of the Appendix for some n < w end 8, E p, IS1l = p, for every i # jESIDomhinDomh, = cfo,...,fn-l}. As (Rangeft\ < h s p, there are 8 c 8, and j, E Range fi (1 < n) such that 181 = p and for every i €8,hi(fo) = joyh,(fl) = jl, . . ., = j,,-,. Clearly 8 is m required. (6) By 3.16(4) each partition W supports s 2IwI elements. By 3.17(1) every element of B(D)is supported by {Ah/D:h E FI(9)). By 3.17(6) IFI(9)I 5 I 91 + K, and every partition of B(D) has cardinality < K. Collecting those faots the result JB(D)Js Jgl*x+ 2<= becomes obvious. LEMMA 3.18: 8uwe D a maximal filter mod& which g* u g is independent, 9*, 9 are disj~id;f E 9* u 9 implies IRange(f)l < cf a: of a > No, 9 = uB FIp* u 9,). (ii) g* V gEis independent mod D,. (E)Di is rrmxhal with re8peCt to (i) and (G). Tb (1) D* = Ut Proof. Part (3) is trivial (aa the family of D’s satisfying (i) and (ii) is closed under amending chains), and part (2) first statement follows by (1). The second statement of (2) holds easily. (1) As 9*is independent mod D, for each i,cleady it is independent mod D*. So we have to prove only the maximality. So let A E I, A # 0 mod B*; now for some S+ c 9,19’1 < CC(B(D)),and A is supported by FI,(9+ u 9*)mod D; 80 for some y < a 9+c g, (as cf a > IRange (f)l for f E 9*U 9,and cf a > No, by 3.17(6) cf a L CC(B(D))hence of a > IS+I). Now [D, u {I - A}] is a filter properly extending D,, and satisfies (i), hence by (iii) should fd to satisfy (ii). That is for 8ome h E FI(9* u 9,)Ah !z A mod D,, hence (by (ii) and 3.16(2)), for some BED,, BID is supported by FI,(9* u 9,)and Ah n B C_ A mod D (by 3.16(2)) SO Ah S A U (A - B)mod D. hth. = k1 u W’, W1 E FI(S*), ha E FI(9’). By 3.17 (4) (where D, A U (A - B), 9*u 9,,9, stands for D, A, 9,Sl).Ahi s A u (A - B)mod D hence A,1 E A modD,. As this holds for every A # $4 modD*, D* is a maximal filter modulo which 9*is independent. (4) Suppose feYp is a counterexample. Then for some A E 1, A # F, modDp and for every t~ Range(f) A n f’(t)= pI modDBso for some B, supported by FI,(Y* U Y@), A n f’(t)c B, modD and I- B,€Dp.By 3.17(2) there is a partition (A,$/D:i< 5) of B(D)on which A is based; by a hypothesis 5 < cfa and w.1.o.g. for some 5 < f[, for i < 5, A,* c A modD, and for i 2 5(i < 5) Ah1n A = F, mod D. So for i < 5, A,( n f’(t) E B, modD. Let hi,o= hir (Y* u g’), so as B, is supported by FI,(Y* U Yp)andfEW = ~-9?’,by 3.17(4) Aht E B, modD. Define A* = Ui..cAht,ornow: h-stly, A c A* modD [as for i < 5 A,* G A,i,Oc A* so A -A* is disjoint to Ahi(i< 5) and for i 2 5 (but < E) (A-A*) fl A,* E A n A,, = Y, modD, so A-A* = Y, modD would contradict the choice of A,,,oc B, modD]. But I-Bt~DB,hence I-A* ED!, hence I-A ED@, a contradiction. Remark. We have shown that B(D) is complete (this holds generally when for some 8,D is maximal such that Y is independent modD). CLAIM 3.19: (1) Let Y be in dependent modD, and (f-'(t)/D: t E Range f) is a partition of B(D)for every f E 8 [which holds if D is mazibl over which Y is independent but also if D, Y are Da, YBfrom 3.18 (see 3.18(4))].If g:I+K, and KI/DI=u < g/D for every a < K, and j~ $9, then d/D tf/D < g/D (hence for every ultrafilter D* 2 D, K'/D* I= f/D* < g/D*). (2) 8wppe CC(B(D))s K, K, p regular, f, : I --+ K for i < p; and KIIDCa <&ID < f!/D,foreverya < K,j < i < p;butforanyg:I-tK if for every a < K d/DCa < g/D then for 80112e i,d/D tg/D > &ID. Then for every dtrafier D* 1 D, lof(K, D) = p (and thi8 iS exempli&?d by the f,/D*'8). (3) Part ow bZd8 for [D u {Ah}]when A E FI(B), f $ Dom A (k for [Du {A}]when A # 0 mod D). Proof. (1) Suppose not, then A = {t E I:f (t) 2 g(t)} # 0 mod D, hence by a hypothesis for some j, < KAn fl(j,,)# $j modD, hence contradioting KIID kjo < g/D. (2) Otherwise there is g/D*(g: I --+ K) contradioting it. Let A, = {t: a > g(t)} and define induotively a(i) < ~(i< K) tm follows: a(0) = 0, a(8) = Ufeba(i),a(i + 1) the first a > a(i) suoh that A, - ANi) # 0 mod D. If a(i)is defined for every i < K, - A,)/D: i < K} contradiots CC(B(D))s K, hence for some i, a(;) is defined but not a(i + 1). Define gl E I -t K, by then gl contradiots the hypothe. (3) Etlay. CLAIM 3.20: 8wppwe 9 is a nun-empty family of facnctions olrto w, D a ddj2termodulo which Y is in&pe. OONSTRVOTWO WLTRAR'ILTIRS (1) IffE Q, thnt, (1 + f(t))/D s lQINo for any draJilter D* 2 D, nt61f(t)/D* = ]{a:ol/D Ca < f/D)] s ]ZSPINo. (2) Iff e Q,, gl : I + w, wl/D t gl/D s f/D (I = 0, 1) adfor every n g;l(n) = g; '(n)mod D thwl/D C go/D = gl/D. Pmf. (1)The phase affer " hence " is trivial, so suppoee nt,(1 + f (t))/D > p =da IQINo, so there me g, : I + w, g,(t) s f (t)suoh that {t: g,(t) # g,(t)) # 0 mod D for i < j < p+. The number of possible sequences (gil(n)/D:n < w) is sp(it is s IB(D)INo;but by 3.17(6) CC(B(D))= N,, hence by 3.17(6) IB(D)I s Po, but pH0 = p). So w.1.o.g. gil(n)/D = gyli(n/D)for i < j < p+ but we get a contradiction by part (2). (2) Suppose not, so A = {t E I:go(t) # gl(t)) # 0 mod D. By 3.17(2) there is n < w suoh that A nf-l(n) # 0 mod D. Let for 1, k s n A,,, = {t E A: f (t) = n, go(t) = k, gl(t) = 1) so clearly A n f -l(n)= U,,, rn Ak,,, hence for some k, E A,,, # 0 mod D. As A,,, s A, olearly k # I; it is also olem that A,,, r gcl(k) and A,,, r gil(l); hence go l(k)n gil(1) # 0 mod D, but g; '(k)is disjoint to gcl(E) and g; '(1) = g; '(1) mod D, contradiotion. CLAIM 3.2 1 : #uppose Q*, 9, g6y W', a, D,, D* are ars in Claim 3.18. Buppose furtheme tlrat f E g* lhge(f)I < K; K > KO or Q* = 0, K= No;~i8regular;f6~~6+l-~6andfOTeVe?'yf< K,{t:[< f4(t)EK)~ DB+ I' Then for every ultraJlter Dl 2 D*, lcf(~,Dl) = of a and thia is exemptifid by f6/D1 (p < a) and d/D1t f6/D1 < fy/D1for y < )8 < a. Proof. By 3.19(1) d/D* b f4/D* < fy/D* for y < < a, and by 3.19(2) (for K > No) and 3.18(2) (for K = No) it suffices to prove (*I If g : I + K, and for every f < K, d/D* t f < g/D* then for some 43 < a d/DS t= g/D* > fs/D*. Suppose g falsifies (*) then for my [ < K, B, = {t: [ < g(t)}E D* hence for some a(& < a, Be E DNO. As of a > K, a(*) = supt,, a([) < a, so Be E DM.) for every f, SO {t: [ < g(t))E D,(.) for every [, henoe by 3.19(1) {t:f,,,,(t) < g(t))= ImodD,,,,, hence {t:f,(,,(t) < g(t))= I mod D*, oontradiotion. CLAIM 3.22: If I J,I s h for j < jo s 2Qhen there are functions f, from h doJ,, and a regular A+ -good@er D, such tlrat Q is independent mod D, and D is a dmaleuchjiaer. 364 SATURATION OF ULTRAPRODUCTS [a.VI, 0 3 Proof. By 3.16(1) and 3.2 there are functions fj(j < j,) from h onto J, and g, from X onto X (for i < 2”) such that Q* u Q is independent mod{A}, where Q* = uj:j< jo},’3 = {g,: i < 2”). Let Do be a maximal filter modulo which Q* u 9 is independent {H,: 0 < i < 2“} a list of all functions from Sw0(A)into (8:8 s A}, each appearing 2” times. We define D,as in 3.18 (letting B, = uj:j < i}).We use foto define a regular D,. If Range (H,) c D,, we work aa in Claim 3.3, using f, to get D’ z D, and extend it to Dt+, by 3.18(3), then UtD, is the required flter. CLAIM 3.23: Let D be a maximal Jilter d& which 3 iaindependent, p = CC(B(D)),forinJinitelymany f E B IRange f I > 1 and D* 2 D an ultraflter. Then D* ie not pi-good. Proof. Choose distinct f, E g and thenj,, E Range f, such that f;l(j,,) 4 D* (possible as IRange fnl > 1). Let A, = n,<,(A- fil(jn))E D* and define g :SN0(p) --t D* by g(4 = A,,,. We suppose 9, :&&) + D* refine g and is multiplicative, and get a contradiction. As gl({a}) E D* (for a < p), g,({a}) f 0 mod D. Hence for some ha E FI(S)Ah. E gl({a}) mod D. By 3.17(6) p > No,and it is regular by definition; so for some S c p, 181 = p and for some k Dom ha n u,: n < w} -C df,: n < k} for every a €8.By 3.17(6) for some 8, c S, I8,l = p, for any n < w, and distinct a(l)~8,(1 < n) nl<,Ahacl, + 0 mod D. Choose 8 = (a(2): 1 s k}, a(2) €8, distinct so g(8) E I - f CLAIM 3.24: 8uppoae D i8 an N,-inmmplete &mal fler Over I dubwhich g u g* ie independed, 9 u Q* a family of functione from I onto o,B* = df:i < S}, of S > KO. DeJine D1a8 theplter generated by (A) tkdW8 Of D, (B)A: = {t: g(t) < f,(t)}when g : I --+ w, and for ewery n < w, g-’(n) i8mqpo9de&byFI,(SU{fj:j< i)),andforanyh~FI(Su{f~:j< i}) for ~cnm?n A, n {t:g(t) 5 n} # 0 mod D. Then (1) g &I independent mod D,. (2) For any UltraJilterD* 2 D,. (i) ox/D*C f,/D* < f,/D* for j < i < 8, OH. VI, 5 31 CONSTRUCTING ULTBAF'ILTEBB 365 (ii) for any a E w'/D*, for 8ome i < 6, d/D* C a < fJD* or fm 8m g : I + W, d/D* C g/D* = a and o'/D C n < g/Dfor every n < w. Proof. (1)Note that by this we prove D, is non-trivial (take A the empty function). So suppose gl, i(Z) (1 < n) as in (B), A E FI(S), and we have to prove Ah n nA;,) # 0 mod D. I Case (i).A # D*. Define Cletlrly A: is one of the generators of D, (by 3.16( 2)), hence wr/D* C a = f/D* = g/D* < fs/D*. Case (ii). A E D*. As D is 8,-incomplete, then are B, E D, nn<,B, = 0; w.1.o.g. B,+, E B,,. Define Then wr/D* C a = g/D*, and wl/D C n < g/D for every n < W. Proof of Tlwore~n3.12. Let D, g be such that 4is a family of p functions from h onto w, independent mod D, D a maximal filter over h modulo which '3 is independent, and D is regular (by 3.22). Let '3 = (fi: i < p~} (pK-ordinal product) = {A: i < p}. We define Dl (i < p~)as in 3.18 (for g* = 8) such that A," = {t:n ng(i)/D* = Idf/D*: oA/D*C f/D* < g/D*}l : Proof of Theorem 3.13. Let D be a maximal filter modulo which 59 is independent, D regular (eee 3.22) suoh that 9 9 UIen9,, 9; = cf:: A, s K < A, + ,, K regular, i c 25~~)(ordinal multiplication) wheref: is onto K. We define by downward induotion, filters D, (Z 5 n)such that (i) 0, = D, D,,, c 0,s (ii) D, is a maximal filter modulo which UL<,qLis independent, (iii) for any ultrafilter D* extending D,, A, s K < K regular lcf(K, D*) = pi. OONSTRVOTWO WLTRAR'ILTIRS If D1+,is defined, we define DlS1(i < 2”~~)&B in 3.18 (with ULeIyPk, ub: A, s K < At+,, K regular, i < b}, 2“ p, for 9*,yPB, a reap.) such that {t E I: f 5 fi(t)}E D1,l+lfor each f < K. Let D, = UfD,,,, so (i) and (ii) holds by 3.18, and by 3.21 (iii) holds. For 1 = 0, for even i’e we do &B above, and for odd i’s we immitate the proof of 3.22 (and 3.1) to gat a hl-good ultrdter Do. Now Do is not ht-good by 3.23 and 3.17(6). Proof of Theorem 3.14. Combine the previous proofs, but use also Claim 3.24. THEOREM 3.26: For every reguhr A s IIl, there W a regular ultra- jilter D Over I such thud (i) D W h-good but not A+-good, (ii) lof(K, D) = Afor every K < A, (iii) every function f :8,&1) + D, IRange (f)l < A, can be rejdto a multiplicative one. Proof. Let 9 = Ua \ 368 SATURATION OF ULTRAPRODUCTS [CH. VI, 6 3 (6) In 3.18 replace Yl,, FI,(9f1 u 9*))}by B,, {B n A,: B el?’, h E FI(9*)} (9* c 9)mp., and wume KO + CC(B(D))< cf a.] EXERCIBE 3.28: (1) Let B be a Boolean algebra, CC(B) 5 A then B is (A, at)-compaot iff B is complete. (2) if A = CC(B(D)),D a A-good filter, then B(D)is complete. EXERCIBE 3.29: Suppose B is a complete Boolean algebra, CC(B) 5 A, IlBll 5 2A. Then for some regular filter over A, B B(D). [Rint: If llBll 5 A, this is easy by Exercise 3.27 (on 3.22) and Exercise 3.28. Otherwise let fi be from A onto A, D a maximal filter modulo which ui:i < 2A}is independent; let B, be the subalgebra of B(D)generated by fil(j)/D,h,, a homomorphism from 5, onto B, which we extend step by step, and get h* : B(D)+ B, and let D* = {A: I(A/D)= l).] EXERCIBE 3.30: Suppose Di (i < A) filters over I, D a filter over A, and let D* = {A:{i < A: A E D,} E D}. (1) D* is a filter; and if p 2 CC(B(D,)),p*-+ (p)! (e.g., p* = (2 EXERCISE 3.32 (G.C.H.): Suppose K < plC5 21'1, regular for each K ES; K ESimplies K 5 II1 is regular, and for inaccessible A {K ES: K < A} is bounded below A. Then for some regular ultrafilter D over I, lcf(K, D) = plCfor each K E S. [Hint: Let {fg: a < 21'1, K ES} is independent mod Do, Do regular, and Range (f,") = K. For regular x, 8, c {K ES:x s K}, let (*)$, there is a filter D = D(S,, x) 2 Do such that (i) {f,": a < 21'1, K < x} is independent mod D, (ii) for every ultrafilter D* 2 D, K ES,, 1Cf(K, D*) = px and this is exemplified byf,"/D*(a < pa). We prove it by induction on the order type of S, (for x a successor of a singular, use x+ and then add x+, if necessary).] DEFINITION3.8: For a model M and a complete Boolean algebra B and ultrafilter D of it, we define the Boolean ultrapower N = M('B)/D as follows (the sup exists by the completeness): (i) On the set {f:Dom f is a partition of B, Range f E M} we define an equivalence relation x :f, x fa iff sup{a, n a,: a, E Rangef, (I = 1, 2) f,(a,) = f,(a,)} E D. By 3.16(2) x is an equivalence relation, and for everyf,, . . . ,f, there are f: x f, which have the same domain. Now IN1 will be the set of equivalence classes of x and RN = {(f,/x,...,f,/x): (similarly for J"). 370 SATURATION OF ULTRAPRODUCTS [CH. VI, $4 EXERCISE 3.34: Prove that Boolean ultrapower is a particulaz case of limit-ultrapower from Definition 4.2, hence the parallel of Exercise 4.10 holds. EXERCISE 3.35: Let D be a filter over I, 5 = B(D), K, = CC(B), D, 2 D an ultrafilter, and D* an ultrafilter of 5, D* = {A/D:A E Dl}. We define a function H:w(m)/D* + wl/D,as follows: iff/ z E w(S)/D*, let Domf = {a,: n c a,, I; w} (as CC(5) = Kl), let a, = A,/D and w.1.o.g. I = Un Remark. By this, the problem on “what can be ateIn,/D: n, c w}” can be reduced to a problem of Boolean ultrapowers. EXERCISE 3.36: In 3.19(2) replace the a < K byf’ (j c K). EXERCISE 3.37: Suppose g is a family of functions from I into the ordinals, and D is a maximal filter over I modulo which Q is inde- pendent. Let p = CC(B(D)),and suppose D is A-good. Then for any ultrafilter D, 2 D, and regular K, x c A satisfying K, x 2 p + 1’31 + and: K > 2X or x > 2’F, or K = x are weakly compact wI/D, has no (K, x)- Dedekind cut (see VII, Definition l.lO(B)). EXERCI8E 3.38: Use Exercise 3.37, Theorem 3.25 and Exercise 3.30 to investigate the possible {(K, x): d/Dhas a (K, X)-Dedekind cut}. Phrase the open problems. QUESTION 3.39: Investigate the problems from Exercise 3.38. VI.4. Keisler’s order DEFM~TION4.1: (1) For every A Keisler’s QA-order on theories is defined aa follows: T, @A T, if when M,is a model of T,(1 = 1,2) and D a regular ultrafilter over A, the A+-compactnessof HG/D implies the A+-compactnessof M:/D. (2) Keisler’s order @ on theories is defined aa follows: TI@ Ta if for every A T, Ta. OH. VI, 8 41 EEISLER'S ORDER 37 1 (3) T,and T, are @.,-equivalent (@-equivalent)if T, @., Ta @., T1 (Ti @ Ta @ Ti). LEMMA 4.1: (1) T7w relations @.,, @ are transitive and re$hAve. (2) @.,- and @-equivalence are equivalence relations. Proof. (1) Easy to check the transitivity by the definition. The re- flexivity follows by 1.9. (2) Follows from (1). Remark. So naturally, @ (@J is an order on its equivalence classes. LEMMA 4.2: (1) T ie @.,-&TTuz~ (i.e., TI@A T for every T,) if fw every model M of T and regular ultraJilter D over A, MA/Di8 A+-COm$MGt+ D is A+-g&. (2) T is @.,-minimal (Le., T @., TIfor every T,)iff for every model M of T and regular ultra$lter D over A, MA/Die A+-mpact. (3) T is @-maximal [-minimal] iff it is @.,-WWX~TTUZ~[-minimal] for every A. Proof. (1) Let M, be a model of T,, M a model of T, and D a regular ultrafilter over A. If T satisfies the condition mentioned in (l),and MA/Dis A+-compact,then D is A+-good, hence by 2.3 and 2.11 Mt/D is A+-compact. As this holds for every M,, D; TI@., T, so T is @.,-maximal. Suppose now T is @.,-maximal, D a regular ultrafilter over A and M a model of T and assume MA/Dis A+-compact. By the @,-maxi- mality of T,Th(Mh)@., T [MAfrom 2.2(3)] hence Mi/Dis A+-compact. By 2.3 this implies that D is A+-good. (2) If T satisfies the condition, clearly it is @.,-minimal. If T is @.,-minimal, then T @., T, [the theory of (A, =)I. But for every regular ultrafilter D over A, II(A, =)h/DII = AA/D= 2A > A, hence (A, =)"D is A+-compact.So clearly T satisfies the condition. (3) Immediate. THEOREM 4.3: Any theory with the strict-order property is @-mtwiml. Proof. Just like 2.6 and then use 2.11 and 4.2. (Alternatively use Exercise 4.6.) 372 SATURATION OF ULTRAPRODUCTS [CH. VI, 8 4 THEOREM 4.4: Let M be a moclel of an urntable theory T, D an ultra- filter over I and suppose that tkre are finite cardinah m,, i E I, ahthat No s n,, m,/D < 2A. Then N = M'/D is not A+-wmpact. Proof. By 11, 4.7 T has the strict order property or the independence property. Case I. T has the strict order property. So there is a formula ~(2,8) which is a partial order, and not reflexive, and for every n there are q,. . ., ?i&l E M such that ktp[sir, Let P, = {ZP:1 < m,},(N, P) = nipr(N, P,); so by assumption IPI < 2A. Notice that each (M, P,), and hence (N, P),satisfies the sentence (VZ)[(3g)[P(g)A (VZ)(P(Z)3 F(2, 2) = q(8,2))l v (VY)(P(B) + 1q(% a))]. Suppose N is A+-compact, then we can define ti,, 5, E P 7 E A22 by induction on Z(q) such that: (i) N t= ~[si,, 5,] and for every n - (N, P)k (360,.. . ,Zn)( A 9(Zl,Zl+l)) A sin = 2, A b, = Zn A A P(ZJ), 1 en 1 en (ii) 7 Case 11. T has the independence property. So some y(x; 8) has the independence property. Hence we can find sequences ti: E IN1 (1 < K) so that k(3z)[AlCk~(z; ?$)l-"lEW)] for each w c k, so let bk E IMI, be such that 'A F(bk,; Zr)i(lew). 1Ck Let p = min{n,,, nJD: n, < w for i E I and n,,, nJD 2 No} < P, P = l?rer nilD. For every ~EIdefine k(i) = [log,n,] if n, # 1, k(i) = 1 if n, = 1. Let P, = {a;(,): 1 < k(i)),&, = {b","):w c k(i)},so clearly n,/2 - 1 I 1Qf1 s 12,. Let (N,P, &) = ILr(M, Pi, QO, 80 clearly IS1 = Riel ISil/D s p, but clearly !PI s 101 and for every 1 < w {id:lP,l 2 1) 2 {id:n,2 2')ED (as nn,/D 2 No).Hence IPI is infinite; hence by p's definition = /PI = p. Let A1 = min(A, p) and choose I s P, 111 = Al. By the de- finition of the Pi's, clearly for every J E I, pr = {y(z;si)ii(dor): si E I) is OH. VI, 5 41 KEISLER’S ORDER 373 a type in N. Now by the definition of the Qr, if pJ is realized in N, it is realized by some element of Q. Hence the number of types pJ,which are realized in N is IIQl = p (because J, # J, implies no element realizes both pJl and pJJ. On the other hand the number of such types is I{J: J c I>I = 21’1 = 2”. Clearly 2” > p, and by hypothesis and de- finition of p, 2A > p; hence 2.,1 > p. So for some J G I N omits pJ,and as IpJI = A, 5 A, N is not A+-compact. EXERCISE] 4.1: Prove that if M = ntsIM,/D is a model of an un- stable theory, D an ultrafilter over I, KO InfoI m,/D < 2.,, then M is not A+ -compact. THEOREM 4.6: Let T be stable and urith the jnite cover property. Let M be a model of T and D an ultrafilter over I; if KO I p = nfsIm,/D then M’/D is not p+-mpact. Proof. Let q(z,y; Z) be as in 11, 4.4, i.e., for every E E IMI q(z,y; a) is an equivalence relation and for n < w, q(x,y; E,J has ~n but < No equivalence classes. Let Neq(E, M) be the number of equivalence classes of q(z,y; 5) in M. Clearly for every family of models M, i E I in which q(z,y; 2) is always an equivalence relation Choose E[i] as En[rl where n[i] = max{Z: Neq(E,, M) Im,} if such exists, n[i] = 1 otherwise. Clearly No I Neq(C, MI/D) Info,m,/D, so we finish. EXERCIBE 4.2: Prove the parallel to Exercise 4.1 for Theorem 4.6. LEMMA 4.6: If T is @A-minimul,p 5 A, then T is @.,-minimal. Proof. Assume T is not @.,-minimal. By 4.2(2) there is a regular ultra- filter on p, and a model ill of T such that M@/Dis not p+-compact. Let D, be a regular ultrafilter on A, D, = D, x D, I = A x p, so D, is a regular ultrafilter on I, 111 = A, and M1/D, z (MA/D,)#/D.By 1.1 M = MA/D,so by 1.9 and 2.11 Mp/Dis p+-compact iff (MA/D,)U/D is p+-compact so MI/D, is not p+-compact, hence not A+-compact. So T is not @.,-minimal. Contradiction. 374 SATURATION OF ULTRAPRODUCTS [OH. VI, 0 4 THEOREM 4.7: If Krdt(T) = 00, D ie a regukcr dtraji2ter over I, M ie an IIl++-eaturatedmodel of T,thn Mr/D is not IIl++-cornl)ccct (eee Definition 111, 7.2). Proof. If T has the strict order property the theorem follows im- mediately from 4.3, 4.2, 2.4 and 1.3. Next aasume K~~(T)= 00 ; w.1.o.g. we can assume (see 111, Definition 7.3, 111, Exercise 7.4 and 1.11) 11M11 > p = aA+where IIl = A and there is a formula cp(Z;g) and a{ E \MI,B < A, a < p, I@{) = Z(g) such that for every fl {cp(f; a:): a < p} is 2-contradictory, and for every 7 E "p {q@; atlE,): < A} is consistent, realized by En. Let P = {a:: /3 < A, a < p}, Q = {En; q E "}. We define equivalence relations E(E, g), Z(Z) = I(g) = I@) as follows E(iZ2, iZ2) iff p1 = pa; -,E(E, ?I{)if 17 $ P, E(E, a) if 8,a $ P. IZ has X equivalence classes in P(M)and IA+ equivalence classes in P(Mr/D)each one of cardinality zp. Let Jf, i < A+ be distinct equivalence classes of E, then for every sequence (a,: a, E J,, i < A+), {cp(E; at): i < A+} is con- sistent, and if Mr/Dis A++-compactit is realized in &(Mr/D).p < p+ s n,<,,,+lJfl s 1Q(M'/D1 = IQ(M)Ir/Ds pA = p, contradiction. So we are left with the case K,,,(T) < 00, ~~~~(2')= 00, hence by 111, 7.11 and 1.11 we can assume there are a formula p(Z; g) and a,, E IMI (7 E 'p)Z(G,,) = Z(J) such that if 7, v E >p are <-incompatible then M C 7(3z)[cp(Z; a,) A ~(2;a,)] and for each 7 E E,, E 1M1 realizes {cp(Z; a,,,,,): n < w}. Let P = {an: r) ~@>p},Q = {En: r) EOUCL), E = {(a,,, a,): for some n < w, 7, v E "p}, and < ={(sin, a,): Z(7) < I(v); 7, v E >p},El = {(a,,-cf), an- Mi l= (E)(W[Q(Z)A (V#)(P(#) A (P@; 8)4 dZ;811 (by the requirements on the p(Z; 4)'s) 80 we 0811 wmme 8 E v,and the rest is easy. Similarly for every q E "+pthere is P E QN recllizing p,,, but s lQMIr/Ds pA = p; the pnysam pairwise explicitly contra- diotory and their number is pA* > p; contradiction. OH. VI, Q 4) KEISLER~S ORDER 375 EXERCISE 4.3: (1) Show that in 4.7 it suffices to assume D is uniform. (2) Show that by 4.7, there is a model Mo of T, 11 Mo11 = 1I1 + ,such that Mo < M implies Mr/D is not I I I ++-compact. DEBTNITION 4.2: Let M be a model, D a filter over I; for a E I Mx/DI, eq(a) = ((8, t) E I x I: a[8] = a[t]} and for Q a filter over I x I, MLJ Q is the submodel of Mx/Dwhose set of elemente is {a E lMx/D1: eq(a)E a). We always aasume {(t, t): t E I) E Q. DEBWITTON4.3: T1 @* T, if for every set I, XI-incomplete ultrafilter D over I, filter Q over I x I, cardinal A, and (A+ + (I1+)-saturated models MI, Ma of T,, T, respectively: if MGDlG is A+ -compact then M',,lQ is A+-compact. DEFINITION 4.4: Let Q) = {vr(Z;2'): y < a}, Q), = {#,(g; 2,): y < p} be indexed sets of formulas (possibly with repetitions) from L(T,) and L(T,) respectively; Z(Z) = ml, l(g) = ma. (a1,m,) 5 (a,,ma) if there is q E such that for every model M, of T, and By E lMll, y < a, there are a model Ma of T, and 6~E 1 Ma],y < /3 such that: for every w c a {g@;BY): y E w} is consistent with M, iff {#,,c,l(g;8nCrl): y E w} is consistent with Ma. We write (@,, m,) 5 (Qi,, ma) by q. Remarke. (1) Clearly by the compactness theorem (a,,m,) 5 (a,,ma) by 3 iff for every finite Q) E @,, (@, m,) < (a,,m,) by q r w, where W, = {y < a: py(E;Ir) E @}. (2) In Dehition 4.4 we can take M,, Ma as ked A-universal models (for large enough A). (3) We write Q) E L (in this section only) to mean @ is an indexed set of formulrts, possibly with repetitions, from L. EXERCISE 4.4: (1) If (Q),, m,) 5 (Q),, m,), @l E Q),, 0, E Oa then (@lS m1) 5 (Psma). (2) If O1[P] is the olosure of @, [a,] under conjunction and dis- junction, then (a,, m,) s (a,,ma) implies (@I, m,) 5 (P,ma) (why do we not eay "negation " 2) (3) If (@lS ml) 5 (@a9 and (@a9 (@a9 %) then (@IS ml) ('J's, 7%). EXERCISE 4.5: If for every 0, E L(T,), 1@,1 = A, there is Q), E L(T,) and % < o such that (Q),, 1) s (Q),, ma) then T, @, TI. 376 SATURATION OF ULTRAPRODUCTS [CH. VI, 8 4 DEFINITION4.6: Let @,, §,, m,, ma be as in Definition 4.4 (@,, m,) S* (Oatma) if there is 7 E ap such that for every model M, of T, and Sir; E 1 M11 y < p, n < W, there are a model Ma of T, and 6; E 1 Mal, y < 8, n < w such that: for every w c a x o {tp,(E; a;) : (y, n) E W} is consistent with M, 8 {#,,Ey,(~; 6:[~1):(y, n) E W} is consistent with Ma. We write (Qs,, m,) r * (@,, ma)by q. EXERCISE 4.6: (1)In Definition 4.6, we can replace o by any a 2 o. (2) (@I, m,) s* (@a, ma) by 7 implies (@I, m,) 5 (02, ma) by 7. (3) (@I, mi) I* (@a, ma) implies (01, mi) 4 (@a, ma). (4) If @,, @' contain the same formulas (with a different number of repetitions) then (6) If @l G @,, Q2 E Oa then (a1,m,) S* (Oa,m,) implies (@I, m,) s* (Ga,ma). (8) (@,, m,) S* (@,, m,) (by the identity map). EXERCIflE 4.7: The following statements about T,, T, are equivalent. (1) For every c L(Tl)there are 0, G L(T,) and ma < w such that (@,; 1) s* (@,, ma). (2) For every §, E L(T,), s (T,I + 1T,1 + there are @, E L(T,) and ma such that (@,, 1) s (@,, ma). (3) For every @, s L(Tl)there are 6, c L(T,) and ma such that (@I, 1) (@a, ma). (4) For every @, c L(Tl), 1@,1 s IT,I there are 0, c L(T,) and ma such that (@,, 1) s * (@,, ma). (6) Let 0, be the set of formulas tp(x;jj) E L(T,) (clearly )§,)= I T1l).There are 0, E L(T,), ma such that (@,, 1) s* (@,, ma). EXERCIflE 4.8: (1) If 0, is the set of all formulas in L(Tl),and for some @, c L(T,), ma < w, (Go, 1) s* (@,, ma)then T, @* T,. (2) In fact it suffices to demand that there are a,, i < i,, such that: if,M, is a non-A+-compact model of T,, then there is a type p over M,, p = ipa(z,7ia): a < a, < A+}, such that for some i < i, every tp,(x, jja) E 0,; and there are @,,, E L(T,), ma,, < o such that (QZ,, 1) r* (@a,is ma,{)- EXERCILSE 4.9: Suppose M, is a A-universal model of T,M, is a A+- compact model of T,and D an ultrafilter over A. If N',/Dis A+ -compact CH. VI, 8 41 KEISLER'S ORDER 377 then Mi/Dis A+-compact.If D is uniform it suffices to demand MI is ( < A)-universal, M2is A-compact. EXERCISE 4.10: For M,I, D, Q as in Definition 4.2, D an ultrafilter; M',la is elementarily equivalent to M, moreover a H (. . ., a, . . .)te,/D is an elementary embedding of M into M',]a. Generalize 1.1. DEFINITION4.6: A complete theory T is simple if (1) L( T) contains only one-place predicates, the equality sign = , and one other two-place predicate E; (2) for every model M of T, EM is an equivalence relation over 1M1; (3) there is a model M of T such that for every ~EWI,[aIM is infinite where [aIM= {b E M:M k bEa, and for every predicate P(x)of L(T) M I. P(a)= P(b)}; (4) there is a model M of T such that for every a E 1M1, there are infinitely many b E (M( from Merent E-equivalence classes which realize the same type over 0. EXERCISE 4.11: Let T be a simple theory. Show that: (1) If M is a model of T,a E IMI, then any permutation of is an automorphism of M. (2) Every formula of L( T) is equivalent to a Boolean combination of formulas of the following forms: (9 x = Y, (ii) xEy, (iii) P(z), (iv) (~Y)[xEYA Aj EXERCISE 4.12: Suppose M is a non-A+-compactmodel of a simple theory T. Show that M omits a type p, which is of one of the following forms: (i) p = {xEa} u {P,(x)n[C1:5 < to Imin(A, IT])} u {x # c,: 5 < lo s A}, (ii) p = {P,(X)"~~:< c0 Imin(A, ITI)} u 1p0 u {TxEc~:5 50 5 A}, 378 SATURATION OF ULTBbPBODUCTS [CH. VI, 8 4 where po consists of formulas of the fourth form from the previous ex- ercise and of negations of such formulas, and r] is a sequence of zeroes and ones. EXERCISE 4.13: If M is a A-compact model of a simple theory T, and N = MblG (see Definition 4.2) is IT1 +-compact, show that N is h-compad. (In fact it is ALIU-compact.) EXERCISE 4.14: Show that: (1) A simple countable theory is @-minimal. (2) If N is a model of simple theory T,D a I TI + -good ultrafilter on p, then Nu/D is Kg/D-compact. Hence if D is p-regular, W/Dis 2,- compact. EXERCISE 4.15: Show that for every theory T,and cardinal h there is a simple theory T2such that Tl@A T, @A T,.If I TII IA then also lTal IA. Moreover if D is a A-regular ultrafilter over p, AZ, a model of TI, M, a model of T2then MU,/Dis A+-compactiff M$/Dis A+-compact. (Hint: See Exercise 4.5.) EXERCISE 4.16: Show that for every set {Tc:&! < to}of theories there is a least upper bound for each of the orderings @, @A. Its cardinality is 5 zcI Tel . EXERCISE 4.17: (1)Show that for every A there is a simple theory Tj,, lThl = h such that TA is @A-maximal.Hence if A < p, TA @ T, but not T, @ TA. So thare is an (uncountable) theory which is not @- minimal nor @-maximal. (2) Show that if there is a countable theory which is not @-minimal nor @-maximal (see 5.9) then there are @-incomparable theories. EXERCISE 4.18: Prove case I of the proof of 4.4 by 4.3. EXERCISE 4.19: (1) Suppose xlnp(T)> lIl+, iK an IIl++-saturatd model of T, and D a regulm ultrafilter over I. Prove that Mr/Dis not III ++-compact(see Theorem 4.7). (2) Prove the parctllel of Exercise 4.3. QUESrTION 4.20: Can we in Exercise 4.19 replace K~~(T)by xcdt(T) ? CH. 8 61 SATURATION OF ULTRAPOWERS 379 EXERCILIE 4.21: Show that T1 Ta for any complete theories Ti, Ta. THEOREM 4.8: For any model M of an umtable theory T and 8,- incomlplete ultrafilter D Over I,K = lcf(w, D), MIlD is not ~+-~~mpact. Proof. Choose an unstable formula ~(z;g) and sequences q (1 c n < o) such that {rp(x, $)u(l>k): 1 < n} is consistent for every k < n < w. Choose a distinct b, E: 1M1 and let: PM = {b,: n < w}, PI(&,kn), * * a). Let N1 = (N, PN,<*, Fry.. .) = (MyPM I, = {Tv(x, Fo(bny GO), . . .): n < O} u {~(x;Po(c,, GO), . . .): O < i < K}. Clearly p is finitely satisfiable, but N omit it (if a realizes it, then the formula P(Y) A (VZ"(4 z Y + ?+, FO(YY CO)F,(Y, CO), - - *)I define a bounded subset of P with no last element). VI.5. Saturation of ultrapowera and categoricity of pseudo-elementary classes THEOREM 6.1 : Let T be a countable theory, MI a rnocEel of T for every i E I, and D an K,-incomp&e ultraJilter mer I. Let N = nlEIMJD. (1) If T does not have tk f.c.p., A = Ki/D, then N is A-saturated. (2) If T is stable and hua the f.c.p., then N is A-saturated, but not A+-saturated, where A = min{n,,, nJD: nlEInllD 2 Xo}. (3) If T does not have the f.c.p., each Aft is p-saturated, and A = pl/D, then N is A-saturated. (4) FOTeveryjinite A s L(T)let &(A) = min{lpl: p is a A-I-type over lM,l in MI which is omitted by Mt} and t L(T),[A1 < H,}. 380 SATURATION OF ULTRAPRODUCTS [m. VI, $6 Let A be an (injnite) cardinal whtlrat A = nisrAi/D for ~omeA:, i E I, wkre for every jEnite A C_ L(T),{i: A: s &(A)} E D. Then if T aoe.9 nd liaVe tk f.c.p., N i8 A-saturated, but not (A*)'-Sat~rate&. Remark. (1) Clearly the results, except (a), m the best possible. For example in (l), if we choose the Mi aa aountable models, then IlNll = K',/D = A, Eo N is not A+-saturated. On (4)see Exercise 6.10. (2) Instead of demanding T countable, we can require D to be ITI + - good. See Exercise 6.2. Procf. Notice that as T is countable, for every model H of T and cardinality K > KO,M is wcompact iff M is K-saturated. Now in part (2), N is not A+-saturated by Theorem 4.6. Similarly we can prove in part (4) that N is not (A*)+-saturated.So it remains to prove that in all the parts N is A-saturated. N is XI-saturated by 2.1, 2.3 and 2.11. By 111, 3.3, 3.9 and 3.10, aa T is countable and stable, it suffices to prove: if {c:: i < o} c IN1 is an indiscernible set, then it can be extended in N to an indiscernible set of cardinality A. For every i E I let us choose a family Sf of subsets of lM,l such that: (i) IS:l = 11~:11, (ii) every finite subset of IHtl belongs to St, (iii) for every finite A c L(T),n < o,if w E 8,is ad-n-indiscernible set, 0 s p s 11H,ll, and there is a A-n-indiscernible set w', w E w' c lMtl, Iw'I = p, then there is wNeSi, Iw"I = p, w E w" c lMtl, and wN is ad-n-indiscernibleset. Let /Mil = {af:j < IIH,II},St = {wi:j < llH,ll}. Let us define the relation E{ on lMil: E' = {(a;, ah): a: E wh}. We shall write z E y instead of ~(z,y). In the language L = L(T)u {E}, clearly, for every finite A c L(T),n < o,there is a formula 'g,,Jz) meaning {y: y E z} is a A-n-indiscernible set. Now for every i E I we define Pi according to the part of the theorem we want to prove; in (1): F = {a:: lwhl z H,,}, in (2): F = {a;: a < 11M,11} = [Mil; in (3): F = {a:: 1w:I 2 p}; in (4): Pf = {a;: lwtl 2 hi); where the A' are defined so that InlE,X/Dl = A, and for every finite A E L(T){i: A: s &(A)}€ D. Now the following hold: OH. 8 61 SATURATION OF ULTRAPOWERS 381 (*I For every finite d c L(T), n < w, there is m = m(d,n) < w, such that the set of i's for which the following holds belongs to D: (**) For every A-n-indiscernible set w;, Iw;l 2 my there is a A-n-indiscernible set wi, wk E wj and a: E P. Let us prove it. In part (2) it is trivial. In the other parts T does not have the f.c.p. so in part (1) it follows from 11, 4.6(3) and in parts (3) and (4) from 11, 4.6(2). Notice that except in part (4) (**) holds for every i. Now clearly (**) is equivalent to a first-order sentence in L' = L u {E} u {P}.Let N' = (N,E~, PN) = nfo,(Mf, cf, P)/D.N' is 8,- saturated (by 2.1, 2.3 and 2.11). By (*), clearly the sentences corre- sponding to (**) (for all finite A, n, m) are satisfied by N'. Remember it suffices to prove that {cf: i < w} can be extended in N to an indiscern- ible set of cardinality A. As {cf: i < w} is an indiscernible set, it is a d-n-indiscernible set for every A, n. Hence every finite subset of p = {c, E x: i < w} u {rpd,,,(x):d E L(T), lAl < No, n < w} u {P(x)}is satisfied in N', hence p is satisfied in N' say by b. As N' C qd.,,(b) for every A, n, clearly w = {u E INI : N' C u E b} is an indiscernible set, and of course {cf: i < w} G w. As N' != P[b],and lwl 2 I{cf: i < w}I = 8,, clearly IwI 2 h (the check for each part is easy). So we have proved the theorem. It would be more satisfactory if in 5.1(4) h = A*. (This is possible if ViEI,M, = M). For this it suffices to prove CONJECTURE 5.1: Let Let (J, <) = (p, <)'ID. (<-the natural order on ordinals.) For a E IJI, let 1.1 = l{b E IJI: b < .}I. Suppose a,, E IJI for n < w, la,,l = lao[.Then there is a E I JI, u 5 a,, for every n < w, and 1.1 = la,[ (when D is an 8,-incomplete ultrafilter). See Exercise 5.11 and 5.12. THEOREM 5.2: Let M be a h-compact model of T,D an ultrujlter over I, IT1 5 111, N = M'/D. If N is (21'I)+-saturatedthen N is hI/D-saturated. Rernurh. (1) For countable T this theorem follows from 4.4, 4.6 and 5.1(3). (2) Here the proof works also for D an 8,-complete ultrafilter. (3) See Exercise 6.13. 382 SATURATION OF ULTRAPRODUOTS [OH. VI, 8 6 Proof. As N in (21rl)+-saturated,by 4.4 and 4.6 T is stable and without the f.c.p., and so clearly every infinite indiscernible set can be extended to one of cardinality 2 (21rl)+. By 111, 3.10 (remembering that by 111, 3.3, K(T)5 I TI + s 111 +), it suffices to prove that: If I is an indiscernible set in N, 111 2 (21rl)+,then there is an in- discernible set J, 11 n JI 2 KO,lJl 2 JAr/Dl.Let {aB:fl < (2111)+} E I. The following statement will be proved later. (*) There is an infinite w c (21'1) + such that for every i E I, {aD[i]:fl E w} is an indiscernible set in M. We can assume A > I TI , as otherwise the conclusion of the theorem is trivial. For every i E I let P' be a maximal indiscernible set such that {a,[i]:flEw}E P c IMI.AsMisA-compact,A > ITI,clearlyIP'I 5 A. Let (N, P) = nfeI(My P)/D. Clearly ]PI = miellPfl 2 Ax/D. Now for every finite A G L(T), n < w, the statement " P is a A-n-indiscern- ible set " is elementary, hence P is an indiscernible set. So {aB: fl E w} E P E INI, hence IPn 11 2 l{aB:fl~w}I2 KO. So P satisfies the re- quirement for J. So we need only prove (*). As T is stable, by 11, 2.13, 1231 I21'1 implies IS(B)I I (21rl)lTl = 2Ir1. It is also clear that for B, c IN], lBtl 5 21'1, for every t €1; I!Xt,rfl(Bt)( = nter ISCBt>l I (2"')''' = 2'"- Define for j 5 lIl+, sets wj c (21'1)+ by induction: (i) wo = 0, wd = uj Remark. We can in fact find such w of cdidity (21'1) + . See Exercise I, 2.9. DEFINJTION6.1 :A model Y is complete if for every n < w, every n-axy relation on and every n-ary function from IN1 to is the inter- CH. VI, 5 61 SATURATION OF ULTRAPOWERS 383 pretation of some predicate symbol or function symbol, respectively, from L(M). Remark. Clearly for every set A there is a complete model M, ]MI = A, IL(M)I = KO + 21-41. DEFINITION 6.2: PC( T,, T) is the class of L(T)-reducts of models of T,, of cardinality 2 lTll, where T is complete, T s T, and T, has in- finite models. The restriction “of cardinality 2 ITl] ” is technical, and without it the class is denoted by PC*(T,, T). THEOREM 6.3: If T is countable, superstable, and doap not hue the f.c.p., then there i8 TI, T E TI, ITl] = 24 8Wh that PC(T1, T) i8 categorical in every cardinality 2 2No. Procf. Let M be a countable model of T.We expand M to a complete model M,, IL(M,)I = 2% Let L, be the language of M, and T, the theory of Ml (ia, the set of sentences from L, that M, satisfies). Clearly T, contains its Skolem functions (see Definition VII, 1.1). Let N, be any uncountable model of T,, and let N be the reduct of N, to L(T). It suffioes to prove that N is saturated (as by I, 1.11 every two saturated models of the same complete theory which are of the same cardinality are isomorphic). So let p be any 1-type in N, 1p1 < IlNll, and it suffices to prove that p is realized in N. Let p1 be any extension of p to a complete type over INI, and let Q(z,~)E~,be such that R[q(~;ii),L,a]= R(pl,L,a). Let 1M1 = {ai:i < w}, and let c,, i c w be individual constants in L, such that cfdi 3: a,. Clearly there is aoE IN,], ao # $1 for i < w. Define A = {FNl[ii,ao]: P is a function symbol in L,}. Clearly the submodel Nf of N,, INfI = A, is an elementary submodel of N, (by the definition of T, and the Tarski-Vaught test I, 1.2 or VII, 1.2). Let N* be the reduct of Nfto L(T). Clearly N* is an elementq submodel of N. We now show (*) Nf is N,-compact hence N* is K,-mturated. So let q be a countable type in Nf and we must prove it is realized in Nf. Let q = {ql(z; a&,. . ., ak): i < w}. As every a: E A, for some Ff, E L,, a: = Fi,jp, ao]. So by substituting we get q = {I,~~(Z;ii, ao):i < w}. Remembering that IMI = {a1: i -c w}, 384 SATURATION OF ULTRAPRODUCTS [CH. VI, 5 6 C~I= a,, M, is complete; it is clear that there is a function symbol Q in L, such that for every a,,,6, bo from lMl, QMi(a,,,5, bo) realizes {t,h,(x;6, bo); i < m} for the maximal possible m In. Clearly for every n M, c (VWY)[ (A Y z ci A (32) A t,h,(x, 2)) -+ A t,h:(Q(y, @,211- l M1 C (VZNVY~** * Yr)[(A Yt + ~j A qd.n(Z) A A ~1 Z) 1 --+ (jy)(qd..(y) A P(Y)A Atsr Y; EY)]. This clearly implies the consistency of q', as {bi: i < w} is an indiscern- ible set (in L(T))and for every c,, . . . , cn E N, there is c E N, such that N, C: (VZ)(Z E c A?-, z = cf). Remarks. (1) TIhas no models in any uncountable cardinal < 2'0 and is categorical in KO. (2) By 5.1(2),VIII, 1.7 and VIII, 2.1 the theorem is the best possible. The following theorems have similar proofs so we omit them. THEOREM 5.4: (1) If T is countable, withut the f.c.p., and stable in KO (i.e., totally transcendental) then there is T,, T G T,, ITl! = No, mhthat PC(T,, T)is categorical in every h 2 KO, and every model in it is saturated. (2) If T has the f.c.p., is countable, and stable in KO,h 5 2N0 then there is T,, T G T,, lTll = h such that PC(T,, T)is categorical in h and every model in it of cardinality h is saturated. THEOREM 5.5: If T is countable and superstable, then there is T,, T C T,, lTll = 2No such that PC(T,, T)is categorical in 2%, and every model in it of cardinality 2w0 is saturated. Remark: We use the following fact: if M, is a complete model which expands (w, c), N, is an uncountable model of the theory of M,, asINl[, [{bE lN1l: b < a)I 2 KO,then I{~E lN1l: b c a}I 2 2% We leave it as an exercise. THEOREM 5.6: Let M be a model of a countable and superstable theory T, M, a complete expansion of M, N E PC(Th(M,), T).Then N is 2~0-saturated. THEOREM 5.7: If T is not @,-minimal, then it is rwt @,,-minimal for every p 2 min{21TI, A}. Proof. If p 2 h the conclusion follows by 4.6 so we can assume h > p 2 2ITI; and by 4.6 again it suffices to prove the theorem for the case p = 2ITI. So let h > p = 2ITl, T is @,-minimal but not @*-minimal. 386 SATTJRJ~TIONOF ULTRAPRODUCTS [CH. VI, f 6 As T is not @,-minimal, by 4.2(2) there is a regular ultrafilter D over A such that for every model N of T,NA/D is not A+-compact. Let M, be a A+-saturated model of T, M < M,, llMll = (TI. Suppose first MA/Dis not IT)+-compact. Then there is A c ]MA/D1, IAI I; IT], such that MA/Domits a type over A. W.1.o.g. (by Ds regularity) there is an equivalence relation, eq, over A, eq G A x A, such that for every a E A, eq c {(i,j) E h x A: a[i] = a[j]}and eq has IT1 equivalence classes. Let N be the submodel of NA/Ddefined by IN1 = {a E IM’/DI: eq c {(i,j):a[i] = arj]}}. Then N is also not ITI+-compact and clearly for some ultrafilter D, over IT1 N is iso- morphic to NITI/D,, so T is not @lTl-minimalhence not @,,-minimal (as we can make D, regular). Assume nowMA/Dis ITI+-saturated.By 1.8,111, 3.10 there is an in- discernible set I = {a,:n < o} in MA/D,dim(I,Mt/D) < A. Without loss of generality there is an equivalence relation eq over h with I; IT1 equivalence classes such that eq E {(i,j):a,,[i] = a,[j]} for every n < o. Clearly {aE wt/DI : eq G {(i,j): a[i]= ao]}}is the universe of an elementary submodel N of Mt/D and I c m. It is also clear that for some ultrafilter D, which we can assume is regular, over ITI, N and MTI/D, are isomorphic. As Mo is A+-saturated, h > 2ITI it suffices to prove N is not (21TI)+-saturated.If it were, it would be A+-saturated by 5.2, hence A 2 dim(I,Mt/D) 2 dim(1,N)2 A+. Con- tradiction. Now we shall try to deduce some results on @. THEOREM 6.8: Let T be countable. (1) T is @-minimal iff T does not have the f.c.p. (2) For h 2 2Mo T is @,-minimal iff T does not have the f.c.p. (3) If No < h < 2N0, T is @,-minimal i,lff T ia stable. Proof. (1) By 4.2(3) and.Exercise 4.21, this follows from (2) and (3). (2) We use the criterion in 4.2(2); i.e., T is @,-minimal if for any model N of T and regular ultrafilter D over A, MA/Dis A+-compact. Suppose T does not have the f.c.p., N is a model of T and D is a regular ultrafilter over A. By 5.1(1) MA/Dis K,”/D-compact and by 1.12 N$/D = N,” 2 A+, hence MAIDis A+-compact. Suppose now T does have the f.c.p.; by 3.12 there is a regular ultra- filter D over A and n, < w (i< A) such that nl<,n,/D =‘2h (aa OH. VI, 5 61 SATURATION OF ULTRAPOWERS 387 2M0 5 A) and by 4.6, for any model M of T,YA/D is not (2h)+-compact hence not A + -cornpaat. (3) Similarly. For M a model of T,T stable; if D is a regular ultra- filter over A, by 6.1(2) MA/Dis p-compact, where p = minm,,, n,/D: nfeAn,/D 2 KO}. But by 1.3(l), (2) D is 24,-incomplete, hexme by Exercise 2.10 pMo = p, hence p 2 2%, hence p > A, hence MA/Dis A + -compact. Now let T be unstable, by 3.12 there is a regular ultrafilter D over A such that lof(No,D) = K,, hence by 4.8 MA/Dis not K,-compeaf, but K, 5 A, so it is not A+-compact. THEOREM 6.9: Anwng countable tWea (1) the thiea with& the f.0.p. fman equivalence class, (2) the stable them&?%with the f.c.p. fman equivahce ch8, (3) the t-ea with the etrict order property are all equivalent, (4) if T,, Ta and T, are a~ in (l), (2) and (3) rqectively, and T 8&8* ni~~of them, th T, @ Ta @ Tad @ T @ T,, but not T, @ T,, and not Thd @ Ta. If, e.g., MA + zn0 > K, holds then not T, @ Thd; mmer K,~~(T)= 00 impliea not T @ Thd. Remarks. (1) The theorem holds for @A instead of @ for any h 2 2no; and for No < h < 2Ho, the only difference is that all the stable theories become equivdent . (2) For (4) see also Exercise 3.26. Proof. Let T,,Ta, T, and T be as in (a), with respective models M,, Ma, bl, and M and ikfhd a model of Thd. (1) This is the previous theorem. (2) M?JD is A+-compact iff n,/D 2 KO * nieAn,/D > A (by 6.1 (2)) hence they me all equivalent. (3) This is 4.3. (4) T, @ T, is clear from the proof of 6.8 and 6.9(2), T, @ Thd is clear by the proof of 6.8(2) and 4.8 (as nf EXERCIBE 5.2: Show that if we replace “T is countable” by “Dis ITI +-good” in the hypothesis of Theorem 6.1 ,the conclusions still hold. EXERCIBE 5.3: Let M be a model of a countable and superstable theory T, D an 8,-incomplete ultralilter over I, U a filter over I x I, such that 11MLlUll > No, MLlU # M. Show that if T does not have the f.0.p. and M is h-compact, then MLlU is ALlB-compact. (See Definition 4.2.) EXERCIBE 5.4: Suppose E(Z, g; 2) is aa in II,4.4, M a countable model of a stable theory T which has the f.c.p., D an N,-incomplete ultra- filter over I and U a filter over I x I. n, = I{Z/E(E,#; a): E E 1M]}1, P = {n,: E E IMl}. Show that there is E E IMLlUl n, = A, iff there is 8 E I(w + 1, < , P)LlUl, such that P(8)and EXERCIBE 5.5: Let M be a model of a stable theory T which haa the f.c.p. D an K,-incomplete ultrafilter over I, U a filter over I x I, and let A c L(T)be finite. Let p be a A-1-type over N = MLlG which is omitted by N; but every q c p, 141 < lpl is realized in N; and lpl is regular. Show there is 8€(11itf11 + 1, <)&lasuch that 1231 = p: CllJfll + 1, 4blU l=t < 811. EXERCISE 5.6: Let M be a model, M, a complete expansion of My T = Th(M), T, = Th(M,). Then {MLIU: D an ultrafilter over I, U a filter over I x I) is equal to PC,(T,, T). EXERCIBE 5.7: Let T be countable and complete. Then: for some countable T,, T E T, and PC(T,, T)is categorical in No iff ID(T)I = No. (Hint: Like 6.3.) EXERCISE 5.8: Use Exercise 1.9 to get a better (2) presentation of the proofs of 6.1 and 6.3 (aee next exercise). OH. VI, 8 51 SATURATION OF ULTRAPOWERS 389 EXERCISE 5.9: Let L E L,, P E L, a one place predicate, and L,- model M,. Let M: be the submodel of the L-reduct of MI with universe P(M,). Then for every theory T, c L,, {Mf; M, C T,, M, an L,- model} is PC(T2, Ta) for some Ta (you can use VII, 1.1 and VII, 1.2). EXERCISE 5.10: In Theorem 5.1(4) let p = &I 11M,11,and A(d) E pr/Dbe (. . ., &(A), . . .)t6j/Dand let A** = I{a E pr/D:for every finite d c L, pr/D C a < A(d)}]. Prove that N is A**-wturated. [Hint: Use Exercise 1.9 and 6.8. There is (in N;) an order of type “w” of the formulas of L (including non-standard ones) and for every “set” I c N there is maximal n = nI such that I is tp,-indiscernible for every k < ni, It suffices to prove that each such I can be extended to an indescernible set of cardinality A**. We define by induction on aE Ni, ii, E N such that nr, is maximal where I, = I u {?is I fi < a}. Then I u {aa:N; C a < A(d) for each finite A} is ae required.] EXERCISE 5.11: The regular ultrafilter D over I satisfies the con- clusion of conjecture 5.1 when (J< ) C w 5 a, for each n iff lcf(K, D) > No,K s IIl where K = lcf(Ko, D). Remark: The conditions (J, < ) C w s a, are satisfied in the case we are interested in. [Hint: Let a, = (. . ., %*, . . .)JD, and w.1.o.g. a: is a cardinal (see Exercise 1.1) and let X, = mink: {t: x < a:} 4 D}; clearly No 5 x,+, 5 x,, so w.1.o.g. X, = xo; and by 1.12 la,l = XI’’. If {t: qn = x} E D for infinitely many n’s, a = (. . ., x, . . .)/D prove the assertion. So w.1.o.g. 4 c x for each n, t. Necessarily x is a limit cardinal, cfx s IIl. Let b, = (. . ., l$, . . .)ts,/D (j < K) exemplify lcf(N,, D) = K; and c, = (. . ., . . .)t,I/D; then (J, <) Cc, s a, A c, < c,fori EXERCIBE 5.12: Show that for good K,-incomplete ultrafilter D conjecture 6.1 holds. [Hint: If (J, <) C a, < w for some n, by 2.3 there is c, (J,< ) C n < c < a, for each n, and use 2.13. Otherwise by 2.3. K I IIl * lcf(x, D) > 111 and uae Exercise 5.11.1 EXERCISE 5.13: Suppose that for some n,, p = nlEInt/D> No and N = M1/D is (I TI + p)+-saturated, and M ia A-compact. Then N ie 390 SATURATION OF ULTRAPRODUCTS [OH. VI, 8 6 AI/D saturated. [Hint: As in 5.2, let A > [TI,T without the f.c.p., I E N indiscernible, 111 = ([TI+ p)+.There are finite Pt E 1M1 such that IP"I I p, 11 n PNI 2 KO, where (N, P") = nt,i(MyPt)/D. There is 7i E I such that stp(a, u I")E Av(1, N),and Qt = It E 1M1 an indiscernible set over u Pt based on stp(a[t], U Pt),and (M, P,&) = nts,(M, P,, Qt)/D. Then Q is indiscernible, /&I = K/D, Av(Q, N) = AV(1, w.1 VI.6. Saturation of ultrslimits For every model M and ultrafilter D over I the natural elementary embedding of M into Mr/D,is defined by a H (a: i E I)/D.Hence we can look at M'/D m an elementary extension of M; and so we get an elementary chain of models by repeatedly taking ultrapowers and taking unions at limit stages. For simplicity, all the ultrapowers will be with the same ultrafilter, which will be assumed Kl-incomplete. DEFINITION6.1 : Let M be a model, D an N,-incomplete ultrafilter over I. We defhe the ultralimit, UL(M, D, a) by induction on a, so that for /3 < a, UL(M, D, /3) is an elementary submodel of UL(M, D, a). (1) UL(M, D, 0) = M. (2) For a a limit ordinal UL(M, D, a) = UBSKUL(M, D, p). (3) For a = /3 + 1, UL(M, D, a)will be isomorphic to UL(M, D, @)'/D, and for each a E IUL(M, D, /3)1, the isomorphism ElB takes (a:i E I)/DE IUL(M, D, /3)'/Dl to a. So UL(M, D, /3) < UL(M, D, a). Notation. As M and D are fixed for most of this section we let M, = UL(M, D, a) and FBbe the isomorphism mentioned in (3) of the above definition. Clearly we can msume that for every a,/3, UL(M, D, a + 8) = UL(M,, D,/3).We write T for Th(M). We shall try to find how compact the ultralimits am, for various properties of the ordinal, the ultrafilter, and the theory of the model. As MK+,is isomorphic to ML/D, we shall restrict ourselves to M6 for limit ordinals 8. THEOREM 6.1: If cf 8 2 p, and for every A < p, D is A-regular, then Md i8 p-COTn$XXCt. Proof. Let p be a type in Md of cardinality < p. Then clearly p is a type in MBfor some /3 < 6. As D is ~p~-regular,p is realized in MB+ by 1.4, OH. VI, f 61 SATUIZATION OF ULTRALIMITS 391 hence p is realized in M,. So every type in itid of cardinality Proof. M, is K,-compact by 2.1, 2.3 and 2.11 (remember D is K,- incomplete). As T is unstable, by 11, 2.13 and 11,2.2; there is a formula ~(z;g) and sequences a,, a,,. . . a,, . . . from M, (an of the length of g) such that: for every m < w, {cp(z; 7in)M(nrm): n < w} is consistent with M,. As cf 6 = p, let 8 = U,<”u(j), wherej < i < p implies 1 < u(j) < u(i) < 8. We shall now define by induction on j sequences such that : (1) lMa(j)+Il,~’#l%j)l. (2) qj = {7cp(z; a,): n < w} U {~(z;a’)} is not rertlized by any element of Ma(j). (3) For every m < w, ppJ”= {cp(z; iE,)if(“Zm): n < w} u {cp(z; 8):i s j) is consistent (with Mw+l). If we succeed in defining the $’a then clearly by (3) p = {-,cp(z; a,): n < w} u {cp(z; Z): i < p} is consistent (with M,), because every finite subset of p is a subtype of ppJ”for some m,j. But ifpis realized in Hd, then it is realized in MB for some /3 < 6, and there is j < of 6, < u(j) < 8. Hence p is realized in Ma(,),in contradiction to (2). Hence p is a consistent type in M,, which M, omits, and IpI = KO + p < p+. So Md is not p + -compact. It remains only to define $, assuming 8 has been defined for i c j. As D is K,-incomplete there are I,, E D, I,,, s I,, I, = I, on<,I, = 0. Let us define 6j E lMk(,JDl: if i E I, - In+l,then @[i] = a,, 8’ = (6’[i]:i E I)/D,and = Ba(j)(8j). Let us check conditions (l), (2) and (3) are satisfied. Clearly 8 E lMa(,)+,1.Now for any n < w, {i €1:6j[i] = a,} = I, - I,+, 4 D so $4 lMml. So (1) is satisfied. For proving (2), suppose, for somej < p, c, E lMa(,,]realizes pi. Then {i E I: Mw) b cp[P&(c,)[i], 8j[i]]}E D, that is, {i E I: M,, b ~[c,, P[i]]} E D. Hence for some n < w Ma(,)k cp[cj, a,], so cj does not realize qj, contradiction, hence (2) holds; and (3) has a similar proof. DEFINITION6.2: p(D) is the least infinite cardinal p such that D is p-descendingly complete, that is, p(D)is the least infinite cardinality p 392 SATURATION OF ULTRAPRODUCTS [OH. VI, 5 6 such that I, E D, a < p, and a c /I * I, c I, implies no<,,I, # 0 (equivalently na<,I, E D). Remark. If D is K-regular then K c p(D);also p(D) I ]I]+.Note also that p(D)is a regular cardinal. Proof. Let p be a type in Md, 1p1 < p, and we shall prove that p is realized in Md, thus proving the theorem. As /pi < p I cf 6, p is a type in Ma for some a < 6. Let lpl = K,. We shall prove by induction on y I p that: (*I Every subtype of p of cardinality IK, is realized in Ma+ y + 1- As /I I K, = 1p1 < p 5 of 6, a + /I + 1 < 8, hence by proving (*)we shall prove that p is realized in Ma. Suppose we have proved (*) for every y1 < y. Hence every subtype of p of cardinality < K, is realized in Ma+, (remember every model is 24,-compact, hence every finite subtype of p is realized in Ma).Clearly we can assume w.1.o.g. that p is a 1-type. So let q = {g~,(z;a,): j < H,} be any subtype of p of cardinality K,; we shall prove q is realized in Ma+,+,. By the induction hypothesis, for every j < K,, there is c, E lMa+,1 which realizes {p,(z;ZJ: i < j}.As K, I IpI < p I p(D)there is a decreasing sequence I,, j < K,, I, E D, n, LEMMA 6.4: (1) If M strongly omitep, IpI = p(D),then M, also strongly omits p. (2) if Ma strongly omits p, 1p1 = p(D), and a < 8, then M, also strongly omita p. (3) In (1) and (2) instead of Ip I = p( D),its suflcea to aesurne that there are no I, EDfor /3 < lpl, /I < y - I, c I,, n,<,,,I, = 0; and lpl is regular. OH. VI, § 61 SATURATION OF ULTRALIMITS 393 Proof. We shall prove that (1) as (2) and (3) have similar proofs. Suppose (1) fails, so c1 E jMll realizes q G p, 1pI = lpl. Let c1 = Po@), p = {(p,(z; a,): I < lpl}. So clearly for every 8 < 191 = 1131 {i E I:211 k tp,[c[i]; a,]} E D. It is also clear that for every i E I, q(i) = {(p,(z; Be): M C (p,[c[i], a,]} is a subtype of q, hence of p, which is realized in M; hence Iq(i)l < 1p1. As lpl = p(D)is regular, for every i E I there is a bound ((i) < 1p1 to (5: M b(p,[c[i];a,]}. Let for 5 < 1p1, I, = {i E I:[(i) 2 l}.Clearly I,, 5 < IpI is a decreasing sequence, and by the definition of [(i), n, THEOREM 6.5: If T is urntable, 6 2 p(D), then M, is not p(D)+- compact. (Moreover there is a type in of cardinality p(D)which M, StrO?& OW&%.) Proof. As it is similar to the proofs of 6.2 and 6.7 we omit it. CONCLUSION 6.6: If T i8 unstable, p = min(cf 6,p(D)), then M, is maximally p-compact. (ie.,p-compact but not p+-compact). Proof. Immediate by 6.2, 6.3 and 6.5. THEOREM 6.7: If K(T)> p = min(cf 6, p(D)),T is stable, then 116, is maximally p-compact. Proof. By 6.3, M, is p-compact so we need to prove only that M, is not p+-compact. By hypothesis T satisfies p < K(T),so there are A,, 4 I p,~E~W,,), Q&, Sc), andai,,n, (I < p, n < w); such that: S < 5 * A, c A,; for every 5 < p, {i&: n < cu} is an indiscernible set over A,, ii,,,, E A,+l and -(pe(x; 7i,,o), (p(x; ~p.Clearly it suffices to prove the theorem for the case L = L(27) is the minimal language containing all the formulas (p,(x;S,); so ILI I p. Choose a([) < 6 for 4 < cf 6 such that 6 = Uc Let H, be the empty function, p(0) = a(0). For f a limit ordinal, He = UC<~HC,rB(O = max(a(€)iU< Remarks. (1) For every 6, there are S,, S such that 6, = S,+S; a,B < 6 =+ a + < 6 and UL(itf, D, 6,) = uL(Md,, D, 6). so the restriction on 6 is natural. (2) Clearly A > I2'1, so it s&ces to prove Md is A-compact. Proof. Let p be a 1-typein itfd, 1p1 < A. It suffices to prove p is realized in M,. Let q be any extension of p in S(lMd)). Notice that if IBI < K(T)s of 6, B E 1Md1, then for Borne a < 6, B G lMal. Hence by 111, 3.2.there is a < 6 and B E lMal such that q does not fork over B, and by 111, 2.16 and I11 4.18, q 1 1Ma1 is station- ary. So by 111, 4.2 there is a set B clHal, IBI s 12'1 such that for every 'p R(q 'p, 'p, 2) = R[(q 1 B) 1 'p, 'p, 21. Now we can define a,, for n < w such that: (1) a,, realizes q 1 (B u {am:m < a}), (2) if 8 > ws an E IJfo+n+ll* This is possible since D is 12'1-regular, by 1.4(2). Hence {a,,:a < w} is an indiscernible set and q = Av({a,,: n < w}, Ma). suppose for a moment 6 > o. Let P = {a,,:n < w} E lMa+al (ae a < 6, o < 6; a + w < 6). Let (Ma,Pd) = UL((M,+,, P),D, 6) (remember 6 = a + o + 6). Clearly Pd extends P and is an indiscernible set. So 'p(z, 6) E p implies 'p(z,6) E q implies {a:a E Pd, I= -+z, 6)) is finite. So all except lpl- 8, < h members of Pdrealizep. As lPdl = UL(Ho, D, 6) = A, the theorem follows. So we remain only with the case 6 = O. But then we can define the an's simultaneously in Ma+ I and the proof goes in the same way. The following exercises will deal with the problem of categoricity in logics with generalized quantifiers. Assumption: Let T be a complete theory in the logic L(Q1,..., Q") (where ifg, is a formula so is (Q'xg,)).Let P,, .. . , P, be unary predicates in L, VxP,(x)€T and let for any cardinals ,ul< < ,um,Al, ..., A,, K(p,,..., A,;..) = {M: M an L-model, that satisfies every $ET when (Qzx)O is interpreted as "there are at least A, x's such that B(x),and IRI = P'Y' Suppose further that ,uF = ,ul, each A, is regular and (Vx < A,)x'o < 4 * Suppose (*)K(p,,.. . , A,, .. .) has a unique model up to isomorphism. 396 SATURATION OF ULTRAPRODUOTS [m. VI, § 6 EXERCISE 6.1: We can assume w.1.o.g. that for any formula q(Z)in I,(()', ...,Q"), forsomepredicateR, (VZ)[~(Z)= R(2)l~T.LetT'bethe set of first order sentences of T. EXERCISE 6.2: Prove T' is superstable. [Hint : It is well known (see [CK 731) that K(pl,..., A,, ...) is closed under ultraproducts for ultrafilters over w. Let M belong to K(p,,..., A,, ...) (exists by a hypothesis), and let D be a non-principal ultrafilter over w. So M, gfM"/D and easily also M, gPUL(M, D, w) belong to the class K(p,,. . . , A,, .. .), hence by a hypothesis they are isomorphic. Now by 2.1, 2.3, 2.11(1), M, is K,-compact. By 6.5 if T is not superstableM, is not K,-compact. We conclude T is superstable.] We now define: W = {(q(z,g), $(g), A,) :p,$ first-order formulas, (Vg)($(g) = 1 (QLx)rp(x,g)) belong to T'} U {(Pz(z),z = x,p;) : I = 1, m}. EXERCISE 6.3 :Suppose T is stable (but not necessarily (*)). Prove there isMEK(p,,..., A,, ...) which is theL-reduct ofM,, IL(M,)(= JL(, M, the Skolem Hull of UIZ:nI,,I,indiscernible &,,I,. [Hint: See VII, §2,5 and use V, 6.10.1 Now we apply the results of V, 96 to analyze the (n+m)-tuples of cardinals for which we get categoricity. EzampZe. Let N = ("2,+ N,P:),<" where : Pr = (7 E "2 : ~(n)= 0}, + is a two-place operation, 7 + Nv = p iff for every n r](n)+ v(n)-p(n) is even. Let M = (WI,PM, QM, EM,+ N,GN, HN, FM, Pf , .. . , Pr , .. . ),