The Morley rank of a Banach space 1. introduction - CiteSeerX
The Morley rank of a Banach space Jose Iovino July 2, 1996 Abstract. We introduce the concepts of Morley rank and Morley degree for structures based on Banach spaces. We characterize !-stability in terms of Morley rank, and prove the existence of prime models for !-stable theories.
1. introduction
A general framework for the model theoretical analysis of structures based on Banach spaces was introduced in the monograph [5]. It was shown there that the space of types of a Banach space theory is naturally endowed with various uniform topologies on it. A general notion of a uniform structure on the space of types of a complete theory was introduced. In [6], we introduced the concept of stable Banach space theory: Let T be a complete Banach space theory, let be an in nite cardinal, and let U be a uniform structure on the space of types of T. The theory T is -stable with respect to U if for every model E of T of density character less than or equal to , the density character of the space of types over E with respect to the uniform topology of U is also less than or equal to . In particular, T is !-stable with respect to U if the space of types over every separable model of T is separable with respect to U. It was shown in [4] that signi cant classes of Banach space structures (e.g., the spaces Lp , for 1 p < 1) are !-stable with respect to natural, metrizable uniform structures. We also showed in [6] that when the uniform structure is metrizable, !-stability implies -stability for any in nite cardinal , and if T has quanti er elimination, stability is equivalent to !-stability. Here, we investigate !-stability further. We de ne a Banach space theoretic Morley rank (more precisely, a family of Morley ranks), and characterize !-stability in terms of it. From this characterization we prove the existence of prime models for theories which are !-stable with respect to the metric d on the space of types introduced in [5]. The shall assume that the reader is familiar with the basic machinery developed in [5] for the model theoretical analysis of Banach space structures, including the concept of uniform structure on the space of types. However, we shall reprint the de nition here for reference. We extend to this paper the general assumptions and notational conventions of [6]. We deal with Banach space structures, positive bounded formulas, and 1
approximate satisfaction j=A. The terms \formula", \theory" and \type" are used as abbreviations of \positive bounded formula", \positive bounded theory" and \positive bounded type", respectively. If and are positive bounded formulas, < means that is an approximation of . The letter T denotes a xed complete positive bounded theory over a countable language L. All the models considered are models of T. We assume that these models are approximately elementary submodels of some large, saturated model E. If a 2 E, kak is an abbreviation of max1in kaik. By the @0 -saturation of E, we have E j=A (a) if and only if E j= (a), for every positive bounded formula and every a 2 E. We write simply j= (a), omitting E. We consider only complete, positive bounded types which are consistent with T. The norm of an n-type p, denoted kpk, is the norm of any n-tuple realizing p. If A is a subset of a model E, we denote by L(A) the result of expanding the language L with constants and appropriate norm bounds for the elements of A, and T(A) is the theory of E in L(A). The set of n-types over A is denoted Sn (A), and S S(A) = n
2. separating pairs of formulas
Let S(T) = AE S(A). The models of T come equipped with a topology, namely, the norm topology. This topology can be transfered onto S(T) as follows. If p(x); q(x) 2 Sn (A), we let d(p; q) = inf f kb ? ck : E j= p(b); q(c) g: Then, d is a metric on Sn (A). We extend d to all of S(T) by letting d(p; q) = 1 when p and q are types in di erent sets of variables or over di erent sets of parameters. For each rational > 0, let U = f (p; q) : d(p; q) g. The family U = f U : 2 Q+ g is an example of a uniform structure on S(T): definition. A uniform structure on on the space of types of T is a family U of subsets of S(T) S(T), called vicinities , such that (1) U is a base for a Hausdor uniform structure (in the standard topological sense. See, for example, [7, Chapter 6]) on S(T). (2) For every vicinity U there exists a set of pairs of formulas D(U) such that (i) If ( ; 0) 2 D(U), then < 0. (ii) D(U) de nes U in the following sense: The pair (p(x); q(x)) is in U \ S(A) S(A) if and only if for every pair ( (x; y); 0(x; y)) 2 D(U) and every a 2 QA (= rational multiples of A), (x; a) 2 p implies 0 (x; a) 2 q; (x; a) 2 q implies 0 (x; a) 2 p: 2
(iii) There exists a vicinity V with the following property: If ( ; 0) 2 D(U) and 00 > 0, there exists ( ; 0 ) 2 D(V ) such that < and 0 < 00 . In the case of the metric d de ned above, we let D(U ) be the set of pairs ( ; 0) such that (x; y) is of the form
8z(kzk ! (x + z; y)); for > , and 0(x; y) is of the form
8z(kzk ? ! 0 (x + z; y)); with 0 > . Another important example of a metrizable uniform structure on the space of types of T is given by the Banach-Mazur metric on types: For 0, let U be the set of pairs of types (p; q) such that there is a 1 + -automorphism E mapping p onto q (a linear isomorphism f is a 1 + -isomorphism if kf k; kf ?1 k 1 + ). The family f U : 2 Q+ g is a metrizable uniform structure on the space of types of T. No theory can be !-stable with respect to this uniform structure. However, if 2@0 , -stability with respect to the metric d implies -stability with respect to the Banach-Mazur metric. See [6] for the details, and for a discussion of further natural uniform structures on the space of types. Hereafter, U will denote a xed uniform structure on the set of types of T, and all the vicinities mentioned are vicinities of U. A notion that recurrently underlies the main arguments of [6] is that of a separating pair of formulas.
definition. Let U be a vicinity of U. A pair of positive bounded formulas
f 1 (x); 2 (x) g is U -separating if (p(x); q(x)) 2= U whenever x) 2 q(x). 2 (
1
(x) 2 p(x) and
If is a set of positive bounded formulas, + denotes the set
f 0 : 0 > for some 2 g: If is a positive bounded formula, we write + instead of f g+. remarks.
(1) If (p(x); q(x)) 2= U, there exists a U-separating pair f 1(x); 2(x) g such that 1(x) 2 p(x) and 2(x) 2 q(x). (2) Let V correspond to U as in (2-iii) of the de nition of uniform structure. Then, if (p(x); q(x)) 2= U, there exists a V -separating pair f 1(x); 2(x) g such that 1 (x) 2 p+ (x) and 2(x) 2 q+ (x). 3
definition. Let (x) be a positive bounded formula and let U be a vicinity.
The U -Morley rank of , denoted MR[ ; U], is either an ordinal or 1. We de ne MR[ ; U] by specifying, for each ordinal , when MR[ ; U] : MR[ ; U] 0 if is consistent (with T). If is a limit ordinal, MR[ ; U] if MR[ ; U] for every < . If = +1, MR[ ; U] if there exist positive bounded formulas i , 0 < i < !, such that (i) MR[ ^ i ; U] , for each i < !; (ii) The pair f i; j g is U-separating for i < j < !. MR[ ; U] = 1 if MR[ ; U] for every ordinal . For a set of formulas , we de ne MR[; U] = minf MR[ ; U] : is a nite conjunction of formulas in + g: remark. If U V and j=A , then MR[ ; V ] MR[ ; U]. In particular, MR[ ^ ; U] MR[ ; U] for any formula , and MR[ ; U] MR[ 0; U] for every 0 > . examples.
(1) Let T is the complete theory of a nite dimensional Banach space, and let U be the uniform structure of the metric d. Then MR[ kxk 1; U] = 0 for every vicinity U by the compactness of the unit ball. (2) Let U be the uniform structure of the Banach-Mazur metric. As we have recalled above, !-stability with respect to U never occurs. Thus, MR[ kxk 1; U] = 1 for every vicinity U by Theorem 9. (3) Let T be the theory of c0. Let ek be the k-th unit vector of co , and let ak = e0 + + ek : Then
2; if k l kek + al k = 1; if k > l:
(In the notation of [6], the positive bounded formulas 0 (x; y; u; v) : ky + uk 1
(x; y; u; v) : ky + uk 2;
order the sequence (ak ek )k
4
(4) The Morley rank is not preserved under distortion of the norm, even by small amounts. For > 0, de ne a new norm k k on `2 as follows. For x = (xi )i2! in `2 , let
kxk = kxk + supf jx2i + x2j ?1j : i < j < ! g Then, if ek is the k-th unit vector, we have p l ke2k + e2l?1 k = p22 ++ ;2; ifif kk < > l: (This example was taken from [1], where the credit is given to B. Bollobas. For a similar example, see [8].) As above, `2 with the new norm is unstable, and the Morley rank of every formula and every vicinity (of every uniform structure) is 1. The following lemma will allow us to extend sets of positive bounded formulas without decreasing the U-Morley rank. lemma 1. If (x) and (x) are formulas, then either MR[ ^ 0 ; U] = MR[ ; U]
for all 0 > ;
or else
MR[ ^ neg(0 ); U] = MR[ ; U] for some 0 > : proof: We prove by induction on that if MR[ ; U] , then MR[ ^ 0; U] for all 0 > , or MR[ ^ neg(0 ); U] for some 0 > . The case = 0 is trivial. Suppose = + 1, and let f i : i < ! g correspond to and U as in the de nition of Morley rank. We consider two separate cases: Case 1. There exist 0 > and an in nite subset X f i : i < ! g such that MR[ ^ i ^ neg(0 ); U] for every i 2 X. In this case, we have MR[ ^ neg(0 ); U] by de nition. Case 2. Case 1 does not hold. Fix 0 > . By the induction hypothesis applied to each of the formulas ^ i ^ 0 , there exists an in nite X f i : i < ! g such that MR[ ^ i ^ 00 ; U] for every i 2 X and 00 > 0 . Since 0 is arbitrarily close to , this means that MR[ ^ 00; U] for every 00 > . Now suppose that is a limit ordinal. Again, we consider two cases. Case 1. There exist < and 0 > such that MR[ ^ 0; U] < . Take < < 0 . By the induction hypothesis, for every < 0 < there exists > such that MR[ ^ neg( ); U] 0 . Since neg( ) j=A neg( ), we have MR[ ^ neg( ); U] 0 for every < 0 < . Hence MR[ ^ neg( ); U] . Case 2. Case 1 does not hold. Then MR[ ^ 0; U] for every < and every 0 > . But then, by de nition, MR[ ^ 0 ; U] for every 0 > : 0
0
0
0
0
5
If and are formulas with < , [ ; ) denotes the set f : < g: The order topology on the language L is de ned as follows. The neighborhoods of a formula are the sets [ ; ), where < . The following concept was introduced in [5]. definition. A quasi-type over A is a set of positive bounded L(A)-formulas whose closure with respect to the order topology is a type. Clearly, if p is a type, p+ is a quasi-type. proposition 2. Let ?(x) be a set of positive bounded L(A)-formulas such that (1) ? is consistent; (2) For some N > 0, the formula kxk N is in ?; (3) For every positive bounded L(A)-formula (x), either or neg( ) is in ?. Then ? is a quasi-type over A. proof: See [5]. proposition 3. Let U be a vicinity. Let (x) be a consistent set of L(A)-formulas, such that for some N > 0, the formula kxk N is in . Then can be extended to a set of formulas ?(x), such that (1) ? is a quasi-type over A; (2) ? is closed under approximations; (3) MR[?; U] = MR[; U]. Proof: Let (x) be a conjunction of formulas in of minimal U-Morley rank. Let f i(x) : 1 < i < card( L(A) ) g be a list of all the L(A)-formulas in the variables x. De ne ?0 = [ ?i = ?j if i is limit ordinal; j
?i [ fg+ ; if MR[ ^ 0 ; U] = MR[ ; U] for every 0 > ?i [ fneg(0 )g; if MR[ ^ neg(0 ); U] = MR[ ; U]: S The set ? = j , ~ > ~ such that (1) f ; neg( ) g is W-separating; (2) f ~ ; neg(~ ) g is W-separating; (3) ^ ~ is inconsistent.
proposition 5. For every vicinity V there exists a vicinity W
proof: Take a vicinity V such that V V V . Take also a vicinity W which corresponds to V as V corresponds to U in (2-iii) of the de nition of uniform
structure. For a type p(x) 2 S(A), let 1p (x) =f 0 (x; a) : ( (x; y); 0(x; y) ) 2 D(W); (x; a) 2 p+ (x); a 2 QA g 2p (x) =f neg( (x; a)) : ( (x; y); 0 (x; y) ) 2 D(W); neg( 0 (x; a)) 2 p+ (x); a 2 QA g: De ne p (x) = 1p (x) [ 2p (x): In the proof, the formulas and ~ will be constructed out of p and q , respectively. step 1. p p. proof: Clearly, 1p p and 2p p: ? step 2. If (x) 2 1p (x) + , there exists < in p+ such that f ; neg( ) g is W-separating. ? proof: Take 2 1p + . Then there exist ( (x; y); 0 (x; y)) 2 D(W) and a 2 QA such that (x; a) 2 p+ (x) and > 0 (x; a). In this case, we take = (x; a): ? step 3. If (x) 2 2p (x) + , there exists < in p+ such that f ; neg( ) g is W-separating. ? proof: Take (x) 2 2p + . Then there exist ( (x; y); 0 (x; y)) 2 D(W) and a 2 QA such that neg( 0 (x; a)) 2 p+ and > neg( (x; a)). Thus (*) neg( ) < (x; a): Let be a formula of p+ such that < neg( 0 (x; a)). First, we notice that ^ 0 (x; a) is inconsistent. Indeed, if q is a type containing , then neg(0 ) 2= q for every 0 > . In particular, neg(neg( 0 (x; a))) 2= q, i.e., 0 (x; a) 2= q. We now show that f ; neg( ) g is W-separating. Let q be a type containing , and r a type containing neg( ). Then (x; a) 2 r (by ()) and 0 (x; a) 2= q (by the preceding paragraph). Hence (q; r) 2= W: 7
step 4. If (x) 2 (p )+ , there exists < in p+ such that f ; neg( ) g is Wseparating. ? ? proof: This follows from Steps 2 and 3, since (p )+ = 1p + [ 2p + : step 5. If is a conjunction of formulas in (p )+ , there exists < in p+ such
that f ; neg( ) g is W-separating.
V proof: Suppose = ki=1 i , where each i is in (p )+ . By Step 4, for each i V
there exists i < i in p+ such that f i; neg( i ) g is W separating. Let be ki=1 i. Then < and f ; neg( ) g is W-separating. step 6. If r(x) is a type extending p , then (p; r) 2 V . proof: Fix ( (x; y); 0(x; y)) 2 D(V ) and a 2 QA. We show that for every 00 > 0, (I) (II)
(x; a) 2 p (x; a) 2 r
implies implies
00 (x; a) 2 r 00 (x; a) 2 p:
Fix 00 > 0 . Find ( ; 0 ) 2 D(W) such that < and 0 < 00 . If (x; a) 2 p, then (x; a) 2 p+ . By de nition, 0 (x; a) 2 1p r. Hence 00 (x; a) 2 r. This proves (I). If 00(x; a) 2= p, then neg( 00(x; a)) 2 p, so neg( 0 (x; a)) 2 p+ . By de nition, neg( (x; a)) 2 2p r. Hence (x; a) 2= r. This proves (II). step 7. Let p(x) and q(x) be types such that (p; q) 2= V . Then p (x) [ q (x) is inconsistent. proof: Suppose that p (x) [ q (x) is consistent, and let r(x) be a type extending it. By Step 6, (p; r) 2 V and (q; r) 2 V . But then, (p; q) 2 V V V , which is a contradiction. Now we prove the proposition. Take two types p(x) and q(x) such that (p; q) 2= V . By Step 7 and the compactness theorem, there exist a conjunction of formulas in (p )+ and conjunction ~ of formulas in (q )+ , such that ^ ~ is inconsistent. By Step 5 we can nd < and ~ < ~ such that 2 p+ and ~ 2 q+ , and the pairs f ; neg( ) g and f ~ ; neg(~ ) g are W-separating. This is precisely what we wished to prove. 4. !-stability and morley rank
The main result of this section is Theorem 9. 8
lemma 6. For every vicinity V there exists a vicinity W with the following property. If X is an in nite subset of Sn (A) such that (p; q) 2= V whenever p and q are two distinct types in X, then there exist types p0; p1; : : : in X, and L(A)-formulas 0 ; 1 ; : : : such that (1) i 2 (pi)+ , for i < !; (2) The pair f i; j g is W-separating, for i < j < !. Proof: Let W correspond to V as in Proposition 5.
For a positive bounded formula , let typ() denote the set of types over A that contain . We construct, inductively, a sequence of types in X p0; p1; : : : sequences of L(A)formulas 0 < 0 ; 1 < 1 and in nite subsets of X X0 X1 : : : such that, for each i < !, (1) pi 2 Xi ; (2) i 2 (pi )+ ; (3) The pair f i; neg( i ) g is W-separating; (4) Xi+1 typ(neg( i )). V The lemma then follows by de ning, for each j < !, j = j ^ i<j neg( i ). Set X0 = X. Suppose that X0 ; X1; : : :; Xi , p0 ; p1; : : :; pi?1, and 0; 0 ; 1; 1 ; : : :; i?1; i?1 have already been de ned, in order to de ne pi, i, i and Xi+1 . Let p and q be two distinct types Xi (these can always be found, for Xi is in nite). Since (p; q) 2= V , we can apply Proposition 5 to nd 2 p+ , ~ 2 q+ , > , and ~ > ~ such that f; neg( )g is W-separating; f~ ; neg(~ )g is W-separating; typ( ) \ typ(~ ) = ;. Since Xi is in nite, either Xi n typ( ) or Xi n typ(~ ) is in nite. If Xi n typ( ) is in nite, we let Xi+1 = Xi n typ( ), pi = p, i = , and i = . If Xi n typ(~ ) is in nite, we let Xi+1 = Xi n typ(~ ), pi = q, i = ~ , and i = ~. The required conditions follow immediately. Let us now introduce some temporary terminology. If U is a vicinity and Y is a set of types, we say that Y is U -large if for every positive integer m there exists Zm Y such that card(Zm ) m; (p; q) 2= U whenever p and q are distinct types in Zm . 9
lemma 7. Let U and V be vicinities such that V V
U. If Y is a U-large set of types, there exists an in nite X Y such that (p; q) 2= V whenever p and q are distinct types in X. proof: For p 2 Y , let V [p] = f q : (p; q) 2 V g. The proof is based in the following observation: if p 2 Y and Zm is as in the de nition of U-large, then card(Zm \ V [p]) 1; hence Y n V [p] is U-large. Let p0 2 Y . The set Y n V [p0] is large and, in particular, nonempty. Pick p1 2 Y n V [p0]. The set Y n (V [p0] [ V [p1]) is U-large and, in particular, nonempty. Iterating this process !-many times, we nd X = f p0; p1; : : : g Y , such that (pi ; pj ) 2= V , for i < j: proposition 8. For every vicinity U there exists a vicinity W with the following property. If (x1 ; : : :; xn; a) is an L(A)-formula with MR[ + ; W] < 1, there is a largest integer m such that there are types p1; : : :; pm satisfying the property PW (p1 ; : : :; pm ) de ned below. 8 p1 ; : : :; pm 2 Sn (B); for some B a > > > < ( x; a) 2 pi ; for i = 1; : : :; m PW (p1 ; : : :; pm) : > MR[pi; W] = MR[ + ; W]; for i = 1; : : :; m > > : (pi ; pj ) 2= U; for i < j: proof: Take V such that V V U, and let W correspond to V as in Lemma 6. Let MR[ + ; W] = . Assume, by way of contradiction, that for each m < ! there exist Bm a and p1 ; : : :; pm 2 Sn (Bm ) such that PW (p1 ; : : :; pm ). Use Corollary 4 to extend every S type p 2 Sn (Bm ) with MR[p; U] = to a type q 2 Sn ( Bm ) with MR[q; U] = . m
By assumption, the set Y is U-large. Applying Lemmas 7 and 6 (in that order), we nd p0; p1; : : : in Y , and formulas 0; 1; : : : such that i 2 (pi )+ and the pair f i ; j g is W-separating, for i < j < !. Now take 0 > such that MR[ 0; U] = MR[ +; U] = . From the fact that 0 ^ i 2 (pi)+ and MR[pi; U] = , we conclude MR[ 0 ^ i ; U] = , for each i. Hence MR[ 0; U] + 1. But this is a contradiction. definition. The number m given by Proposition 8 is called (U; W)-Morley degree of , and written deg[ ; U; W]. If is a set of formulas, deg[; U; W] = minf deg[; U; W] : is a conjunction of formulas in , and MR[+ ; W] = MR[; W] g We will express the fact that W corresponds to U as in Proposition 8 by saying that \deg[?; U; W] is de ned". 10
theorem 9. Suppose that U is countable. Then the following conditions are equivalent. (1) T is !-stable with respect to U. (2) MR[ ; U] < 1 for every positive bounded formula and every vicinity U. (3) MR[p; U] < 1 for every type p and every vicinity U. Proof: (1))(2) is proved exactly as in classical model theory: Suppose that
MR[ ; U] = 1. Let
= supf MR[; U] : is an L-formula and MR[; U] < 1g: For any formula , we have MR[; U] = 1 if and only if MR[; U] > . Now, since MR[ ; U] +2, there exist 1(x) and 2(x) such that f 1 (x); 2(x)g, is Useparating and MR[ ^ 1; U]; MR[ ^ 2; U] +1. But then MR[ ^ 2 ; U] + 2 by the choice of , so we can repeat the argument. Iterating this process ! many times, we obtain an uncountable set of types over a countable set of parameters such any two distinct types in the set are separated by U. Thus, T is not !-stable. (2))(3) is trivial. (3))(1): Take a countable set A, in order to prove that S(A) is separable. For every vicinity U x W U such that deg[?; U; W] is de ned. For every L(A)-formula , nd types q1 ; : : :; qdeg[ ;U;W ] such that PW (q1 ; : : :; qdeg[ ;U;W ] ), where PW is the property de ned in Proposition 8. We can assume that these types are in S(A). claim. The set [ f q1 ; : : :; qdeg[ ;U;W ] g 2L(A) U 2U
is dense in S(A). Since L(A) and U are countable, this will prove that S(A) is separable. proof of the claim: Fix p 2 Sn (A). Take also 2 p be such that MR[ + ; W] = MR[p; W]. By the maximality of deg[ ; U; W], there exists 1 i MR[ + ; W] such that (p; qi ) 2 U: 5. principal types
If p is a type and U is a vicinity, we denote by U[p] the set f q : (p; q) 2 U g. If is a positive bounded formula, Sn (A)[ ] denotes the set f p 2 Sn (A) : 2 p g. 11
definition. A type p 2 Sn (A) is principal with respect to U if for every vicinity U there exist an L(A)-formula consistent with T(A) and an approximation 0 of such that Sn (A)[ 0 ] U[p]: The concept of principal type in Banach space model theory was studied in [2]. If is an L(E)-formula, let [ ] = f p 2 S(T) : 2 p g The logical topology on S(T) is de ned as follows. The basic closed sets are the sets of the form [ ]. This topology is Hausdor but not compact.T(For m < !, let m (x) be the formula kxk m. Then [ 0 ] [ 1 ] : : :, but m
(1) If p and q are types, d(p; q) < if and only if (x) 2 p
implies
9z(kzk ^ (x + z)) 2 q:
Hence, a type p(x) 2 S(A) is principal with respect to the metric d if and only if for every > 0 there exist L(A)-formulas (x) < 0(x), consistent with T(A), such that for every (x) 2 p, T(A); (x)0 j=A 9z( kzk ^ (x + z) ): (2) A type p 2 S(A) is principal with respect to the Banach-Mazur metric if and only if for every > 1 there exist L(A)-formulas < 0, consistent with T(A), such that T(A); 0 j=A p , where p is the set of -approximations of formulas in p. (See [3] or [5].) (3) No type can be principal with respect to the discrete uniform structure: Suppose that p(x) 2 S(A) is such a type. Then there exist formulas (x) < 0(x), consistent with T(A), such that T(A); 0 j=A p. Take formulas < < 0 < 0 : Take also a model E and a 2 E such that E j=A (a). By the perturbation lemma(see [5]), there exists > 0 such that E j=A 0 (b), for every b 2 E such that ka ? bk < . Thus E j=A p(b) for kb ? ak < . But this is impossible, for every neighborhood of a contain elements of di erent norm, and hence di erent type. (4) A type over A which is principal is almost realized in every model containing A i.e., if p 2 S(A) is principal and E A is a model, then for every 12
vicinity U there exists q 2 S(A) such that (p; q) 2 U and q is realized in every model of T. (5) A type over A which is principal with respect to the metric d is realized in every model containing A. Let p 2 S(A) be principal, and take a model E A in order to prove that p is realized in E. First, recall that if d(q; q0) , then for every realization c of q there exists a realization c0 of q0 such that kc ? c0 k . Using this fact and the preceding remark iteratively, we nd a sequence (cn)n2! 2 E and a sequence (dn)n2! 2 E such that for every n 2 ! (i) cn realizes p; (ii) kcn ? dn k 2?n; (iii) kdn ? cn+1 k 2?n. By (ii) and (iii), the sequences (cn ) and (dn) are Cauchy and asymptotically equivalent. Thus, their unique limit is in E. But, by (i) and the perturbation lemma [5], this limit must realize p. This argument is due to C. W. Henson. In [2], the author proves the converse for A countable. (Omitting Types Theorem for the metric d.) (6) A type which is principal with respect to the Banach-Mazur metric need not be realized in every model of the theory. Take a sequence 1 p 0 < p1 < < 1 such that pk ! 1 as k ! 1. Let M E = 2 `pn (m); n
where `p (m) denotes the m-dimensional `p space and 2 is an in nite `2 sum. Let p(x; y) be the type of a pair of vectors generating 2-dimensional space isometric to `1 (2). Then p is principal, but is not realized in E. lemma 10. Let U and V be vicinities such that deg[?; U; V ] is de ned.
Suppose MR[p(x); V ] = , and let (x) be a formula in p such that MR[ + ; V ] = MR[p; V ] and deg[ ; U; V ] = deg[p; U; V ]. Then, if f 1; 2 g is a U-separating pair, either = p; or else 1 2 MR[( ^ 2 ^ kxk N)+ ; V ] < for every N > 0: Proof: Suppose that 1 2 p and MR[( ^ 2 ^ kxk N)+ ; V ] , in order to reach a contradiction. Let m = deg[p; U; V ] = deg[ ^ 1; U; V ]. Find types p1; : : :pm 2 Sn (A) such that ^ 1 2 p1 ; : : :; pm; MR[pi; V ] = , for i = 1; : : :; m; (pi ; pj ) 2 U, for i 6= j. 13
Since MR[( ^ 2 ^ kxk N)+ ; V ] = , we can apply Corollary 4 to nd a type q such that ^ 2 2 q and MR[q; V ] = . Since f 1 ; 2 g is U-separating, (pi; q) 2= U for i = 1; : : :; m. But then have deg[ ; U; V ] m + 1, which is a contradiction. lemma 11. Suppose that T is !-stable with respect to U. Let (x1; : : :; xn) and 0 (x1 ; : : :; xn) be consistent L(A)-formulas such that < 0 . Then there exist p(x) 2 Sn (A) and formulas < 0 in p+ such that Sn (A)[ 0 ] Sn (A)[]0 \ U[p]: Proof: Find a vicinity V
Let
U such that deg[?; U; V ] is de ned.
= minf MR[p; V ] : p 2 Sn (A) and 0 2 p+ g: Take p 2 Sn (A) such that 2 p+ and MR[p; V ] = . Choose formulas < 0 in p+ such that MR[ 0; V ] = MR[ + ; V ] = MR[p; V ] = ; deg[ 0; U; V ] = deg[p; U; V ]: We may, further, assume that 0 is of the form 0 ^(: : :). ThusSn (A)[ 0] Sn (A)[]0 . We prove that Sn (A)[ 0 ] U[p]. Suppose that this is false. Then there exists q 2 Sn (A) such that 0 2 q and (p; q) 2= U. Therefore there is a U-separating pair f 1 (x); 2(x)g such that 1 2 p 2 2 q. By the preceding lemma, MR[q; U; V ] MR[( ^
2
^ kxk kqk)+ ; V ] < :
But this contradicts the minimality of : proposition 12. Suppose that U is countable and T is !-stable. Then for any consistent L(A)-formula (x1 ; : : :xn) there exists a principal type p(x) 2 Sn (A) such that 2 p. Proof: Take a decreasing sequence of vicinities U0 U1 : : : which is co nal in U, i.e., every vicinity contains some Um . Take also a sequence of formulas (m )m 1 > > and limm!1 m = (where the limit is taken in the order topology). Using Lemma 11 iteratively, we construct a sequence of types p0 (x); p1(x); : : : and sequences of L(A)-formulas consistent with T(A) 0 < 00 1 < 01 .. . 14
such that
Sn (A)[ 00] Sn (A)[0] \ U0 [p0]; Sn (A)[ 0m+1 ] Sn (A)[m+1 ^ m ] \ Um [pm+1 ]: By the compactness theorem, there exists p 2 Sn (A) such that m 2 p for all m. We have 2 p and (p; pm ) 2 Um for all m. It is easy to see that then p must be principal. 6. prime models theorem 13. Suppose that U is countable and T is !-stable with respect to U.
Then for every set A of cardinality there exists a rst-order model M A such that (1) M = A [ f bi : i < g, for some !; (2) tp( bj jA [ fbi : i < j g ) is principal for each j < . proof: The theorem is proved as in the rst-order case, using Lemma 12. Now we investigate the implications of Theorem 13 for a concrete uniform structure, namely, that of the metric d. Thanks to the fact that principal types relative to the metric d are realized in every model (see Remark 5 after the de nition of principal type), the existence of prime models for theories which are !-stable with respect to d can be proved, almost literally, as in rst-order logic. theorem 14. Suppose that T is !-stable with respect to d. Then for every set A, there is a model E A with the following property: For any model F A there exists an elementary embedding f : E A F, xing A pointwise. If A is separable, the existence of such a model E can be proved from the Omitting Types Theorem for the metric d (see [2]) under an assumption weaker that !stability with respect to d, namely, that S(A) is separable with respect to d. Proof: Let E be the completion of the model M of Theorem 13. Fix F A to de ne f : E A F. Clearly, it suces to de ne f on M. For j < , we de ne f(bj ) inductively. Suppose that we have de ned f jA [ f bi : i < j g, in order to de ne f(bj ). Let p = tp( bijA [ f bi : i < j g ). Fix > 0. Since p is principal, there exists (x) 2 L(A [ f bi : i < j g) and 0 > such that for every (x) 2 p, T(A); 0(x) j=A 9z( kzk ^ (x + z) ); Hence, T(A); f( 0 (x)) j=A 9z( kzk ^ f((x + z)) ): Since is arbitrary, we conclude that f(p) is principal. De ne f(bi ) as any realization of f(p) in F: 15
Bibliography 1. D. H. J. Garling, Stable Banach spaces, random measures and Orlicz functions, Probability Measures on Groups, Lecture Notes in Math., 928, Springer-Verlag, Berlin-New York, 1982. 2. C. W. Henson, Banach space model theory, III: Separable spaces under isometric equivalence, In preparation. 3. S. Heinrich and C. W. Henson, Banach space model theory, II: Isomorphic equivalence, Math. Nachr. 125 (1986), 301{317. 4. C. W. Henson, Some examples of stable Banach spaces, In preparation. 5. C. W. Henson and J. Iovino, Banach space model theory I: Basics, In preparation. 6. J. Iovino, Stable Banach spaces, I: Fundamentals, Submitted. 7. J. Kelley, General Topology, Van Nostrand, Princeton, 1955. 8. Y. Raynaud, Separabilite uniforme de L'espace des types d'un espace de Banach { Quelques exemples, Seminaire de Geometrie des espaces de Banach. Univ. Paris 7 2 (1983). Department of Mathematics Carnegie-Mellon University Pittsburgh, PA 15213 E-mail : [email protected]
16
1. introduction
A general framework for the model theoretical analysis of structures based on Banach spaces was introduced in the monograph [5]. It was shown there that the space of types of a Banach space theory is naturally endowed with various uniform topologies on it. A general notion of a uniform structure on the space of types of a complete theory was introduced. In [6], we introduced the concept of stable Banach space theory: Let T be a complete Banach space theory, let be an in nite cardinal, and let U be a uniform structure on the space of types of T. The theory T is -stable with respect to U if for every model E of T of density character less than or equal to , the density character of the space of types over E with respect to the uniform topology of U is also less than or equal to . In particular, T is !-stable with respect to U if the space of types over every separable model of T is separable with respect to U. It was shown in [4] that signi cant classes of Banach space structures (e.g., the spaces Lp , for 1 p < 1) are !-stable with respect to natural, metrizable uniform structures. We also showed in [6] that when the uniform structure is metrizable, !-stability implies -stability for any in nite cardinal , and if T has quanti er elimination, stability is equivalent to !-stability. Here, we investigate !-stability further. We de ne a Banach space theoretic Morley rank (more precisely, a family of Morley ranks), and characterize !-stability in terms of it. From this characterization we prove the existence of prime models for theories which are !-stable with respect to the metric d on the space of types introduced in [5]. The shall assume that the reader is familiar with the basic machinery developed in [5] for the model theoretical analysis of Banach space structures, including the concept of uniform structure on the space of types. However, we shall reprint the de nition here for reference. We extend to this paper the general assumptions and notational conventions of [6]. We deal with Banach space structures, positive bounded formulas, and 1
approximate satisfaction j=A. The terms \formula", \theory" and \type" are used as abbreviations of \positive bounded formula", \positive bounded theory" and \positive bounded type", respectively. If and are positive bounded formulas, < means that is an approximation of . The letter T denotes a xed complete positive bounded theory over a countable language L. All the models considered are models of T. We assume that these models are approximately elementary submodels of some large, saturated model E. If a 2 E, kak is an abbreviation of max1in kaik. By the @0 -saturation of E, we have E j=A (a) if and only if E j= (a), for every positive bounded formula and every a 2 E. We write simply j= (a), omitting E. We consider only complete, positive bounded types which are consistent with T. The norm of an n-type p, denoted kpk, is the norm of any n-tuple realizing p. If A is a subset of a model E, we denote by L(A) the result of expanding the language L with constants and appropriate norm bounds for the elements of A, and T(A) is the theory of E in L(A). The set of n-types over A is denoted Sn (A), and S S(A) = n
2. separating pairs of formulas
Let S(T) = AE S(A). The models of T come equipped with a topology, namely, the norm topology. This topology can be transfered onto S(T) as follows. If p(x); q(x) 2 Sn (A), we let d(p; q) = inf f kb ? ck : E j= p(b); q(c) g: Then, d is a metric on Sn (A). We extend d to all of S(T) by letting d(p; q) = 1 when p and q are types in di erent sets of variables or over di erent sets of parameters. For each rational > 0, let U = f (p; q) : d(p; q) g. The family U = f U : 2 Q+ g is an example of a uniform structure on S(T): definition. A uniform structure on on the space of types of T is a family U of subsets of S(T) S(T), called vicinities , such that (1) U is a base for a Hausdor uniform structure (in the standard topological sense. See, for example, [7, Chapter 6]) on S(T). (2) For every vicinity U there exists a set of pairs of formulas D(U) such that (i) If ( ; 0) 2 D(U), then < 0. (ii) D(U) de nes U in the following sense: The pair (p(x); q(x)) is in U \ S(A) S(A) if and only if for every pair ( (x; y); 0(x; y)) 2 D(U) and every a 2 QA (= rational multiples of A), (x; a) 2 p implies 0 (x; a) 2 q; (x; a) 2 q implies 0 (x; a) 2 p: 2
(iii) There exists a vicinity V with the following property: If ( ; 0) 2 D(U) and 00 > 0, there exists ( ; 0 ) 2 D(V ) such that < and 0 < 00 . In the case of the metric d de ned above, we let D(U ) be the set of pairs ( ; 0) such that (x; y) is of the form
8z(kzk ! (x + z; y)); for > , and 0(x; y) is of the form
8z(kzk ? ! 0 (x + z; y)); with 0 > . Another important example of a metrizable uniform structure on the space of types of T is given by the Banach-Mazur metric on types: For 0, let U be the set of pairs of types (p; q) such that there is a 1 + -automorphism E mapping p onto q (a linear isomorphism f is a 1 + -isomorphism if kf k; kf ?1 k 1 + ). The family f U : 2 Q+ g is a metrizable uniform structure on the space of types of T. No theory can be !-stable with respect to this uniform structure. However, if 2@0 , -stability with respect to the metric d implies -stability with respect to the Banach-Mazur metric. See [6] for the details, and for a discussion of further natural uniform structures on the space of types. Hereafter, U will denote a xed uniform structure on the set of types of T, and all the vicinities mentioned are vicinities of U. A notion that recurrently underlies the main arguments of [6] is that of a separating pair of formulas.
definition. Let U be a vicinity of U. A pair of positive bounded formulas
f 1 (x); 2 (x) g is U -separating if (p(x); q(x)) 2= U whenever x) 2 q(x). 2 (
1
(x) 2 p(x) and
If is a set of positive bounded formulas, + denotes the set
f 0 : 0 > for some 2 g: If is a positive bounded formula, we write + instead of f g+. remarks.
(1) If (p(x); q(x)) 2= U, there exists a U-separating pair f 1(x); 2(x) g such that 1(x) 2 p(x) and 2(x) 2 q(x). (2) Let V correspond to U as in (2-iii) of the de nition of uniform structure. Then, if (p(x); q(x)) 2= U, there exists a V -separating pair f 1(x); 2(x) g such that 1 (x) 2 p+ (x) and 2(x) 2 q+ (x). 3
definition. Let (x) be a positive bounded formula and let U be a vicinity.
The U -Morley rank of , denoted MR[ ; U], is either an ordinal or 1. We de ne MR[ ; U] by specifying, for each ordinal , when MR[ ; U] : MR[ ; U] 0 if is consistent (with T). If is a limit ordinal, MR[ ; U] if MR[ ; U] for every < . If = +1, MR[ ; U] if there exist positive bounded formulas i , 0 < i < !, such that (i) MR[ ^ i ; U] , for each i < !; (ii) The pair f i; j g is U-separating for i < j < !. MR[ ; U] = 1 if MR[ ; U] for every ordinal . For a set of formulas , we de ne MR[; U] = minf MR[ ; U] : is a nite conjunction of formulas in + g: remark. If U V and j=A , then MR[ ; V ] MR[ ; U]. In particular, MR[ ^ ; U] MR[ ; U] for any formula , and MR[ ; U] MR[ 0; U] for every 0 > . examples.
(1) Let T is the complete theory of a nite dimensional Banach space, and let U be the uniform structure of the metric d. Then MR[ kxk 1; U] = 0 for every vicinity U by the compactness of the unit ball. (2) Let U be the uniform structure of the Banach-Mazur metric. As we have recalled above, !-stability with respect to U never occurs. Thus, MR[ kxk 1; U] = 1 for every vicinity U by Theorem 9. (3) Let T be the theory of c0. Let ek be the k-th unit vector of co , and let ak = e0 + + ek : Then
2; if k l kek + al k = 1; if k > l:
(In the notation of [6], the positive bounded formulas 0 (x; y; u; v) : ky + uk 1
(x; y; u; v) : ky + uk 2;
order the sequence (ak ek )k
4
(4) The Morley rank is not preserved under distortion of the norm, even by small amounts. For > 0, de ne a new norm k k on `2 as follows. For x = (xi )i2! in `2 , let
kxk = kxk + supf jx2i + x2j ?1j : i < j < ! g Then, if ek is the k-th unit vector, we have p l ke2k + e2l?1 k = p22 ++ ;2; ifif kk < > l: (This example was taken from [1], where the credit is given to B. Bollobas. For a similar example, see [8].) As above, `2 with the new norm is unstable, and the Morley rank of every formula and every vicinity (of every uniform structure) is 1. The following lemma will allow us to extend sets of positive bounded formulas without decreasing the U-Morley rank. lemma 1. If (x) and (x) are formulas, then either MR[ ^ 0 ; U] = MR[ ; U]
for all 0 > ;
or else
MR[ ^ neg(0 ); U] = MR[ ; U] for some 0 > : proof: We prove by induction on that if MR[ ; U] , then MR[ ^ 0; U] for all 0 > , or MR[ ^ neg(0 ); U] for some 0 > . The case = 0 is trivial. Suppose = + 1, and let f i : i < ! g correspond to and U as in the de nition of Morley rank. We consider two separate cases: Case 1. There exist 0 > and an in nite subset X f i : i < ! g such that MR[ ^ i ^ neg(0 ); U] for every i 2 X. In this case, we have MR[ ^ neg(0 ); U] by de nition. Case 2. Case 1 does not hold. Fix 0 > . By the induction hypothesis applied to each of the formulas ^ i ^ 0 , there exists an in nite X f i : i < ! g such that MR[ ^ i ^ 00 ; U] for every i 2 X and 00 > 0 . Since 0 is arbitrarily close to , this means that MR[ ^ 00; U] for every 00 > . Now suppose that is a limit ordinal. Again, we consider two cases. Case 1. There exist < and 0 > such that MR[ ^ 0; U] < . Take < < 0 . By the induction hypothesis, for every < 0 < there exists > such that MR[ ^ neg( ); U] 0 . Since neg( ) j=A neg( ), we have MR[ ^ neg( ); U] 0 for every < 0 < . Hence MR[ ^ neg( ); U] . Case 2. Case 1 does not hold. Then MR[ ^ 0; U] for every < and every 0 > . But then, by de nition, MR[ ^ 0 ; U] for every 0 > : 0
0
0
0
0
5
If and are formulas with < , [ ; ) denotes the set f : < g: The order topology on the language L is de ned as follows. The neighborhoods of a formula are the sets [ ; ), where < . The following concept was introduced in [5]. definition. A quasi-type over A is a set of positive bounded L(A)-formulas whose closure with respect to the order topology is a type. Clearly, if p is a type, p+ is a quasi-type. proposition 2. Let ?(x) be a set of positive bounded L(A)-formulas such that (1) ? is consistent; (2) For some N > 0, the formula kxk N is in ?; (3) For every positive bounded L(A)-formula (x), either or neg( ) is in ?. Then ? is a quasi-type over A. proof: See [5]. proposition 3. Let U be a vicinity. Let (x) be a consistent set of L(A)-formulas, such that for some N > 0, the formula kxk N is in . Then can be extended to a set of formulas ?(x), such that (1) ? is a quasi-type over A; (2) ? is closed under approximations; (3) MR[?; U] = MR[; U]. Proof: Let (x) be a conjunction of formulas in of minimal U-Morley rank. Let f i(x) : 1 < i < card( L(A) ) g be a list of all the L(A)-formulas in the variables x. De ne ?0 = [ ?i = ?j if i is limit ordinal; j
?i [ fg+ ; if MR[ ^ 0 ; U] = MR[ ; U] for every 0 > ?i [ fneg(0 )g; if MR[ ^ neg(0 ); U] = MR[ ; U]: S The set ? = j , ~ > ~ such that (1) f ; neg( ) g is W-separating; (2) f ~ ; neg(~ ) g is W-separating; (3) ^ ~ is inconsistent.
proposition 5. For every vicinity V there exists a vicinity W
proof: Take a vicinity V such that V V V . Take also a vicinity W which corresponds to V as V corresponds to U in (2-iii) of the de nition of uniform
structure. For a type p(x) 2 S(A), let 1p (x) =f 0 (x; a) : ( (x; y); 0(x; y) ) 2 D(W); (x; a) 2 p+ (x); a 2 QA g 2p (x) =f neg( (x; a)) : ( (x; y); 0 (x; y) ) 2 D(W); neg( 0 (x; a)) 2 p+ (x); a 2 QA g: De ne p (x) = 1p (x) [ 2p (x): In the proof, the formulas and ~ will be constructed out of p and q , respectively. step 1. p p. proof: Clearly, 1p p and 2p p: ? step 2. If (x) 2 1p (x) + , there exists < in p+ such that f ; neg( ) g is W-separating. ? proof: Take 2 1p + . Then there exist ( (x; y); 0 (x; y)) 2 D(W) and a 2 QA such that (x; a) 2 p+ (x) and > 0 (x; a). In this case, we take = (x; a): ? step 3. If (x) 2 2p (x) + , there exists < in p+ such that f ; neg( ) g is W-separating. ? proof: Take (x) 2 2p + . Then there exist ( (x; y); 0 (x; y)) 2 D(W) and a 2 QA such that neg( 0 (x; a)) 2 p+ and > neg( (x; a)). Thus (*) neg( ) < (x; a): Let be a formula of p+ such that < neg( 0 (x; a)). First, we notice that ^ 0 (x; a) is inconsistent. Indeed, if q is a type containing , then neg(0 ) 2= q for every 0 > . In particular, neg(neg( 0 (x; a))) 2= q, i.e., 0 (x; a) 2= q. We now show that f ; neg( ) g is W-separating. Let q be a type containing , and r a type containing neg( ). Then (x; a) 2 r (by ()) and 0 (x; a) 2= q (by the preceding paragraph). Hence (q; r) 2= W: 7
step 4. If (x) 2 (p )+ , there exists < in p+ such that f ; neg( ) g is Wseparating. ? ? proof: This follows from Steps 2 and 3, since (p )+ = 1p + [ 2p + : step 5. If is a conjunction of formulas in (p )+ , there exists < in p+ such
that f ; neg( ) g is W-separating.
V proof: Suppose = ki=1 i , where each i is in (p )+ . By Step 4, for each i V
there exists i < i in p+ such that f i; neg( i ) g is W separating. Let be ki=1 i. Then < and f ; neg( ) g is W-separating. step 6. If r(x) is a type extending p , then (p; r) 2 V . proof: Fix ( (x; y); 0(x; y)) 2 D(V ) and a 2 QA. We show that for every 00 > 0, (I) (II)
(x; a) 2 p (x; a) 2 r
implies implies
00 (x; a) 2 r 00 (x; a) 2 p:
Fix 00 > 0 . Find ( ; 0 ) 2 D(W) such that < and 0 < 00 . If (x; a) 2 p, then (x; a) 2 p+ . By de nition, 0 (x; a) 2 1p r. Hence 00 (x; a) 2 r. This proves (I). If 00(x; a) 2= p, then neg( 00(x; a)) 2 p, so neg( 0 (x; a)) 2 p+ . By de nition, neg( (x; a)) 2 2p r. Hence (x; a) 2= r. This proves (II). step 7. Let p(x) and q(x) be types such that (p; q) 2= V . Then p (x) [ q (x) is inconsistent. proof: Suppose that p (x) [ q (x) is consistent, and let r(x) be a type extending it. By Step 6, (p; r) 2 V and (q; r) 2 V . But then, (p; q) 2 V V V , which is a contradiction. Now we prove the proposition. Take two types p(x) and q(x) such that (p; q) 2= V . By Step 7 and the compactness theorem, there exist a conjunction of formulas in (p )+ and conjunction ~ of formulas in (q )+ , such that ^ ~ is inconsistent. By Step 5 we can nd < and ~ < ~ such that 2 p+ and ~ 2 q+ , and the pairs f ; neg( ) g and f ~ ; neg(~ ) g are W-separating. This is precisely what we wished to prove. 4. !-stability and morley rank
The main result of this section is Theorem 9. 8
lemma 6. For every vicinity V there exists a vicinity W with the following property. If X is an in nite subset of Sn (A) such that (p; q) 2= V whenever p and q are two distinct types in X, then there exist types p0; p1; : : : in X, and L(A)-formulas 0 ; 1 ; : : : such that (1) i 2 (pi)+ , for i < !; (2) The pair f i; j g is W-separating, for i < j < !. Proof: Let W correspond to V as in Proposition 5.
For a positive bounded formula , let typ() denote the set of types over A that contain . We construct, inductively, a sequence of types in X p0; p1; : : : sequences of L(A)formulas 0 < 0 ; 1 < 1 and in nite subsets of X X0 X1 : : : such that, for each i < !, (1) pi 2 Xi ; (2) i 2 (pi )+ ; (3) The pair f i; neg( i ) g is W-separating; (4) Xi+1 typ(neg( i )). V The lemma then follows by de ning, for each j < !, j = j ^ i<j neg( i ). Set X0 = X. Suppose that X0 ; X1; : : :; Xi , p0 ; p1; : : :; pi?1, and 0; 0 ; 1; 1 ; : : :; i?1; i?1 have already been de ned, in order to de ne pi, i, i and Xi+1 . Let p and q be two distinct types Xi (these can always be found, for Xi is in nite). Since (p; q) 2= V , we can apply Proposition 5 to nd 2 p+ , ~ 2 q+ , > , and ~ > ~ such that f; neg( )g is W-separating; f~ ; neg(~ )g is W-separating; typ( ) \ typ(~ ) = ;. Since Xi is in nite, either Xi n typ( ) or Xi n typ(~ ) is in nite. If Xi n typ( ) is in nite, we let Xi+1 = Xi n typ( ), pi = p, i = , and i = . If Xi n typ(~ ) is in nite, we let Xi+1 = Xi n typ(~ ), pi = q, i = ~ , and i = ~. The required conditions follow immediately. Let us now introduce some temporary terminology. If U is a vicinity and Y is a set of types, we say that Y is U -large if for every positive integer m there exists Zm Y such that card(Zm ) m; (p; q) 2= U whenever p and q are distinct types in Zm . 9
lemma 7. Let U and V be vicinities such that V V
U. If Y is a U-large set of types, there exists an in nite X Y such that (p; q) 2= V whenever p and q are distinct types in X. proof: For p 2 Y , let V [p] = f q : (p; q) 2 V g. The proof is based in the following observation: if p 2 Y and Zm is as in the de nition of U-large, then card(Zm \ V [p]) 1; hence Y n V [p] is U-large. Let p0 2 Y . The set Y n V [p0] is large and, in particular, nonempty. Pick p1 2 Y n V [p0]. The set Y n (V [p0] [ V [p1]) is U-large and, in particular, nonempty. Iterating this process !-many times, we nd X = f p0; p1; : : : g Y , such that (pi ; pj ) 2= V , for i < j: proposition 8. For every vicinity U there exists a vicinity W with the following property. If (x1 ; : : :; xn; a) is an L(A)-formula with MR[ + ; W] < 1, there is a largest integer m such that there are types p1; : : :; pm satisfying the property PW (p1 ; : : :; pm ) de ned below. 8 p1 ; : : :; pm 2 Sn (B); for some B a > > > < ( x; a) 2 pi ; for i = 1; : : :; m PW (p1 ; : : :; pm) : > MR[pi; W] = MR[ + ; W]; for i = 1; : : :; m > > : (pi ; pj ) 2= U; for i < j: proof: Take V such that V V U, and let W correspond to V as in Lemma 6. Let MR[ + ; W] = . Assume, by way of contradiction, that for each m < ! there exist Bm a and p1 ; : : :; pm 2 Sn (Bm ) such that PW (p1 ; : : :; pm ). Use Corollary 4 to extend every S type p 2 Sn (Bm ) with MR[p; U] = to a type q 2 Sn ( Bm ) with MR[q; U] = . m
By assumption, the set Y is U-large. Applying Lemmas 7 and 6 (in that order), we nd p0; p1; : : : in Y , and formulas 0; 1; : : : such that i 2 (pi )+ and the pair f i ; j g is W-separating, for i < j < !. Now take 0 > such that MR[ 0; U] = MR[ +; U] = . From the fact that 0 ^ i 2 (pi)+ and MR[pi; U] = , we conclude MR[ 0 ^ i ; U] = , for each i. Hence MR[ 0; U] + 1. But this is a contradiction. definition. The number m given by Proposition 8 is called (U; W)-Morley degree of , and written deg[ ; U; W]. If is a set of formulas, deg[; U; W] = minf deg[; U; W] : is a conjunction of formulas in , and MR[+ ; W] = MR[; W] g We will express the fact that W corresponds to U as in Proposition 8 by saying that \deg[?; U; W] is de ned". 10
theorem 9. Suppose that U is countable. Then the following conditions are equivalent. (1) T is !-stable with respect to U. (2) MR[ ; U] < 1 for every positive bounded formula and every vicinity U. (3) MR[p; U] < 1 for every type p and every vicinity U. Proof: (1))(2) is proved exactly as in classical model theory: Suppose that
MR[ ; U] = 1. Let
= supf MR[; U] : is an L-formula and MR[; U] < 1g: For any formula , we have MR[; U] = 1 if and only if MR[; U] > . Now, since MR[ ; U] +2, there exist 1(x) and 2(x) such that f 1 (x); 2(x)g, is Useparating and MR[ ^ 1; U]; MR[ ^ 2; U] +1. But then MR[ ^ 2 ; U] + 2 by the choice of , so we can repeat the argument. Iterating this process ! many times, we obtain an uncountable set of types over a countable set of parameters such any two distinct types in the set are separated by U. Thus, T is not !-stable. (2))(3) is trivial. (3))(1): Take a countable set A, in order to prove that S(A) is separable. For every vicinity U x W U such that deg[?; U; W] is de ned. For every L(A)-formula , nd types q1 ; : : :; qdeg[ ;U;W ] such that PW (q1 ; : : :; qdeg[ ;U;W ] ), where PW is the property de ned in Proposition 8. We can assume that these types are in S(A). claim. The set [ f q1 ; : : :; qdeg[ ;U;W ] g 2L(A) U 2U
is dense in S(A). Since L(A) and U are countable, this will prove that S(A) is separable. proof of the claim: Fix p 2 Sn (A). Take also 2 p be such that MR[ + ; W] = MR[p; W]. By the maximality of deg[ ; U; W], there exists 1 i MR[ + ; W] such that (p; qi ) 2 U: 5. principal types
If p is a type and U is a vicinity, we denote by U[p] the set f q : (p; q) 2 U g. If is a positive bounded formula, Sn (A)[ ] denotes the set f p 2 Sn (A) : 2 p g. 11
definition. A type p 2 Sn (A) is principal with respect to U if for every vicinity U there exist an L(A)-formula consistent with T(A) and an approximation 0 of such that Sn (A)[ 0 ] U[p]: The concept of principal type in Banach space model theory was studied in [2]. If is an L(E)-formula, let [ ] = f p 2 S(T) : 2 p g The logical topology on S(T) is de ned as follows. The basic closed sets are the sets of the form [ ]. This topology is Hausdor but not compact.T(For m < !, let m (x) be the formula kxk m. Then [ 0 ] [ 1 ] : : :, but m
(1) If p and q are types, d(p; q) < if and only if (x) 2 p
implies
9z(kzk ^ (x + z)) 2 q:
Hence, a type p(x) 2 S(A) is principal with respect to the metric d if and only if for every > 0 there exist L(A)-formulas (x) < 0(x), consistent with T(A), such that for every (x) 2 p, T(A); (x)0 j=A 9z( kzk ^ (x + z) ): (2) A type p 2 S(A) is principal with respect to the Banach-Mazur metric if and only if for every > 1 there exist L(A)-formulas < 0, consistent with T(A), such that T(A); 0 j=A p , where p is the set of -approximations of formulas in p. (See [3] or [5].) (3) No type can be principal with respect to the discrete uniform structure: Suppose that p(x) 2 S(A) is such a type. Then there exist formulas (x) < 0(x), consistent with T(A), such that T(A); 0 j=A p. Take formulas < < 0 < 0 : Take also a model E and a 2 E such that E j=A (a). By the perturbation lemma(see [5]), there exists > 0 such that E j=A 0 (b), for every b 2 E such that ka ? bk < . Thus E j=A p(b) for kb ? ak < . But this is impossible, for every neighborhood of a contain elements of di erent norm, and hence di erent type. (4) A type over A which is principal is almost realized in every model containing A i.e., if p 2 S(A) is principal and E A is a model, then for every 12
vicinity U there exists q 2 S(A) such that (p; q) 2 U and q is realized in every model of T. (5) A type over A which is principal with respect to the metric d is realized in every model containing A. Let p 2 S(A) be principal, and take a model E A in order to prove that p is realized in E. First, recall that if d(q; q0) , then for every realization c of q there exists a realization c0 of q0 such that kc ? c0 k . Using this fact and the preceding remark iteratively, we nd a sequence (cn)n2! 2 E and a sequence (dn)n2! 2 E such that for every n 2 ! (i) cn realizes p; (ii) kcn ? dn k 2?n; (iii) kdn ? cn+1 k 2?n. By (ii) and (iii), the sequences (cn ) and (dn) are Cauchy and asymptotically equivalent. Thus, their unique limit is in E. But, by (i) and the perturbation lemma [5], this limit must realize p. This argument is due to C. W. Henson. In [2], the author proves the converse for A countable. (Omitting Types Theorem for the metric d.) (6) A type which is principal with respect to the Banach-Mazur metric need not be realized in every model of the theory. Take a sequence 1 p 0 < p1 < < 1 such that pk ! 1 as k ! 1. Let M E = 2 `pn (m); n
where `p (m) denotes the m-dimensional `p space and 2 is an in nite `2 sum. Let p(x; y) be the type of a pair of vectors generating 2-dimensional space isometric to `1 (2). Then p is principal, but is not realized in E. lemma 10. Let U and V be vicinities such that deg[?; U; V ] is de ned.
Suppose MR[p(x); V ] = , and let (x) be a formula in p such that MR[ + ; V ] = MR[p; V ] and deg[ ; U; V ] = deg[p; U; V ]. Then, if f 1; 2 g is a U-separating pair, either = p; or else 1 2 MR[( ^ 2 ^ kxk N)+ ; V ] < for every N > 0: Proof: Suppose that 1 2 p and MR[( ^ 2 ^ kxk N)+ ; V ] , in order to reach a contradiction. Let m = deg[p; U; V ] = deg[ ^ 1; U; V ]. Find types p1; : : :pm 2 Sn (A) such that ^ 1 2 p1 ; : : :; pm; MR[pi; V ] = , for i = 1; : : :; m; (pi ; pj ) 2 U, for i 6= j. 13
Since MR[( ^ 2 ^ kxk N)+ ; V ] = , we can apply Corollary 4 to nd a type q such that ^ 2 2 q and MR[q; V ] = . Since f 1 ; 2 g is U-separating, (pi; q) 2= U for i = 1; : : :; m. But then have deg[ ; U; V ] m + 1, which is a contradiction. lemma 11. Suppose that T is !-stable with respect to U. Let (x1; : : :; xn) and 0 (x1 ; : : :; xn) be consistent L(A)-formulas such that < 0 . Then there exist p(x) 2 Sn (A) and formulas < 0 in p+ such that Sn (A)[ 0 ] Sn (A)[]0 \ U[p]: Proof: Find a vicinity V
Let
U such that deg[?; U; V ] is de ned.
= minf MR[p; V ] : p 2 Sn (A) and 0 2 p+ g: Take p 2 Sn (A) such that 2 p+ and MR[p; V ] = . Choose formulas < 0 in p+ such that MR[ 0; V ] = MR[ + ; V ] = MR[p; V ] = ; deg[ 0; U; V ] = deg[p; U; V ]: We may, further, assume that 0 is of the form 0 ^(: : :). ThusSn (A)[ 0] Sn (A)[]0 . We prove that Sn (A)[ 0 ] U[p]. Suppose that this is false. Then there exists q 2 Sn (A) such that 0 2 q and (p; q) 2= U. Therefore there is a U-separating pair f 1 (x); 2(x)g such that 1 2 p 2 2 q. By the preceding lemma, MR[q; U; V ] MR[( ^
2
^ kxk kqk)+ ; V ] < :
But this contradicts the minimality of : proposition 12. Suppose that U is countable and T is !-stable. Then for any consistent L(A)-formula (x1 ; : : :xn) there exists a principal type p(x) 2 Sn (A) such that 2 p. Proof: Take a decreasing sequence of vicinities U0 U1 : : : which is co nal in U, i.e., every vicinity contains some Um . Take also a sequence of formulas (m )m 1 > > and limm!1 m = (where the limit is taken in the order topology). Using Lemma 11 iteratively, we construct a sequence of types p0 (x); p1(x); : : : and sequences of L(A)-formulas consistent with T(A) 0 < 00 1 < 01 .. . 14
such that
Sn (A)[ 00] Sn (A)[0] \ U0 [p0]; Sn (A)[ 0m+1 ] Sn (A)[m+1 ^ m ] \ Um [pm+1 ]: By the compactness theorem, there exists p 2 Sn (A) such that m 2 p for all m. We have 2 p and (p; pm ) 2 Um for all m. It is easy to see that then p must be principal. 6. prime models theorem 13. Suppose that U is countable and T is !-stable with respect to U.
Then for every set A of cardinality there exists a rst-order model M A such that (1) M = A [ f bi : i < g, for some !; (2) tp( bj jA [ fbi : i < j g ) is principal for each j < . proof: The theorem is proved as in the rst-order case, using Lemma 12. Now we investigate the implications of Theorem 13 for a concrete uniform structure, namely, that of the metric d. Thanks to the fact that principal types relative to the metric d are realized in every model (see Remark 5 after the de nition of principal type), the existence of prime models for theories which are !-stable with respect to d can be proved, almost literally, as in rst-order logic. theorem 14. Suppose that T is !-stable with respect to d. Then for every set A, there is a model E A with the following property: For any model F A there exists an elementary embedding f : E A F, xing A pointwise. If A is separable, the existence of such a model E can be proved from the Omitting Types Theorem for the metric d (see [2]) under an assumption weaker that !stability with respect to d, namely, that S(A) is separable with respect to d. Proof: Let E be the completion of the model M of Theorem 13. Fix F A to de ne f : E A F. Clearly, it suces to de ne f on M. For j < , we de ne f(bj ) inductively. Suppose that we have de ned f jA [ f bi : i < j g, in order to de ne f(bj ). Let p = tp( bijA [ f bi : i < j g ). Fix > 0. Since p is principal, there exists (x) 2 L(A [ f bi : i < j g) and 0 > such that for every (x) 2 p, T(A); 0(x) j=A 9z( kzk ^ (x + z) ); Hence, T(A); f( 0 (x)) j=A 9z( kzk ^ f((x + z)) ): Since is arbitrary, we conclude that f(p) is principal. De ne f(bi ) as any realization of f(p) in F: 15
Bibliography 1. D. H. J. Garling, Stable Banach spaces, random measures and Orlicz functions, Probability Measures on Groups, Lecture Notes in Math., 928, Springer-Verlag, Berlin-New York, 1982. 2. C. W. Henson, Banach space model theory, III: Separable spaces under isometric equivalence, In preparation. 3. S. Heinrich and C. W. Henson, Banach space model theory, II: Isomorphic equivalence, Math. Nachr. 125 (1986), 301{317. 4. C. W. Henson, Some examples of stable Banach spaces, In preparation. 5. C. W. Henson and J. Iovino, Banach space model theory I: Basics, In preparation. 6. J. Iovino, Stable Banach spaces, I: Fundamentals, Submitted. 7. J. Kelley, General Topology, Van Nostrand, Princeton, 1955. 8. Y. Raynaud, Separabilite uniforme de L'espace des types d'un espace de Banach { Quelques exemples, Seminaire de Geometrie des espaces de Banach. Univ. Paris 7 2 (1983). Department of Mathematics Carnegie-Mellon University Pittsburgh, PA 15213 E-mail : [email protected]
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