aa r X i v : . [ m a t h . L O ] N ov A nonstructure theorem for countable, stable,unsuperstable theories
Michael C. Laskowski ∗ Department of MathematicsUniversity of MarylandS. Shelah † Department of MathematicsHebrew University of JerusalemDepartment of MathematicsRutgers UniversityNovember 15, 2018
Abstract
A trichotomy theorem for countable, stable, unsuperstable theoriesis offered. We develop the notion of a ‘regular ideal’ of formulas andstudy types that are minimal with respect to such an ideal.
By definition, a stable unsuperstable theory admits a type that is not basedon any finite subset of its domain. From this one sees that such a theoryadmits trees of definable sets. That is, there is a sequence h ϕ n ( x, y ) : n ∈ ω i ∗ Partially supported by NSF grant DMS-0600217 † Partially supported by U.S.-Israel Binational Science Foundation Grant no. 2002323and Israel Science Foundation Grant no. 242/03. Publication no. 871.
1f formulas such that for any cardinal κ there are definable sets { ϕ n ( x, a ν ) : ν ∈ <ω κ } giving rise to κ ℵ partial types { p µ : µ ∈ ω κ } where each p µ forksover { a µ | k : k < n } for all n ∈ ω . In [10] the second author used thesetrees to count the number of uncountable models or to find the maximal sizeof a family of pairwise nonembeddable models of a fixed cardinality of anystable, unsuperstable theory. However, for other combinatorial questions,such as computing the Karp complexity of the class of uncountable modelsof such a theory, the existence of these trees does not seem to be sufficient.Here we prove that when the language is countable, any strictly stable theoryexhibits one of three more detailed nonstructural properties. This trichotomyis used in [7], but it is likely to be used in other contexts as well. Two of thealternatives, the Dimensional Order Property (DOP) or a theory being deepappear in [10] and are compatible with superstability. The third alternativeis new and is captured by the following definition: Definition 1.1 An abelian group witness to unsuperstability is a descendingsequence h A n : n ∈ ω i of abelian groups with [ A n : A n +1 ] infinite for each n such that the intersection A = T n A n is connected and whose generic typeis regular.The existence of such a sequence readily contradicts superstability as forany cardinal κ one immediately obtains a tree { C µ : µ ∈ ω κ } of cosets of A .As well, with Theorems 4.1 and 4.4 we see that one can frequently say moreabout the generic type of A . This added information is used in [7].In order to establish these results, the bulk of the paper discusses thenotion of a regular ideal of formulas (see Definition 2.3). The origins of theseideas date back to Section V.4 of [10] and have been reworked and expandedin [1] and [8].Our notation is standard, and complies with either [8] or [10]. For astable theory T κ r ( T ) denotes the least regular cardinal κ such that there isno forking chain of length κ . Thus, a stable theory is superstable if and only if κ r ( T ) = ℵ and κ r ( T ) = ℵ when T is countable and strictly stable. We calla model ‘a-saturated’ (a-prime) in place of ‘ F aκ r ( T ) -saturated’ ( F aκ r ( T ) -prime). Throughout the whole of this paper we assume ‘T = T eq .’ That is, T is a stable theory in a multi-sorted language, C is a large, saturated modelof T , and the language L is closed under the following operation: If E (¯ x, ¯ y )is a definable equivalence relation then there is a sort U E and a definable2urjection f E : C lg (¯ x ) → U E ( C ) in the language L . In particular, the set ofsorts is closed under finite products. Thus any finite tuple of elements fromvarying sorts can be viewed as an element of the product sort. With thisidentification, every formula can be considered to have a single free variable.As notation, L ( C ) denotes the set of formulas with parameters from C andfor a specific sort s , L s ( C ) denotes the L ( C )-formulas ϕ ( x ) in which the freevariable has sort s . Definition 2.1 An invariant ideal ID is a subset of L ( C ) containing allalgebraic formulas, closed under automorphisms of C , and for any sort s andany ϕ, ψ ∈ L s ( C )1. If ϕ, ψ ∈ ID then ϕ ∨ ψ ∈ ID ; and2. If ϕ ⊢ ψ and ψ ∈ ID , then ϕ ∈ ID .A partial type Γ (i.e., a subset of L s ( C ) for some sort s ) is ID -small if itentails some element of ID ∩ L s ( C ).Many times we will make use of the fact that formulas in ID may have‘hidden’ parameters. Lemma 2.2
Let ID be any invariant ideal.1. A complete type p ∈ S ( A ) is ID -small if and only if p ∩ ID 6 = ∅ .2. For any A and a , stp( a/A ) is ID -small if and only if tp( a/A ) is ID -small.3. If A ⊆ B and tp( a/B ) does not fork over A , then tp( a/A ) is ID -smallif and only if tp( b/A ) is ID -small. Proof. (1) Right to left is immediate. For the converse, assume p entails ψ ∈ ID . By compactness there is ϕ ∈ p such that ϕ ⊢ ψ , hence ϕ ∈ ID .(2) Right to left is clear. If stp( a/A ) entails ψ ( x, b ) ∈ ID , then by com-pactness and the finite equivalence relation theorem there is an A -definableequivalence relation E ( x, y ) with finitely many classes such that tp( a/A ) ∪ E ( x, c ) } ⊢ ψ ( x, b ) for some c . Choose A -automorphisms { σ i : i < n } of C such that { E ( x, σ i ( c )) : i < n } includes all the E -classes. Since ID is aninvariant ideal W i An invariant ideal ID is regular if, for all L ( C )-formulas ψ ( y ) and θ ( x, y ), IF ψ ∈ ID and θ ( x, b ) ∈ ID for every b ∈ ψ ( C ) THEN ∃ y ( ψ ( y ) ∧ θ ( x, y )) ∈ ID .We call a strong type stp( a/A ) ID -internal if there is a set B ⊇ A independent from a over A , a B -definable function f , and elements ¯ c suchthat tp( c/B ) is ID -small for each c ∈ ¯ c and a = f (¯ c ). The strong typestp( a/A ) is ID -analyzable if there is a finite sequence h a i : i ≤ n i fromdcl( Aa ) such that a n = a and stp( a i /A ∪ { a j : j < i } ) is ID -internal for each i ≤ n . Since ID is a collection of formulas, this definition of analyzability isequivalent to the usual one, see e.g., [8].In order to iterate the defining property of a regular ideal, we need thefollowing notion, whose terminology is borrowed from [4]. Definition 2.4 A formula ϕ ( x, c ) is in ID , provably over B if there is some θ ( y ) ∈ tp( c/B ) such that ϕ ( x, c ′ ) ∈ ID for every c ′ realizing θ . Lemma 2.5 For all sets B and every n ∈ ω , if ϕ ( x, y , . . . , y n − ) is B -definable and a, c , . . . , c n − satisfy:1. tp( c i /B ) is ID -small for each i < n ;2. ϕ ( x, c , . . . , c n − ) ∈ ID provably over B ; and3. ϕ ( a, c , . . . , c n − ) then tp( a/B ) is ID -small. roof. Fix any set B . We argue by induction on n . If n = 0 theformula ϕ ( x ) itself witnesses that tp( a/B ) is ID -small. Assume the re-sult holds for n and fix a formula ϕ ( x, c , . . . , c n ) and a, c , . . . , c n as inthe hypotheses. Choose a formula θ ( y , . . . , y n ) ∈ tp( c . . . c n /B ) such that ϕ ( x, c ′ , . . . , c ′ n ) ∈ ID for all c ′ . . . c ′ n realizing θ and, using Lemma 2.2, choose ψ ( y n ) ∈ tp( c n /B ) ∩ ID .Let θ ∗ ( y , . . . , y n ) := θ ( y , . . . , y n ) ∧ ψ ( y n ), θ ′ ( y , . . . , y n − ) := ∃ y n θ ∗ and ϕ ′ ( x, y , . . . , y n − ) := ∃ y n ( ϕ ( x, y , . . . , y n ) ∧ θ ∗ ( y , . . . , y n ))We argue that Conditions (1)–(3) are satisfied by ϕ ′ and a, c , . . . , c n − .Conditions (1) and (3) are clear. We claim that the formula θ ′ witnessesthat ϕ ′ ( x, c , . . . , c n − ) ∈ ID provably over B . Indeed, it is clear that θ ′ ∈ tp( c . . . c n − /B ), so choose c ′ . . . c ′ n − realizing θ ′ . Since ψ ∈ ID and θ ∗ ( c ′ , . . . , c ′ n − , y n ) ⊢ ψ , θ ∗ ( c ′ , . . . , c ′ n − , y n ) ∈ ID . As well, for any c ′ n such that θ ∗ ( c ′ , . . . , c ′ n ) holds, we have θ ( c ′ , . . . , c ′ n ) holding as well, so ϕ ( x, c ′ , . . . , c ′ n ) ∈ ID . Thus ϕ ′ ( x, c ′ , . . . , c ′ n − ) ∈ ID since ID is a regularideal. Proposition 2.6 If stp( a/A ) is ID -internal, then tp( a/A ) is ID -small. Proof. Choose B ⊇ A independent from a over A , a B -definable for-mula ϕ ( x, ¯ y ), and a tuple of elements ¯ c such that each tp( c/B ) is ID -smallfor each c ∈ ¯ c , ϕ ( a, ¯ c ) holds, and ∃ =1 xϕ ( x, ¯ c ). But the formula ϕ ( x, ¯ c ) ∈ ID provably over B via the formula ∃ =1 xϕ ( x, ¯ y ), so tp( a/B ) is ID -small byLemma 2.5. That tp( a/A ) is ID -small follows from Lemma 2.2.The reader is cautioned that while ID -internal types are ID -small, thisresult does not extend to ID -analyzable types. In fact, the theory and typementioned in Remark 8.1.6 of [8] gives rise to an example of this. Much of themotivation of this section, and in particular how it differs from treatmentsin [1] and [8], revolves around how we handle ID -analyzable types that arenot ID -small. Definition 2.7 A strong type p is foreign to ID , written p ⊥ ID , if p ⊥ q for every ID -small q . Lemma 2.8 The following are equivalent for any regular ideal ID and anystrong type p : . p ⊥ ID ;2. p ⊥ q for every ID -internal strong type q ;3. p ⊥ q for every ID -analyzable strong type q ;4. If p = stp( a/A ) then there is no a ′ ∈ dcl( Aa ) such that tp( a ′ /A ) is ID -small. Proof. (1) ⇒ (2) follows immediately from Proposition 2.6. (2) ⇒ (3)follows by induction on the length of the ID -analysis, using the fact that p ⊥ tp( b/B ) and p ⊥ tp( a/Bb ) implies p ⊥ tp( ab/B ). (3) ⇒ (4) is trivial,and (4) ⇒ (1) follows immediately from (say) Corollary 7.4.6 of [8].The reader is cautioned that when the regular ideal is not closed under ID -analyzability, these definitions differ from those in [8]. Definition 2.9 A partial type Γ is ID -large if it is not ID -small. Γ is ID -minimal if it is ID -large, but any forking extension of Γ is ID -small. Γis ID ⊥ -minimal if it is ID -large, but any forking extension Γ ∪ { θ ( x, c ) } is ID -small whenever stp( c/ dom(Γ)) ⊥ ID .Clearly ID -minimality implies ID ⊥ -minimality, but one of the applica-tions in Section 4 will use ID ⊥ -minimal types that are not ID -minimal. Lemma 2.10 Let ID be any regular ideal. If a strong type p is both ID ⊥ -minimal and foreign to ID , then p is regular. Proof. The point is that a counterexample to the regularity of p can befound within the set of realizations of p . If M is a-saturated and p = tp( a/M )is not regular then there are a tuple ¯ c = h c , . . . , c n i realizing p ( n ) for some n and a realization b of p such that tp( a/M ¯ c ) forks over M , tp( b/M ¯ c ) does notfork over M , and tp( b/M ¯ ca ) forks over M ¯ c . Let q = tp( a/M ¯ c ) and choose an L ( M )-formula θ ( x, ¯ c ) ∈ q such that p ∪ { θ ( x, ¯ c ) } forks over M . As p ⊥ ID , p ( n ) ⊥ ID , so the ID ⊥ -minimality of p implies tp( a/M ¯ c ) is ID -small.But, since p is foreign to ID , tp( b/M ¯ c ), which is a nonforking extensionof p would be orthogonal to q by Lemma 2.8(2). In particular, tp( b/M ¯ ca )would not fork over M ¯ c .The following easy ‘transfer result’ will be used in the subsequent sections.6 emma 2.11 Assume that B is algebraically closed, p = tp( a/B ) is foreignto ID , q = tp( b/B ) , and b ∈ acl( Ba ) \ B . Then q is foreign to ID . If,in addition, p is ID -minimal ( ID ⊥ -minimal) then q is ID -minimal ( ID ⊥ -minimal) as well. Proof. If q were not foreign to ID , then by Lemma 2.8(4) there is c ∈ dcl( Bb ) \ B such that tp( c/B ) is ID -small. Since tp( c/B ) is not algebraicit is not orthogonal to p , which, via Lemma 2.8(2), contradicts p being foreignto ID . Thus q ⊥ ID .Next, suppose that p is ID -minimal. Since p q and p ⊥ ID , q cannotbe ID -small. To see that q is ID -minimal, choose C ⊇ B such that tp( b/C )forks over B . Then tp( a/C ) forks over B , so tp( a/C ) is ID -small. Thustp( b/C ) is ID -small by Lemma 2.5. Throughout this section ID always denotes a regular ideal. Definition 3.1 We say A is an ID -subset of B , written A ⊆ ID B , if A ⊆ B and stp( b/A ) ⊥ ID for every finite tuple b from B . When M and N aremodels we write M (cid:22) ID N when both M (cid:22) N and M ⊆ ID N . A set A is ID -full if A ⊆ ID M for some (equivalently for every) a-prime model M over A . Lemma 3.2 Let ID be any regular ideal and assume M is a-saturated.1. If M (cid:22) N are models then M (cid:22) ID N if and only if ϕ ( N ) = ϕ ( M ) forall ϕ ∈ L ( M ) ∩ ID .2. If M ⊆ ID A , then M (cid:22) ID M [ A ] , where M [ A ] is any a-prime modelover M ∪ A . Proof. (1) First suppose M (cid:22) ID N and choose ϕ ∈ L ( M ) ∩ ID . If c ∈ ϕ ( N ) then tp( c/N ) is ID -small. If tp( c/M ) were not algebraic, it wouldbe nonorthogonal to an ID -small type, contradicting tp( c/M ) ⊥ ID . Sotp( c/M ) is algebraic, hence c ∈ ϕ ( M ). Conversely, if there were c ∈ N suchthat tp( c/M ) 6⊥ ID , then by Lemma 2.8(4) there is c ′ ∈ dcl( M c ) \ M suchthat tp( c ′ /M ) is ID -small. Then ϕ ( N ) = ϕ ( M ) for any ϕ ∈ tp( c ′ /M ) ∩ ID .72) Recall that because M is a-saturated, M [ A ] is dominated by A over M .Choose any tuple c from M [ A ]. If tp( c/M ) were not foreign to ID , then as M is a-saturated, there is an ID -small type q ∈ S ( M ) such that tp( c/M ) q ,hence tp( c/M ) is not almost orthogonal to q . Since c is dominated by A over M , there is a from A such that tp( a/M ) is not almost orthogonal to q , whichcontradicts M ⊆ ID A . Definition 3.3 A saturated chain is an elementary chain h M α : α < δ i ofa-saturated models in which M α +1 realizes every complete type over M α foreach α < δ . An ID -chain is a sequence h M α : α < δ i of a-saturated modelssuch that M α (cid:22) ID M β for all α < β < δ and M α +1 realizes every type over M α foreign to ID . A chain (of either kind) is ID -full if the union S α<δ M α is an ID -full set.In general, a saturated chain need not be ID -full. However, if ID iseither the ideal of algebraic formulas or superstable formulas (both of whichare regular), then any a-saturated chain is ID -full, since types are basedon finite sets. A more complete explanation of this is given in the proof ofLemma 4.2. By contrast, the following Lemma demonstrates that ID -chainsare always ID -full. Lemma 3.4 Every ID -chain is full. That is, if h M α : α < δ i is an ID -chain, δ is a nonzero limit ordinal, and M δ is a-prime over S α<δ M α , then M α (cid:22) ID M δ for all α < δ . Proof. By the characterization of M (cid:22) ID N given by Lemma 3.2(1),the first sentence follows from the second. So fix an ID -chain h M α : α < δ i .Let N = S α<δ M α and let M δ be a-prime over N . Fix any α < δ . Since M α ⊆ ID M β for all α < β < δ , M α ⊆ ID N , so M α (cid:22) ID M δ by Lemma 3.2(2). Definition 3.5 A formula θ is weakly ID -minimal (weakly ID ⊥ -minimal) if { θ } is ID -minimal ( ID ⊥ -minimal).We now state offer two complementary propositions. The main point ofboth is that they produce regular types that are ‘close’ to a given regularideal. The advantage of (1) is that one obtains ID -minimality at the cost ofrequiring the chain to be ID -full. In (2) the fullness condition is automati-cally satisfied by Lemma 3.4, but one only gets ID ⊥ -minimality.8 roposition 3.6 Fix a regular ideal ID , a countable, stable theory T , andan ID -large formula ϕ .1. Either there is a weakly ID -minimal formula ψ ⊢ ϕ or for every ID -full saturated chain h M n : n ∈ ω i with ϕ ∈ L ( M ) , there is an ℵ -isolated, ID -minimal p ∈ S ( S n M n ) with ϕ ∈ p and p ⊥ ID .2. Either there is a weakly ID ⊥ -minimal formula ψ ⊢ ϕ or for every ID -chain h M n : n ∈ ω i with ϕ ∈ L ( M ) , there is an ℵ -isolated, ID ⊥ -minimal p ∈ S ( S n M n ) with ϕ ∈ p and p ⊥ ID .Moreover, in either of the two ‘second cases’ the type p is regular. Proof. Assume that there is no weakly ID -minimal ψ ⊢ ϕ . Fix an ID -full saturated chain h M n : n ∈ ω i with ϕ ∈ L ( M ), let N = S n ∈ ω M n ,and let M ω be ℵ -prime over N . Let ∆ ⊆ ∆ ⊆ . . . be finite sets of formulaswith L = S n ∈ ω ∆ n . We inductively construct a sequence h ϕ n : n ∈ ω i of ID -large formulas as follows: Let ϕ be our given ϕ . Given ϕ n ⊢ ϕ that isan ID -large L ( M n )-formula A n = { ψ ∈ L ( M n +1 ) : ψ ⊢ ϕ n , ψ is ID -large and forks over M n } . As M n +1 realizes every type over M n foreign to ID and ϕ n is not weakly ID -minimal, A n is nonempty. Choose ϕ n +1 ∈ A n so as to minimize R ( ψ, ∆ n , { ϕ n : n ∈ ω } . We first argue that Γ has a unique extension to acomplete type in S ( N ). Claim. Γ ⊢ ¬ ψ ( x, b ) for all ψ ( x, b ) ∈ ID ∩ L ( N ). Proof. If the Claim were to fail, then Γ ∪ { ψ ( x, b ) } would be consistent,hence would be realized in M ω , say by an element c . As the chain is ID -full, c ∈ N . Choose n such that b, c ∈ M n . But ϕ n +1 was chosen to fork over M n ,yet is realized in M n , which is impossible.Now let ψ ( x, b ) be any L ( N )-formula. Choose n such that ψ ( x, y ) ∈ ∆ n .As ϕ n +1 was chosen to be of minimal R(–,∆ n , ϕ n +1 ∧ ψ ( x, b ) and ϕ n +1 ∧ ¬ ψ ( x, b ) to be in A n . As ID is an ideal, atleast one of the two of them is ID -large, so is an element of A n , thus theother one is ID -small or inconsistent. Using the Claim, either Γ ⊢ ψ ( x, b ) orΓ ⊢ ¬ ψ ( x, b ). Thus Γ implies a complete type in S ( N ), which we call p .9y construction p is ℵ -isolated and is ID -large by the Claim. Since M ω is ℵ -saturated and p is ℵ -isolated, there is a realization c of p in M ω . If p were not foreign to ID then by Lemma 2.8(4) there would be c ′ ∈ dcl( N c ) \ N with c ′ /N ID -small, directly contradicting ID -fullness.It remains to show that any forking extension of p is ID -small. let θ ( x, a ∗ )be any L ( C )-formula such that p ∪ θ ( x, a ∗ ) forks over M ω . Then for some n , ϕ n +1 ∧ θ ( x, a ∗ ) ∆ n -forks over M n . As M n +1 realizes all types over M n thereis a ′ ∈ M n +1 such that tp( a ′ /M n ) = tp( a ∗ /M n ). But then ϕ n +1 ∧ θ ( x, a ′ )∆ n -forks over M n , contradicting the minimality of R(–,∆ n , 2) rank of ϕ n +1 .As for (2) assume that there is no ID ⊥ -minimal formula implying ϕ .Choose an ID -chain h M n : n ∈ ω i , which is automatically ID -full byLemma 3.4. The definition of { A n } n ∈ ω and the constructions of Γ and p remain the same. All that is affected is that in the final paragraph, aswe only need to establish ID ⊥ -minimality, one chooses a formula θ ( x, a ∗ )with tp( a ∗ /N ) ⊥ ID . By Lemma 2.8(4) this implies tp( a ∗ /M n ) ⊥ ID for all n ∈ ω , so choosing n as above, one obtains a ′ ∈ M n +1 satisfyingtp( a ′ /M n ) = tp( a ∗ /M n ) and a similar contradiction is obtained.In both cases, the regularity of p follows immediately from Lemma 2.10.Recall that a stable theory has NDIDIP if for every elementary chain h M n : n ∈ ω i of models, every type that is nonorthogonal to some a-primemodel over S n ∈ ω M n is nonorthogonal to some M n . Relationships betweenNDIDIP and NDOP are explored in [6]. Proposition 3.7 Fix a countable, stable theory T with NDIDIP and a reg-ular ideal ID such that the formula ‘ x = x ’ 6∈ ID .1. If there is an an ID -full, saturated chain h M n : n ∈ ω i , but there is noweakly ID -minimal formula then there is an abelian group witness tounsuperstability, where in addition the generic type of the intersectionis both ID -minimal and foreign to ID .2. If there is no weakly ID ⊥ -minimal formula then there is an abeliangroup witness to unsuperstability where the generic type of the intersec-tion is ID ⊥ -minimal and foreign to ID . Proof. (1) Fix an ID -full, saturated chain h M n : n ∈ ω i and let N = S n ∈ ω M n . Using Proposition 3.6(1) choose p ∈ S ( N ) to be ℵ -isolated,10oreign to ID , and ID -minimal, hence regular. Since T has NDIDIP, p M n .Since p is regular and M n is a-saturated, by Claim X 1.4 of [10] there is aregular type r ∈ S ( M n ) nonorthogonal to p . Let r denote the nonforkingextension of r to N . As p and r are nonorthogonal there is an integer m such that p ( m ) is not almost orthogonal to r ( ω ) . Since p is ℵ -isolated and M n is a-saturated, N a is dominated by N over M n for any a realizing p . Thus p (1) is not almost orthogonal to r ( ω ) over N . Choose k ≥ p ( k ) is almost orthogonal to r ( ω ) over N and choose ¯ c realizing p ( k ) . Let B = acl( N ¯ c ) and choose a realization a of the nonforking extension of p to B . By Theorem 1 of [2], there is b ∈ dcl( Ba ) \ B and a type-definable,connected group A with a regular generic type q (so A is abelian by Poizat’stheorem [9]) and a definable regular, transitive action of A on p ( C ), where p = tp( b/B ). By Lemma 2.11 the type p and hence q are both foreign to ID and ID -minimal. By Theorem 2 of [3] there is a definable supergroup A ⊇ A . By an easy compactness argument we may assume A is abelianas well. Furthermore, by iterating Theorem 2 of [3] we obtain a descendingsequence h A n : n ∈ ω i of subgroups of A with A = T n ∈ ω A n .Thus far we have not guaranteed that A n +1 has infinite index in A n . Inorder to show that there is a subsequence of the A n ’s with this propertyand thereby complete the proof of the Proposition, it suffices to prove thefollowing claim: Claim For every n ∈ ω there is m ≥ n such that [ A n : A m ] is infinite. Proof. By symmetry it suffices to show this for n = 0. Assume thatthis were not the case, i.e., that [ A , A m ] is finite for each m . Then A hasbounded index in A . We will obtain a contradiction by showing that thedefinable set A is weakly ID -minimal. First, since q is ID -large, the formuladefining A is ID -large as well. Let ϕ ( x, e ) be any forking extension of theformula defining A and let E ⊆ A be the set of realizations of ϕ ( x, e ). Let { C i : i < κ ≤ ℵ } enumerate the A -cosets of A . For each i , E ∩ C i isa forking extension of C i . Since every C i is a translate of A whose generictype is ID -minimal, this implies that E ∩ C i is ID -small for each i . Thus ϕ ( x, e ) ∈ ID by compactness (and the fact that ID is an ideal). Thus, theformula defining A is weakly ID -minimal, contradiction.The proof of (2) is identical, choosing an ID -chain satisfying the hypothe-ses and using Proposition 3.6(2) in place of 3.6(1).11 Applications Our first application gives a ‘trichotomy’ for strictly stable theories in acountable language. It uses the ideal of superstable formulas. Let R ∞ denotethe ideal of Definition 4.1 R ∞ denotes the ideal of superstable formulas (i.e., all for-mulas ϕ with R ∞ ( ϕ ) < ∞ ).Equivalently, ϕ ∈ R ∞ if and only if for all cardinals κ ≥ | T | , for anymodel M of size κ containing the parameters of ϕ , there are at most κ complete types over M extending ϕ . Lemma 4.2 R ∞ is a regular ideal, any elementary chain h M n : n ∈ ω i of a-saturated models is R ∞ -full, and there are no weakly R ∞ -minimal formulas. Proof. Invariance under automorphisms of C is clear and R ∞ being anideal follows by counting types. To show regularity, choose ψ ( y ) ∈ R ∞ and θ ( x, y ) ∈ L ( C ) such that θ ( x, b ) ∈ R ∞ for every b realizing ψ . Choose κ ≥ | T | and a model M of size κ containing the hidden parameters of both ψ and θ .Then there are at most κ types p ( x, y ) ∈ S ( M ) extending θ ( x, y ) ∧ ψ ( y ), sothe projection ∃ y ( θ ( x, y ) ∧ ψ ( y )) ∈ R ∞ as only κ types q ( x ) ∈ S ( M ) extendit. To establish fullness, fix an elementary chain h M n : n ∈ ω i of a-saturatedmodels. Let N = S n ∈ ω M n and choose an a-prime model M ω over N . Becauseof Lemma 2.8(4), in order to show that N ⊆ ID M ω it suffices to show that noelement of c ∈ M ω \ N is R ∞ -small. So choose c ∈ M ω such that tp( c/N ) is ID -small and we will show that c ∈ N . On one hand, since tp( c/N ) containsa superstable formula there is a finite n such that tp( c/N ) is based on M n .On the other hand, since M ω is a-prime over N , tp( c/N ) is a-isolated. Thustp( c/M n ) is a-isolated as well (see e.g., Theorem IV 4.3(1) of [10]). Since M n is a-saturated, this implies c ∈ M n ⊆ N .To show that there are no weakly R ∞ -minimal formulas, suppose that aformula ϕ has the property that any forking extension of ϕ is R ∞ -small. Wewill show that ϕ ∈ R ∞ by counting types. Fix a cardinal κ ≥ | T | and amodel M of size κ containing the parameters of ϕ . Let M (cid:22) M have size | T | that also contains the parameters containing ϕ . It suffices to show that every p ∈ S ( M ) extending ϕ has at most κ extensions to types in S ( M ). Clearly,12here is a unique nonforking extension of p and any forking extension of p contains an L ( M )-formula witnessing the forking. Each such forking formula ψ ∈ R ∞ , so there are at most κ q ∈ S ( M ) extending ψ . So, since the totalnumber of ψ ∈ L ( M ) is at most κ , p has at most κ extensions to types in S ( M ). Theorem 4.3 Let T be a stable, unsuperstable theory in a countable lan-guage. Then at least one of the following three conditions occurs:1. T has the dimensional order property (DOP); or2. T has NDOP, but is deep (i.e., there is a sequence h M n : n ∈ ω i suchthat tp( M n +1 /M n ) ⊥ M n − for all n ≥ ); or3. There is an abelian group witness to unsuperstability (see Definition 1.1)in which the generic type of the intersection is both R ∞ -minimal andforeign to R ∞ . Proof. To begin, Corollary 1.12 of [6] asserts that any such theory T hasNDIDIP. Since T is not superstable the formula ‘ x = x ’ R ∞ . As well, byLemma 4.2 there are no weakly R ∞ -minimal formulas, so Proposition 3.7(1)asserts that an abelian group witness to unsuperstability exists, whose generictype is regular and both R ∞ -minimal and foreign to R ∞ .Our second application comes from an attempt to solve the ‘Main Gapfor ℵ -saturated models.’ As in the previous theorem, the relevant settingis where a countable theory T is stable, unsuperstable, with NDOP, and isshallow. The main open question is whether, for such a theory every nonal-gebraic type r is nonorthogonal to a regular type. The following result shedssome light on this issue. In order to analyze this problem, fix a nonalgebraic,stationary type r over the empty set. Let ID r = { ϕ ∈ L ( C ) : r ⊥ ϕ } Verifying that ID r is an invariant ideal is straightforward. To see that it isa regular ideal, fix L ( C )-formulas ψ ( y ) ∈ ID r and θ ( x, y ) such that θ ( x, b ) ∈ID r for every b realizing ψ . Choose an a-saturated model M containingthe parameters of ψ and θ , pick a realization c of the nonforking extensionof r to M , and let M [ c ] be any a-prime model over M c . To show that13 ( x ) := ∃ y ( θ ( x, y ) ∧ ψ ( y )) ⊥ r it suffices to prove that any realization of ϕ in M [ c ] is contained in M . So choose any a ∈ ϕ ( M [ c ]). Choose b ∈ M [ c ]such that θ ( a, b ) ∧ ψ ( b ) holds. Since r ⊥ ψ , b ∈ M . But then θ ( x, b ) is over M and is ⊥ r , so a ∈ M as well. Thus ID r is a regular ideal. Theorem 4.4 Assume that a countable theory T is stable, unsuperstable, hasNDOP, and is shallow. If a nonalgebraic, stationary type r is orthogonal toevery regular type, then there is an abelian group witness to unsuperstabilityin which the generic type of the intersection A = S n A n is both ( ID r ) ⊥ -minimal and foreign to ID r . Proof. Fix such a type r . By naming constants we may assume that r is over the empty set. Note that any formula ϕ ∈ r is not an element of ID r , so ‘ x = x ’ 6∈ ID r . Claim. There is no weakly ( ID r ) ⊥ -minimal formula. Proof. Assume that ϕ were ( ID r ) ⊥ -minimal. We construct a regulartype p r as follows: Choose an a-saturated model M containing the pa-rameters in ϕ , pick a realization c of the nonforking extension of r to M , andchoose an a-prime model M [ c ] over M c . Since ϕ is ID r -large we can find an a ∈ M [ c ] \ M realizing ϕ . Choose such an a and let p = tp( a/M ). Clearly, p r . To see that p is regular, first note that p is ( ID r )-minimal since p is ID r -large and extends ϕ . As well, p is foreign to ID r , since if it were not,then by Lemma 2.8(4) there would be b ∈ dcl( M a ) with tp( b/M ) ID r -small.But then tp ( c/M b ) would fork over M , implying that r is nonorthogonalto an ID r -small type, which is a contradiction. So p is ( ID r )-minimal andforeign to ID r , hence is regular by Lemma 2.10.The theorem now follows immediately from Proposition 3.6(2). References [1] E. Hrushovski, Kueker’s conjecture for stable theories, J. Symbolic Logic (1989), no. 1, 207–220.[2] E. Hrushovski, Almost orthogonal regular types, in Stability in modeltheory, II (Trento, 1987) Ann. Pure Appl. Logic (1989), no. 2, 139–155. 143] E. Hrushovski, Unidimensional theories are superstable, Ann. PureAppl. Logic (1990), no. 2, 117–138.[4] E. Hrushovski and S. Shelah, A dichotomy theorem for regular types, inStability in model theory, II (Trento, 1987). Ann. Pure Appl. Logic (1989), no. 2, 157–169.[5] M.C. Laskowski, Descriptive set theory and uncountable model theory, Logic Colloquium ’03 , 133–145, Lect. Notes Logic, , Assoc. Symbol.Logic, La Jolla, CA, 2006.[6] M.C. Laskowski and S. Shelah, Decompositions of saturated models ofstable theories, Fund. Math. (2006), no. 2, 95–124.[7] M.C. Laskowski and S. Shelah, The Karp complexity of stable theories,in preparation.[8] A. Pillay, Geometric Stability Theory , Oxford University Press, 1996.[9] B. Poizat, Groupes stables, avec types g´en´eriques r´eguliers (French) [Sta-ble groups with regular generic types]