Arch. Math. Logic (1992) 31 : 163-169 Archive for

Mathematical Logic

�9 Springer-Verlag 1992

Countable models of 1-based theories

Anand Pillay*

Departments of Mathematics, Notre Dame University, Notre Dame, IN 46556, USA

Received February 13, 1991/in revised form July 23, 1991

In this paper T denotes a countable complete stable theory. We prove

Theorem 1. Suppose T is 1-based and there is a nonisolated regular type p ~ S(a) which is orthogonal to 0 and also orthogonal to any forking extension of a conjugate of itself. Then T has 2 ̀0 countable models.

Theorem 2. Suppose T is trivial, 1-based, and unsuperstable. Then T satisfies the hypothesis of Theorem 1.

In [LP] it is proved that a countable superstable non co-stable theory which does not interpret an infinite group has 2`0 countable models. So we conclude:

Corollary 3. I f T is trivial, 1-based and not co-stable then T has 2 ~ countable models.

Note that this yields Vaught's conjecture for trivial 1-based theories, as the co-stable case is dealt with by ESHM].

1 Introduction

There are several notions of geometric simplicity of a stable theory. One is the notion of 1-based or weakly normal. T is said to be 1-based if for any a and A, Cb(stp(a/A)) c__ acl(a). Another is triviality, where T is said to be trivial if any set which is pairwise independent is independent. There exist 1-based theories which are not trivial (for example modules) and trivial theories which are not 1-based (for example the "free pseudoplane" (see [P2]), or the trivial, non totally trivial theories of [BH] and rPo]). The class of stable theories which are both trivial and 1-based encompasses the "naive" nonalgebraic examples of stable theories, but also some less naive examples such as the pairing function without cycles (see [-BoPo]).

Here we prove Vaught's conjecture for theories in this class. In [B] an example of Shelah is given of an co-stable theory with ENI-depth > 1 but with only

* Supported by NSF grant DMS90-06628

164 A. PiUay

countably many countable models. Theorem I says, among other things, that this cannot happen in the 1-based situation.

Thanks are due to Bernhard Herwig for comments on an earlier version of this paper.

For basic facts on stability theory which are assumed in this paper, we refer the reader to [P3] and [La]. For the machinery of internality and analysability, used in Sect. 4, the reader is referred to [Hr2].

2 Preliminaries on small 1-based theories

We show that, under the assumptions of small and 1-based, there are many regular types and that every type has finite weight. The first remarks repeat some observations in [Hrl] .

Lemma 2.1. Assume T is small and 1-based, and p e S(A). Then there is no dense chain of forking extensions of p. Namely there do not exist complete types p~ for i ~ Q such that Po = P and i<j implies pj is a forking extension of p~.

Proof Suppose otherwise. Let p~ e S(A~). For r e ~ let Pr = u{p~: i < r}. So r < s implies Ps is a forking extension ofpr and thus for some finite A, rkA(Ps ) < rka(pr ). As T is 1-based, for each r there are a, and b~ such that stp(a,/br) is parallel to p, and so for all finite A rkA(pr ) = rkn(tp(aJbr)). Thus the a~b, realise different types over 0, contradicting countability of S(T). []

Corollary 2.2. ( T small, 1-based.) For any a and A there is no infinite A-independent set B such that a forks with b over A for all b in B.

Proof Without loss B = {b~" i e Q}. Let B~ = {b, : t < i}. Then if the corollary fails we have Pi = tp(a/AB~) a dense forking chain. []

We now define a notion of dimension for types, similar to those notions studied by Ziegler [Z] and Prest EPr] for pp-types in modules. By a type I will mean a stationary complete type over some set in a big model. (Note that the following definition makes sense in an arbitrary stable theory.)

Definition2.3. a) Let plCp2 be types. We define dim(pl/P2)>~ for ~ > - 1 , inductively

i) dim(pl/p2)= - 1 iff P2 is a nonforking extension of Pl- ii) dim(pl/p2)> ~+ 1 if there are ri for i<co with Pl CriCri+l CP2 for all i, and

dim(rJri+l)>~ for all i. iii) for limit 6, dim(pl/p2)>6 if dim(pl/P2)>=~ for all ~<6. iv) By dim(p) we mean dim(p/p') where p' is an algebraic extension of p over a

very saturated model. b) For ~ > 0, we define U,(p) > fl for fl > 0 inductively. i) Always U~(p) => 0.

ii) U~(p)>fl+ 1 if there is r3p with dim(p/r)>~ and U~(r)>fl. Note that dim(p)= dim(pl) if Pl is a nonforking extension of p.

Corollary 2.4. (T small, 1-based). For any p, dim(p) < ~ .

Proof This follows from Lemma 2.1. []

Remark 2.5. i) dim(p) = 0 iff p is nonalgebraic and has U-rank. Moreover in this case Uo(p)= U(p).

Countable models of l-based theories 165

ii) U~(p)=0 iffdim(p)<~. U~(p)=oo iff dim(p)>a. iii) If p' is a nonforking extension of p then U~(p) = U~(p'). The following lemma is left to the reader.

Lemma 2.6. For any o~ we have i) if b ~ acl(Aa) then U~(tp(ab/A)) = U~(tp(a/A)).

ii) U~(tp(a/bA)) + U~(tp(b/A)) < U~(tp(ab/A)).

Proposition 2.7. (T small and 1-based.) Let tp(a/A) be nonalgebraic. Then there is b E acl(aA) such that stp(b/A) is regular.

Proof. Without loss A = 0. Let b ~ act(a) be nonalgebraic such that dim(tp(b)) = a is minimal, and also U~(tp(b))= fl is minimal.

Claim. Let tp(b/B) be a forking extension of tp(b). Then U~(tp(b/B))< ft.

Proof of claim. Otherwise we have equality. Let e ~ Cb(tp(b/B)) be nonalgebraic. So c ~ acl(b) (by 1-basedness). Moreover U~(tp(b/c))= ft. By 2.6i) and ii) we thus have

U~(tp(b/c)) + U~(tp(c)) <_<_ U~(tp(bc)) = ft.

This forces U~(tp(c))=O. By 2.5ii) dim(tp(c))< a. As c ~ act(a), this contradicts the least choice of a, and proves the claim. []

Now if tp(b) were not regular, there would be a set B independent from b, and some c realising a forking extension over B of tp(b) such that b and c fork over B. Let dECb(b/eB) be nonalgebraic. So d~acl(b)c~acl(cB). By the claim, U~(tp(c/B)) < ft. Thus by 2.6i) U~(tp(d/B))< ft. On the other hand, b is independent from B and thus also d is independent from B. By 2.5iii), U~(tp(d)) < ft. As d ~ acl(a), this again contradicts minimality in choice of a. This proves the proposition. []

Proposition 2.8. ( T small and 1-based.) Any type p has finite weight, and is in fact domination equivalent to a finite product of regular types.

Proof. Without loss p is over an a-model (i.e. an Nl-saturated model in the unsuperstable case) M. Let a realise p and let N be the model a-prime over M and a. Let B be a maximal set of elements of N which realise regular types over M and are M-independent. By Corollary 2.2, B is finite. We claim that B dominates a over M. If not let c be such that c is independent fl~om B over M, but forks with a over M. By l-basedness there is some d Eacl(Ma)nacl(Mc), d r M . By 2.7 we may assume tp(d/M) is regular. Thus d ~ N and d is independent from B over M, contradicting maximality of B. Thus a and B are domination equivalent over M, proving the proposition. []

We finally in this section point out a crucial fact about small 1-based theories, which has some connection with our method of proof in [P1 ] that a 1-based non co-categorical theory has infinitely many countable models. We say that b is semi- isolated over a if there is a formula (p(x, y) over 0 such that ~ q~(a, b) and ~ (p(a, y) ~p(y) where p = tp(b).

CB will denote Cantor-Bendixson rank on types. We write CB(a) for CB(tp(a)). Note that as T is small, any type (over 0 or over a finite set) has CB-rank < m. We recall

Fact 2.9. i) If b C act(a) then CB(ab) = CB(a). ii) If CB(ab)= CB(a) then tp(b/a) is isolated.

166 A. Pillay

Proposition 2.10. Let tp(a) = tp(b) and suppose that b is semi-isolated over a. Then a is semi-isolated over b.

Proof. By induction on CB(a). If it is 0 this is immediate. So suppose CB(a) = a > O. Let A=Cb(stp(b/a)). As b is semi-isolated over act(a), by the Open mapping theorem, b is semi-isolated over A. Let c ~ A be such that

i) b is semi-isolated over c. As ii) c ~ acl(a)c~acl(b), we have by 2.9 that iii) CB(ac) = CB(bc) = ~. Now if CB(c) = a then by iii) and 2.9 tp(a/c) is isolated. As c ~ acl(b), a is semi-

isolated over b. Otherwise CB(c)< ct. Choose d such that tp(ad)= tp(bc). So by i) we have that iv) a is semi-isolated over d. In particular, as c ~ acl(a), c is semi-isolated over d. By the induction hypothesis,

d is semi-isolated over c. As c ~ acl(b), we conclude using iv) that a is semi-isolated over b. []

3 Proof of Theorem 1

So let T and p e S(a) satisfy the assumptions of Theorem 1. We want to show that T has 2 ~ countable models. So we may assume T to be small, and thus can use the results of Sect. 2. We may clearly assume p to be stationary.

Lemma 3.1. There is p'e S(ab) a nonforking extension of p, such that tp(a/b) is stationary with weight 1, and p' is orthogonal to b.

Proof. First as in Lemma 1.6 of [SHM], we can find a-models M and N such that M < N, a ~ N, w(N/M) = 1 and p is orthogonal to M. (This makes use of 2.8 above). As T is 1-based and M is an a-model, there is b E M such that tp(a/M) is definable over b. So tp(a/b) has weight 1. Let p' be the nonforking extension ofp to ab. As p is orthogonal to M, p' is orthogonal to b. []

We may add b to the language and we let p now denote the p' of Lemma 3.1. So we have p ~ S(a) is regular nonisolated, orthogonal to 0 and also to any forking extension of a conjugate of p. Also tp(a)~ q is stationary of weight I.

Lemma 3.2. I f tp(a) is nontrivial then T has 2 ̀0 countable models.

Proof. This is the standard construction. Let G be a countable disjoint union of finite graphs G~, for i ~ I. Let r be the type of a fixed triple of elements realising q which are dependent, but pairwise independent. Let A = wAi be an independent set ofrealisations of q, corresponding to the vertices of G. For b, c ~ Ai which are joined by an edge of G (i.e. of Gi) let dbc be such that bcdbc realises r and such that Aw{dbc: b, c joined by an edge} is as independent as possible. Let D be this set, and with respect to a fixed enumeration of D let Mo be a countable model which is "almost atomic" over D, namely for every tuple e from Mo, tp(e/D') is isolated for all sufficiently large initial segments D' of D. (This is possible as T is small. See [K]). For d realising q let Pd denote the copy of p over d.

Claim 1. For every d ~ D, dim(p d, Mo) is finite.

Proof. Write D i for Aiu{db~ :b, c ~ Ai}. So D~ is finite. Let de D i, and suppose that dim(pd, Mo) is infinite. As tp(DJd) has finite weight (by 2.8), there is c ~ Mo realising

Countable models of 1-based theories 167

Pd such that c is independent from D~ over d. So tp(c/Di) is nonisolated and orthogonal to 0. But tp(e/E) is isolated for some finite E C D which we may assume contains D~. As D~ and E - D i are independent, the orthogonality considerations imply that tp(c/D3 is isolated, a contradiction. []

For every b e Mo such that Pb is orthogonal to Pa for all d e D, let Ib be a countable Morley sequence in pblM o. Moreover assume the set of I b is independent over M 0. Let M1 be a countable model containing M o u I (where I = u/b) such that for every c e M1, c forks with I over M 0.

Claim 2. For every dED, Pa is not realised in M 1 - M o .

Proof. If c ~ M 1 - Mo realises Pd, then either tp(c/Mo) does not fork over d, or it does fork over d. The first case is ruled out by the fact that Pd is weakly orthogo- nal to tp(I/Mo) and the second by the fact that p [and thus tp(I/Mo)] is weakly orthogonal to any forking extension of a conjugate of p. []

By Claim 2 and a union of chain argument, we build a countable model Ma containing D such that for all d~D, dim(pd, M~) is finite, and for any b~M~ realising q such that Pb is orthogonal to Pa for all d ~ D, dim(pb, MG) is infinite. As in [HM-I (Lemma 2.2), or [B] (XVI.3.1), for suitable G one can recover G from the isomorphism type of M G. Thus T has 2 ̀o countable models. []

Lemma 3.3. I f q is trivial then T has 2`o countable models.

Proof Here the 1-based assumption is used rather more strongly. Let A = {a(i):i < co} be an independent set of realisations of q. For each i let Bi be a Morley sequence of length i in Pa~i), such that tp(BJA)u {Bj :j 4= i} does not fork over a(i). For X C co let Mx be a countable model which is almost atomic over AxuBx where A x = {a(i): i ~ X} and B x = u {B~: i ~ X}. We will prove

Claim. If X - {0} 4= Y - {0} then m x and My are not isomorphic.

Proof Let k e Y - X, k 4= 0. Suppose f is an isomorphism between M r and Mx. We obtain a contradiction. Let dC denote f(a(k)Bk). So C is a Morley sequence in Mx in Pa of length k, and Moreover tp(dC) is not isolated. If d is independent from Ax, then the fact that Pa is orthogonal to 0 implies that dC is independent from AxuBx. So tp(dC/E) is nonisolated for any finite subset IEI of AxuBx, which is impos- sible. Thus d forks with Ax. By triviality of the weight I type q and independence of Ax there is unique a = a(i)~ Ax such that d forks with a. Let B denote B i. As above we have

i) d is independent from A x - {a}, and thus dC is independent from (AxuBx) - a B (over 0).

Let E C A x u B x be finite such that tp(dC/E) is isolated, and we may assume that E contains aB. Let E'= E - a B . So by i) dC is independent from E'.

Subelaim. k < i.

Proof of subclaim. Otherwise k > i. Let C' be a (proper) subsequence of C of length i. So tp(dC') = tp(aB). As this type is stationary and each of dC, aB is independent from E', we see that

ii) tp(dC'/E')=tp(aB/E'). As tp(dC'/E'aB) is isolated, by 2.10 and ii), also iii) tp(aB/E'dC') is isolated.

168 A. Pillay

[We use here the easy fact that if tp(x/y) is isolated and y is semi-isolated over x then tp(y/x) is isolated.]

Putting iii) together with the fact that tp(dC/E'aB) is isolated yields iv) tp(dC/dC'E') is isolated.

However, for c ~ C - C', tp(c/dC') is nonisolated, and moreover dC is independ- ent from E'. This is in contradiction with iv) and so proves the subclaim. []

To conclude the proof we carry out the same argument in Mr. Let a'B' be f- l(aB). So as a forks with d, a' forks with a(k). Again we conclude that a'B' is independent from (AruBr)-a(k)Bk over 0, and as in the subclaim we conclude that i < k. So i = k after all. This proves the Claim. []

Clearly T has 2 ~ countable models. []

4 Proof of Theorem 2

So T is trivial, 1-based and nonsuperstable. We seek a regular type with the stated properties. (Our result is related to a result of Tsuboi IT], which states that if T is unsuperstable and there is no infinite definable group in a model of T, then there is a regular type orthogonal to 0. Note Tsuboi's hypotheses are weaker than ours.) Let H be the class of superstable definable sets, namely definable sets with ordinal valued D-rank. We assume familiarity with the notions of internality and analysability from [Hr2]. In any case, we use the following facts:

i) tp(a/A) is/-/-internal iff R~(a/A)< ~ , and ii) a is H-analysable over A [or equivalently tp(a/A) is H-analysable] iffthere is

some n<co and ao, . . . ,a,=a such that tp(ai/ai-lA) is/-/-internal for each i<n. Moreover such ai can be found in dcl(aA).

iii) From i) and ii) it follows that if a is H-analysable over A, then there is some finite A1 CA such that a is H-analysable over A r

Now as T is unsuperstable there is some element a such that

U(tp(a)) = ~ . Let A 0 = {b e acl(a): U(tp(b)) < ~ } .

Claim I. a is not H-analysable over A o.

Proof. Otherwise there are ao,..., a, = a with R~(a,/ai_ 1Ao)< ~ for all i. There is clearly some finite BCAo such that R~(aJa,_~B)<oo for all i. But then U(tp(B))<o% and U(aJai_lB)<~ for all i. By subaddivity of U-rank, U(tp(a)) < ~ , a contradiction. []

Now, by iii) let tp(a/A) be an extension of tp(a/Ao), such that tp(a/A) is not H-analysable but every forking extension is. (This can easily be accomplished using ii).)

Claim II. tp(a/A) is orthogonal to every H-analysable type.

Proof. If not, there is a nonforking extension tp(a/B) of tp(a/A) and some b such that tp(b/B) is H-analysable and a and b fork over B. So tp(a/bB) is H-analysable. Thus tp(a/B) is H-analysable. By Remark 7(c) of [Hr2], tp(a/A) is H-analy- sable, a contradiction. []

Claim III. tp(a/ A) orthogonal to every forking extension of a conjugate of itself (and so is regular), and moreover has ordinal valued U-rank.

Countable models of 1-based theories 169

Proof The first part follows from Claim II and the choice of tp(a/A). Now any forking extension oftp(a/A) is//-analysable, and thus as in the proof of Claim I has U-rank < oo. So U(tp(a/A)) < oo. []

Claim IV. tp(a/A) is orthogonal to O.

Proof Let C= Cb(stp(a/A) and note that AoCacl(C ) [as Ao C acl(a)], and by 1-basedness also C C acl(a). If the claim fails, then by triviality of T there is some b such that tp(b/C) does not fork over 0 and a and b fork over C. Let c ~ Cb(tp(b/aC))

- acI(C). As c e acl(aC), by Claim III, U(tp(c/C)) < or. On the other hand c e acl(b), whereby tp(c/C) does not fork over 0. Thus c ~ acl(a), and U(tp(c)) < oo. But then c e Ao (acl(C), a contradiction. []

Now by l-basedness let d be a finite tuple such that stp(a/A) is parallel to stp(a/d).

Claim V. tp(a/d) is not isolated.

Proof By Claim III, U(tp(a/d))<oo. On the other hand by Claim II, R~(tp(a/d)) = oo. So clearly tp(a/d) can not be isolated. []

Claims III, IV, and V prove Theorem 2.

Postscript. P. Tanovic has improved our Theorem 1 to: if T is 1-based and there is a nonisolated p e S(a) orthogonal to 0, then T has 2 ~' countable models.

References

[B]

[BH] [BoPo]

[HM]

[Hrl] [Hr2]

[K] [La] [LP] [Pi j

[P2]

[P3] [Po] [Pr] [SHM]

[T]

[Z]

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