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1. THE TSIRELSON SPACE 1.1. The Tsirelson space of Figiel and Johnson. Let us start this section by recalling the con- struction of the Tsirelson space of Figiel-Johnson [FJ74]. (The reader is referred to [CS89] for a comprehensive treatment of Tsirelson-like spaces.) If E,F are finite non-empty subsets of ℕ = {1, 2, 3, …}, we write E ≤ F if max E ≤ min F , and E ∑ Ex = anen. n∈E É A collection of m ≥ 1 nonempty sets E1, E2, … , Em of natural numbers is admissible if m ≤ E1 < ⋯ ⋅ ∞ E2 < < Em. Define a sequence of norms { k}k=0 on c00 inductively as follows. For fixed ∞ x ∈ c00, say x = n=1 anen, let x 0 =‖ max‖ an ∑ n and, for k ≥ 0, ‖ ‖ ð ð m 1 x = max x , max E x ∶1 ≤ m ≤ E < E < ⋯ < E , k+1 k 2 i k 1 2 m H T i=1 UI É ô ô where the‖ innermost‖ maximum‖ ‖ is taken over theô setô of all admissible collections of any size m ≥ 1. ∞ For x = n=1 anen, we have ∑ ∞ x ≤ x ≤ a = x l ; k k+1 n 1 n=1 É thus, each norm ⋅ is 1-Lipschitz‖ ‖ with‖ respect‖ to ⋅ð l ð. ‖ ‖ k 1 1. Definition. The Tsirelson norm ⋅ on c is defined as ‖ ‖ T 00 ‖ ‖ x T = lim x k . (x ∈ c00.) ‖ ‖ k→∞ ⋅ Tsirelson’s space, denoted , is the norm-completion of (c00, T ). (As customary, the extension ‖ ‖ ‖ ‖ ⋅ ⋅ of the Tsirelson norm to Tsirelson’s space is still denoted T .) The norm k will be called be the k-th iterate in the construction of the Tsirelson norm. ‖ ‖ ‖ ‖ ‖ ‖ ⋅ 1.2. Non-uniform convergence of the Tsirelson approximants. As in section 1.1 above, k will denote the k-th iterate in the construction of the Tsirelson norm on c00. (m) As a preliminary step, we need to construct certain functions of rapid growth. Recall that‖g‖ denotes the m-fold iteration g◦ … ◦g of a function g on a set X, with the usual convention that g(0) is the identity map on X.) 2. Definition. Let functions f1, f2, … , fk, … on natural numbers be as follows. For all m ∈ ℕ, let 2 (m) ℕ f1(m) = m , and recursively define fk+1(m) = fk (m) for k ∈ . 2m−1 Thus, f2(m) = m , but fk for k > 2 admits no explicit algebraic expression. THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 3 Bellenot had found essentially the same rapid-growth functions in connection, not with the con- vergence of approximants norms, but with James’ non-constructive proof that the standard basis 1 (en) of Tsirelson’s space is not subsymmetric [Jam64, Bel84]. ℕ The support {n ∈ ∶ an ≠ 0} of an element x = n∈ℕ anen ∈ c00 will be denoted supp(x). For c < d ∈ ℕ, we write [c, d) for {c, c +1, … , d − 1}. ∑ ⋅ 3. Theorem. For k ≥ 1, let k be the k-th iterate in the definition of the Tsirelson norm. For ≥ n every n 2 there are vectors {xi}i=1 in c00 such that: ‖ ‖ (j−1) (j) n (1) xj issupportedonasubsetof fk (n), 2 fk (n) ,andthesum x = j=1 xj is supported 4 t 1 ∑ on a subset of n, 2 fk+1(n) , (2) x = 1 for i =1, 2, … ,n, i k 2 √ (3) x k ≤ 1, (4) x ≥ n , and ô ôk+1 4 ô ô k−1 (5) x l ≤ 2 n. ‖ ‖ 1 We will‖ ‖ prove Theorem 3 by induction on k. ‖ ‖ ≥ (i−1) 2i−1 Proof of Theorem 3 for k =1 (Base step). Fix n 2. For i =1, 2, … ,n, let mi = f1 (n) = n , and Ei = [mi, 2mi)={mi,mi +1, … , 2mi − 1}. n Define vectors {xi}i=1 by 2m −1 1 i xi = ej. mi j=m Éi 2 Clearly, supp(xi) = Ei = [mi, 2mi) = mi, 2 f1(m) (since f1(m) = m ≥ 2m for m ≥ 2). (n) Thus, x is supported on a subset of m1, 2 f1(√mn) = n, 2 f1 (n) = n, 2 f2(n) , proving 4 1 n √ n 2mit−1 1 √ assertion (1). We also have x l = i=1 xi l = i=1 j=m = nmi∕mi = n, proving (5). For 1 1 i mi i =1, 2, … ,n, we have x = 1∕m , hence (2) follows from i 0 ∑i ∑ ∑ ‖ ‖ ô ô ô 2ômi−1 ô ô 1 1 mi 1 ô ô xi 1 = = = . 2 j=m mi 2mi 2 Éi ô ô By disjointness of the supports ofô theôxi, we have n n 1 1 1 n x 2 ≥ xi 1 = = , 2 i=1 2 i=1 2 4 É É ô ô proving (4). It remains to prove‖ (3).‖ Since theô normsô xi 0 = 1∕mi decrease with i, it follows that x 0 = x1 0 = 1∕m1 = 1∕n. Inordertofindanupper boundfor x 1,wefix q and any admissible ô ô ô ô 1Moreô specifically,ô Bellenot’s theorem shows that (e ) is not equivalent to the standard basis (e ) of Tsirelson’s ‖ ‖ ô ô nk ‖ ‖ n space if and only if (nk) asymptotically grows faster than (fk(k)) with fk as in definition 2 below. This answers a question of Casazza as to whether “reasonable” sequences (n ) yield (e ) equivalent to (e ). k nk n THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 4 ⋯ collection q ≤ F1 Fj ⊂ Ej, j=1 j=1 Í Í since, otherwise, Fj is unnecessarily capturing zero coefficients of x. Without loss of generality we may assume that q ∈ F1. (Otherwise, Σ is possibly enlarged when one appends to F1 the integers ≤ ≤ ≤ 2 between q and min F1.) Henceforth, we fix 1 p n such that mp q 2 b−1 b−1 We adopt the standard conventions j=b Zj = ç and j=b zj =0. ⋃ ∑ THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 5 ≤ ≤ n (1) supp(xij) ⊂ mij, 2 fk(mij) for 1 j mi, and the sum xi = j=1 xij is supported on a subset of mi, 2 f√k+1(mi) ; ∑ 1 (2) xij = for 1√≤ j ≤ mi; k 2 (3) xi k ≤ 1, (4) ôx ô ≥ mi , and ô i ôk+1 4 ô ô k−1 (5) ôxiôl ≤ 2 mi. ô ô 1 ô ô For eachô ôi, let ô ô ô ô xi yi = . 2 xi k+1 As shown above, the elements y (1 ≤ i ≤ n) together with their sum y = n y satisfy (1) of i ô ô i=1 i Theorem 3 with k +1 in place of k (y is supportedô onô a subset of n, 2 f (m ) , and f (m ) = k+1 ∑n k+1 n f (n) (n) = f (n)). Next, we have k+1 k+2 √ 1 y = , i k+1 2 xi 1 2 ô ôy = k ≤ = , ô ôi k 2m ∕4 m 2 xi k+1 i i ôn ô n ô ô 1 ô ô 1 1 n yô ô ≥ ô ô y = = , k+2 ô ô i k+1 2 i=1 2 i=1 2 4 É É ô ô proving properties (2) and (4)‖ for‖ k +1. Propertyô ô (5) follows from n n n k−1 xi l 1 1 1 2 mi k y l = yi l = ≤ =2 n. 1 1 2 2 m ∕4 i=1 i=1 xi k+1 i=1 i É É ô ô É ô ô ô ô It remains to show‖ ‖ that propertyô ô (3) holds, i.e., that y k+1 ≤ 1. Fix q and any admissible ⋯ ô ô collection q ≤ F1 < F2 < < Fq. As in theô ô proof of the case k = 1, we may assume that q = min F1, and that for some p we have mp ≤ q < 2 fk+1‖(m‖p). For each i,j, let Eij be the support mi l ≤ l ≤ of yij, and let Ei = j=1 Eij be the support of yi. There exists a unique (1 n) such that l √ −1 ≠ l ≤ ≤ (i) i=1 Fi ⊂ Ep, (ii) Fl ∩ Ep ç, and (iii) if < n, then Fl+1 ∩ Ep = ç. For 1 i n and ⋃ l 1 ≤ j ≤ q, let zij = Fjyi = (Ei ∩ Fj)y. We have zij =0 if i (by⋃ (iii)). We seek an upper bound for q q n 1 1 Σ ≔ Fjy = zij . k 2 j=1 2 j=1 i=p ô ôk É É ôÉ ô ô ô ô ô By the triangle inequality (and omitting zeroô terms):ô ô ô ô ô ô ô ô ô n q l n q 1 1 1 Σ ≤ zij = zpj + zij k k k 2 i=p j=1 2 j=1 2 i=p+1 j=l É É ô ô ¨É ô¨¨ ô É É ô ô ô ô = Σ + Σô . ô ô ô ô ô ô ô ô ô THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 6 On the one hand, l ¨ 1 1 Σ ≔ zpj ≤ yp = . k k+1 2 j=1 2 On the other hand, assuming (otherwiseÉ ô theô quantityô ôΣ¨¨ is zero), since = by definition p 2n fk+1(môp) ô ô ô ≤ ô ô q < 2 fkô+1(ômp) √mp+1 t 2n 2n 2 2 1 ≤ ≤ = ≤ < (m ≥ f (m ) ≥ f (n) ≥ n8). 3 3 p+1 k+1 p k+1 8 n 3 2 fk+1(n) n ≥ ≥ ¨¨ ≠ When n = 2, since√ k 1, we√ have m1 = 2 and m2 = fk+1(m1) f2(2) = 16. If Σ 0, then the range of index i in the outer sum defining Σ¨¨ is simply i = 2 = n = p +1, so we may replace nq by q on the first line of the estimate above: q 2 m2 2 2 1 Σ¨¨ ≤ < = ≤ = , m m 2 2 √2 m2 16 (2) ≥ (2) since m2 = fk+1(m1) = fk+1(2) = fk (2) f1 (2)√ = 16. √ It follows that q 1 ≤ Fiy k 1, 2 i=1 É and hence y k+1 ≤ 1. This proves property (3)ô forô k +1, completing the inductive step, and the proof of Theorem 3. ô ô 4. Proposition.‖ ‖ There exist sequences ∞ ∙ (xi)i=1 in c00, and ∞ ℕ ∙ (kj)j=1 strictly increasing in , such that ∙ x ≤ 1∕3 for j ≤ i, i kj ∙ xi k ≥ 2∕3 for j>i, and ô ô j ∙ limô ôj→∞ xi k = xi T =1 for all i. ô ô j The proofô ô of Proposition 4 hinges on the following Lemma. ô ô ô ô ô ô ô ô ¨ 5. Lemma. Given k ∈ ℕ, there exist k > k and x ∈ c00 such that x k ≤ 1∕3, x k¨ ≥ 2∕3, x T =1. ‖ ‖ ‖ ‖ ‖ ‖ THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 7 Proof. By Theorem 3 with n = 12, given k there exists y ∈ c00 such that y k ≤ 1 and y k+1 ≥ 3. ¨ 2 Since N ≔ y = sup y ≥ y =3, there exists k with y ¨ ≥ max N, 3 . The vector T l l k+1 k 3 x = y∕N satisfies: ‖ ‖ ‖ ‖ ∙ x k‖=‖ y k ∕N‖≤‖1∕N‖≤‖1∕3, ‖ ‖ 2 ∙ x ¨ = y ¨ ∕N ≥ N∕N = 2∕3, and k k 3 ∙ ‖x‖T = ‖y‖ T ∕N = N∕N =1. ‖ ‖ ‖ ‖ ∞ ¨ Since the sequence ( x k)k=1 is nondecreasing, the first two properties above imply that k > k. ‖ ‖ ‖ ‖ ¨ Proof of Proposition 4. Given k ∈ ℕ, let xk ≔ x and f(k) ≔ k be those given by Lemma 5. ‖ ∞‖ ℕ Define a sequence (kj)j=1 in recursively by ∙ k1 =1, and ∙ kj+1 = f(kj ) for j ∈ ℕ, i.e., k = f (j−1)(1). We claim that the sequences (x )∞ and (k )∞ satisfy the properties stated j ki i=1 j j=1 in Proposition 4. Since f is strictly increasing per Lemma 5, so is the sequence (kj); in particular, k → ∞ as j → ∞. Write N(i,j) for x . By Lemma 5, the monotonicity of the family { ⋅ } j ki k kj and of the sequence (k ), we have j ô ô ô ô ‖ ‖ ∙ For j ≤ i: N(i,j) ≤ N(i,i) ≤ô1∕3ô, ∙ For j>i: N(i,j) ≥ N(i,i + 1) ≥ 2∕3, ∙ For all x: lim x = x (since k → ∞ as j → ∞), and j→∞ kj T j ∙ For all i: xi T =1. ‖ ‖ ‖ ‖ 6. Proposition. There exist sequences (x )∞ in c , and (k )∞ strictly increasing in ℕ, such that ô ô i i=1 00 j j=1 ô ô ∙ the limit ≔ lim → x exists for each j ∈ ℕ, and j i ∞ i kj ∙ the limit Λ ≔ limj→∞ j exists, satisfies Λ ≤ 1∕3, and ∙ lim →∞ x = x ô =1ô for all i ∈ ℕ. j i kj i Tô ô Proof. ∞ ∞ Let (yi)i=1ô inô c00 andô ô(lj)j=1 be sequences satisfying the conclusions of Proposition 4. Let N(i,j) ≔ y ô.ô For fixedô ôj, the sequence {N(⋅,j)} eventually takes values in [0, 1∕3]; by se- i lj ∞ quential compactness of [0, 1∕3], it has a convergent subsequence, say {N(ni,j)}i=1. The choice of indexes (n )ô∞ ôrealizing the subsequence depends on the fixed choice of j; we will write n instead i ôi=1 ô ij of ni in order to exhibit the dependence explicitly. Thus, j ≔ limi→∞ N(nij,j) exists for each j, and ≤ 1∕3. Let x = y for each i, so (x ) is the “diagonal” subsequence of the family of se- j i nii i quences {(y ) ∶ j ∈ ℕ}. Once more, by sequential compactness, there is a subsequence (l )∞ nij i mj j=1 of (lj) such that the limit Λ ≔ limj→∞ j exists, and necessarily Λ ≤ 1∕3. The sequences (xi) and (k ) with k = l satisfy the following properties: For fixed j ∈ ℕ, (n ) is a subsequence of (n ) , j j mj ii i ij i so j = lim N(nij,j)= lim N(nii,j)= lim xi , i→∞ i→∞ i→∞ kj where the latter (subsequential) limit necessarily exists because (x )∞ is a subsequence of (y )∞ . iôi=1ô nij i=1 ô ô Finally, since (kj) is a subsequence of (lj), lim x = lim y = y =1. i kj nii nii j→∞ j→∞ lj T ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 8 2. FIRST-ORDER NON-DEFINABILITY OF THE TSIRELSON SPACE 2.1. Uniformly Multinormed Structures. Let V be any real vector space endowed with a point- wise bounded collection N of seminorms, i.e., such that for each x ∈ V there exists C ≥ 0 such that N(x) ≤ C for all N ∈ N. The (concrete) uniformly multi-normed (UmN) structure associated to (V, N) is the structure N ⋅ Norm cM = , Vn, 0n, +n, t,n,m,n, k ∶ m,n ∈ ℕ, m ≤ n, t ∈ ℝ , N where and Vn (for ∈ ℕ) are distinct sorts of and for all m ≤ n ∈ ℕ: ⟨ N ⟩ ∙ Vn ={x ∈ V ∶ N(x) ≤ n for all N ∈ }; ∙ 0n is the zero of V , ∙ m,n is the inclusion Vm ↪ Vn; ⋅ ∙ +n and t,n are (the restrictions of) the sum and scalar product-by-t on V to functions Vn × Vn → V2n and Vn → Vl where l = tn (the least integer ≥ tn ). Norm N ∙ n(N,x) = N(x) for each N ∈ and x ∈ Vn. The language for UmN structures is the first-order⌈ð ð⌉ (real-valued) languageð ð LUmN for general struc- Norm tures whose vocabulary consists of the sorts, constants, functions, and the predicates n above. We shall allow a slightly more general notion of concrete UmN structure, so as to obtain a first- order axiomatizable class (up to canonical identifications3) of general structures in Keisler’s sense. N We shall only require that V1 = N∈N BN (rN ) where, for each N ∈ , we have 0 < rN ≤ 1 and BN (rN ) is either the open or closed ball of radius rN relative to the seminorm N (some balls may 4 ⋂ be open and others closed) and, accordingly, letting Vn ≔ nV1 for all n ∈ ℕ. However, in this more ¨ general setting, V ≔ n Vn may be a proper subset of V (in which case, nevertheless, one obtains exactly the same UmN structure from (V ¨, N ↾ V ¨)). Using this more general notion of concrete UmN structure, a given⋃ nontrivial (V, N) typically admits many formally different associated UmN structures (i.e., in different Keisler isomorphism classes). The first-order theory (in real-valued logic) LUmN-theory of concrete UmN structures is de- noted ThUmN. Its models are (abstract) uniformly multinormed (UmN) structures, and consist of the following: ∙ A set (sort) N, called the norms sort. ∙ For all natural numbers m,n: – a set (“sort”) Vn and an element 0n ∈ Vn; Norm N – a real-valued predicate n on × Vn; ⋅ – for every t ∈ ℝ, a function t,n ∶ Vn → V tn ; – an operation +n,m ∶ Vn × Vm → Vn+m; ⌈ ⌉ – if n ≤ m, a function n,m ∶ Vn → Vm; and are models of ThUmN. Indeed, the zero elements and operations implicitly define a vector space V and a collection N of seminorms therein (one may construct these explicitly from the direct limit V ≔ lim V —relative to the identifications —with operations + and ⋅ induced by ,,,,,,,,,,→ n m,n ⋅ N the operations +m and t,n, plus one seminorm corresponding to each element N ∈ —it is induced Norm ⋅ by the functions n(N, ) as n varies). The theory of UmNstructures is necessary and sufficient N to ensure that V1 is the intersection of open or closed balls (in the seminorms N ∈ ) of radii at most 1. 3See Remarks 7 4 The sorts Vn first described above are obtained when the balls BN (1) are all of radius rN =1 and closed. THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 9 7. Remarks. (1) Whenever sensible, if is an UmN structure, N an element of N, and Norm x ∈ Vn , we write N(x) as an alias for the syntactically clumsy n (N,x). (2) If is a concrete UmN structure obtained, so is its reduction Red ≔ ∕ ≡ obtained under identification of elements of each Vn by the equivalence relation x ≡ y ⇔ N(y − x)=0 for all N ∈ N, and the correspondingly induced operations and interpretations of the Norm predicates. Moreover, and Red isomorphic in Keisler’s sense, although we prefer to say that they are in the same isomorphism class (one of the peculiarities of the theory of general classes is that there need not exist a bijective isomorphism between the universes of of Red —the quotient maps a ↦ a∕≡ from the universe of onto the universe of Red need not be injective). In summary, every UmN structure is concrete, up to reduction and isomorphism. Although the technical definition of UmN structures is needed to apply the Keisler formal- ism to spaces endowed with multiple norms, the preceding remarks essentially allow abstract- ∗ ing the sorts Vn and regard a uniformly multinormed structure as a “metastructure” = ⋅ N Norm ⋅ N V, 0, +, , , with a single vector sort V = n Vn with zero 0, operations + and , and a set of formal names (labels) for seminorms on N realized via the single predicate Norm (the latter ∗ ⋃ ingredients⟨ of are⟩ naturally induced from those of ). A critical feature of the metapredicate Norm Norm ⋅ ≤ N is its local boundedness on V (actually, (N, ) 1 on V1 for all N ∈ ) as alluded by the adverb “uniformly” (although Norm(N, ⋅) is typically unbounded on all of V ). In what follows, we mostly pretend away the technicalities of the multisorts Vn, and treat UmN metastructures in the above sense as though they are bona fide Keisler general structures. In par- ticular, we shall treat UmN structures as concrete ones (with Vk ⊆ Vl for k ≤ l, in particular). The phenomenon that many different (say, concrete) UmN structures may yield the same metastructure hints at an external notion of isomorphism classes (i.e., isomorphism classes of ≡-reduced UmN metastructures) typically larger than Keisler isomorphism classes. However, we shall not pursue this avenue of research presently as it is not relevant to the considerations of this manuscript. Still, when the multisorted nature of UmN structures is obscured by the metastructural viewpoint, we shall comment accordingly. 8. Remarks. ∙ The peculiar sort N whose elements formally name seminorms on V (rather than the seemingly more natural viewpoint of regarding each individual seminorm as a predicate) allows us to use the theory of stability and definability of types to study norms regarded as objects in the universe of the (meta)structure. ∙ Given a UmN structure (say, concrete, at least for purposes of exposition) and any col- ≤ lection S of seminorms on V such that N(x) 1 for all N ∈ S and x ∈ V1 , one obtains an elementary extension ⪰ by keeping the same vector sort V ≔ V , letting N ≔ S ⊔ N, and extending Norm to N × V by Norm(N,x) ≔ N(x) for N ∈ S and x ∈ V . For the rest of this section, will be a fixed multi-normed structure associated to (V, N) (thus V is a real vector space and and N is a pointwise bounded collection of seminorms on V ) and we will extend all the notational convention introduced in this subsection. THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 10 2.2. Types and Definability in UmN Structures. For our applications, we shall focus exclusively on mixed-sort types in UmN structures; these types describe potential properties of object(s) of one sort (V vs. N) in terms of parameters in the other sort. Fix a formula '(N; z) onan m-tuple N of formal variables of sort N, and an n-tuple z of variables of sort V . Given an UmN-structure and a set A ⊆ V , the '-type of N ∈ (N)m (with parameters in A) is the function ' ∶ An → ℝ N (2.1) (x1, … ,xn) ↦ ' (N; x1, … ,xn). The notion of ̃'-types, which are dual to the '-types above, is obtained by formally exchanging the roles of the variables N and x, i.e., ̃' types are types for the formula ̃'(x; N) ≡ '(N; x). Explicitly, the ̃'-type of the n-tuple x of elements of the universe V of an LNseq-structure , with parameters in B ⊆ N, is the function m ̃'x ∶ B → ℝ N ↦ '(N; x). The '-type of a tuple N ∈ (N)m is said to be realized in . The collection of these is denoted ' (A), where A ⊆ V is the set of parameters of these types. The collection '(A) of all '- types (with parameters in A) consists of the types of all m-tuples N of elements of sort N in all elementary extensions , a ∶ a ∈ A ⪰ , a ∶ a ∈ A . The set '(A) inherits the subspace topology from the product space RAm . The formally multisorted nature of (the metasort) V implies ℕ 5 that A ⊆ Vk for some⟨ k ∈ , and⟩ furthermore⟨ '(A) is⟩ compact (by logical compactness). These considerations translate mutatis mutandis to the sets ̃'(B) of all dual types (realized in a 6 fixed model , in the case of ̃' (B)). 2.2.1. Semidefinable Global Predicates and Definable Types. Fix a formula '(N; x). Throughout N this section is an arbitrary LUmN-structure and (N k) is a fixed sequence of m-tuples in . The n global ̃'-predicate P of kind V semidefined by the ultrafilter ∈ ℕ and (N k) is the class of all real-valued functions P ∶ (V )n → ℝ x ↦ lim ' (N k; x) ,k as ranges over elementary extension of . Concretely, each P is obtained in all ⪰ as the same ultralimit of the sequence (' ) of types realized in . Nk A semi-definable global ̃'-predicate P is definable over a set of parameters B ⊆ N if P is equal to the uniform limit of some sequence of types (with parameters in V ) that are realized in (though not necessarily of '-types). More concretely, P is indeed so definable precisely if 5 Formally, the variables in the tuple x are necessarily of sort Vk for some k, so ' sensu stricti only involves param- N N eters on A ⪯ Vk for (any) such k. On the other hand, since sort is discrete by definition, the parameter set B ⊆ is otherwise completely arbitrary. 6 k The spaces of dual types a priori each consist of types ̃'x with x of sort V for some k depending only on ̃' itself. THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 11 each P is obtained as the same continuous combination of the collection ' ∶ N ∈ Bm , i.e., N if there exists a continuous function C ∶ ℝBm → ℝ such that P (x) = C '(N; x) ∶ N ∈ Bm for all x ∈ (V ⟨)n. ⟩ A type p ∈ ' is (semi)definable over a set of parameters B if there exists a (semi)definable ⟨ ⟩ global ̃'-predicate Pp over B such that n p(x) = Pp (x) for all x ∈ (V ) . (2.2) Such predicate Pp defines p. 9. Remark. A definable ̃'-predicate P over admits an extension to a continuous function ̃' → ℝ, which we also call P by an abuse of notation. Via its definition Pp, a definable '-type p over admits a natural extension to a continuous function on ̃' . For details on definability and stability in the context of real-valued logic, the reader is referred to the literature [BYU10, BY14, Kei]. 2.2.2. Norm-Sequence Structures. The language LNseq for norm sequences expands LUmN with constant symbols (of the approximant norms) ∙ k (k ∈ ℕ) and ∙ (the symbol for the master norm) of sort N. Norm Norm We shalluse thesyntacticaliases x k (resp.,‖ x‖ ) for ( ∙ðððkððð,x) (resp., for ( ∙ ,x)). ↦ Norm ⋅ ⋅ The function x ( ∙ k ,x) on V will be denoted k (or just k, if is clear), and similarly for ⋅ (= ⋅ ). ‖ ‖ ððð ððð ‖ ‖ ððð ððð Given an Nseq structure ‖ and‖ an n-tuple x elements of the vector‖ ‖ sort of ‖ ‖, by formally iden- tifying ∙ with its index k, the ̃'-type ̃' —with parameters in the collection N ≔ { ∙ ∶ k ððð ððð ððð ððð x ℕ k k ∈ ℕ} of approximant norms—is a bounded real sequence (tk)k∈ℕ, tk = ' ( ∙ k ; x). Accord- N ℝℕ ingly, we‖ may‖ regard ̃' ( ℕ ) as a topological subspace of (with the product topology,‖ i.e.,‖ the topology of pointwise convergence). ‖ ‖ 10. Remarks. An Nseq structure may have additional functions, constants, predicate interpreta- tions, and possibly even other sorts. However, as long as the vocabulary L⊇LNseq for such a structure is fixed and countable, Keisler’s Expansion Theorem for general such L-structures im- plies that any L-theory T has a pre-metric expansion with a pseudo-metric approximate distance d. Although d is not canonical, the metric topology induced in models of T is. Thus, models of such a theory T are (multisorted) metric spaces, up to canonical reduction and metric equivalence. For structures with the exact vocabulary LNseq and any LNseq-theory T including the sentences sup Norm ( ∙ ,x) ≤̇ sup Norm ( ∙ ,x) for k, n ∈ ℕ, one may regard the sort N as x∈Vk k n x∈Vk k a discrete space, and the sorts Vk as metrized by d(x, y) ≔ y − x in models of T . Thus, for Tsirelson and Schlumprecht‖ ‖ structures as definedððð ððð below, the formalism of metric structures à la Ben Yaacov-Berenstein-Henson-Usvyatsov [BYU10, ?] sufficesððð for ourððð purposes. T 2.3. Tsirelson Structures. The classical Tsirelson structure isthe LNseq structure obtained from Tsirelson’s space , with ∙ k interpreted as the k-th Tsirelson approximant, and ∙ interpreted as the Tsirelson norm (via the Norm predicate, of course). A Tsirelson structure is‖ a‖ model of Th , the LNseq-theory of the classicalððð Tsirelsonððð struc- ture T . In general, the N-sort N of a Tsirelson structure contains elements not named by N the symbols ∙ , ∙ k (see Remark 8); still, any element N ∈ whatsoever still serves as the Norm name of a seminorm x ↦ N(x) ≔ (N,x) on the vector sort V . Th also ensures that ⋅ ⋅ N( ) ≤ ðððeverywhere.ððð ‖ ‖ ððð ððð THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 12 ⋅ In every Tsirelson structure we define the (external) Tsirelson norm T on in the obvious way, namely x ≔ lim x for all x ∈ V . (2.3) T k→∞ k ‖ ‖ The existence (and finiteness!) of the limit in (2.3) follows from the observation that is a model ̇ ̇ of Th , which includes the sentences‖ ‖ (∀x ∈‖V1‖) x k ≤ x l ≤ x for k ∙ a sequence {xi ∶ i ∈ ℕ} in Tsirelson’s space, and ∙ a sequence { ⋅ ∶ j ∈ ℕ} of approximants to the Tsirelson norm ⋅ , kj T such that ‖ ‖ lim xi = lim lim xi ≠ lim lim xi , ‖ ‖ (2.4) i→∞ T i→∞ j→∞ kj j→∞ i→∞ kj where all limits involved exist and are finite. ô ô ô ô ô ô ô ô ô ô ô ô Proof. With {xi} and {kj} chosen so the conclusions of Proposition 6 hold, all limits in the state- ment of Theorem 12 exist, and 1= lim xi = lim lim xi , i→∞ T i→∞ j→∞ kj but ô ô ô ô limôlimô xi ≤ 1∕3ô. ô j→∞ i→∞ kj 13. Proposition. Given a structure , the followingô ô properties are equivalent: ô ô (1) A formula ' is stable over a structure . 7 ⋅ In fact, T is the unique such limit point (ultralimit) when models Th . ‖ ‖ THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 13 ∞ (2) If a '-type p ∈ '( ) is an accumulation point of a sequence (pi)i=1 of '-types realized in , then p is definable over , and there exists a subsequence (p )∞ whose '-types ij j=1 converge point-wise on ̃'() to a definition of p. Furthermore, if such is the case, then every '-type is a pointwise limit (i.e., a limit in the logic topology) of realized types: ' = ' . Proof. The assertion is a particular case of the equivalence between properties (i) and (iii)in [BY14, Theorem 5]. ⋅ 14. Theorem (First-Order Non-Definability of the Tsirelson Norm). The Tsirelson norm T is not definable over LNseq-structures that satisfy either of the following properties: ∙ is an elementary extension of the classical Tsirelson space T , or ‖ ‖ ∙ is a countably saturated model of Th . Proof. As shown in section 2.3, Th implies that the Tsirelson norm (2.3) is the only global type Norm ⋅ semidefinable over the sequence of types k ≔ k of the approximant norms (independently of the choice of ∈ ℕ).8 In particular, the Tsirelson norm is the only accumulation point of any sequence ( ⋅ )∞ of the Norm-types of the approximant norms. ki i=1 ‖ ‖ Assume first that is an elementary extension of T . By Theorem 12, Norm is not stable in T , T hence neither‖ ‖ in ⪰ ; thus, by Proposition 13, the Tsirelson norm is not definable over , but we can be more specific. With (xi) and (kj) as in Theorem 12, let be any nonprincipal Norm∼ ultrafilter on ℕ, and let q = lim be the -ultralimit type of the realized dual types i, xi of (x ). If ⋅ were definable, then it would be equal to the pointwise limit of the types ⋅ for i T lj 9 ⋅ some subsequence (lj) of (kj), and the value (of the extension to dual types ) of T at q would be ≔ lim‖ ‖→ (q) = lim → lim x = lim → lim → x (since the latter limits‖ ‖ exist j ∞ lj j ∞ i, i lj j ∞ i ∞ i kj and equal the respective subsequential limit and ultralimit). On theother hand, again‖ by‖ definability, ⋅ Norm∼ T would be a continuous function onô dualô types, so its valueô at qô= limi, would equal ô ô ô ô xi ≔ lim q = lim x = lim → lim → x = lim → lim → x . However, i, T i, i T i ∞ k ∞ i k i ∞ j ∞ i kj ‖ ‖≠ per Theorem 12. Thus, in the sense explained, the dual type q witnesses the non-definability of ⋅ . ô ô ô ô ô ô T ‖ ‖ ô ô ô ô ô ô Next, let be a countably saturated model of Th . Consider the full dual type q (in a countable tuple z = (z )∞ of variables) of the sequence x = (x )∞ in V T above. This full dual type is the ‖ ‖ i i=1 i i=1 1 ⋅ collection ( ̃'x ∶ ' ∈ Φ) of all dual types ̃'x = '( ; x) as '(N; z) varies over the collection Φ of all L-formulas in which only finitely many of the variables zi appear free. The dual types 'x ∞ considered here are functions on the collection ⋅ T ∪ ⋅ T of all interpretations of the k k=1 norm symbol N in T (rather than functions defined only on the collection of norms ⋅ T ).10 Since k q is a type on countably many variables, countableððð saturationððð ‖of‖ ensures that it is realized by a sequence y = (y )∞ in V . For ease of exposition, let us assume that (x ) and (k ) were chosen i i=1 i j‖ ‖ 11 with the properties stated in Proposition 4. For each i, the element yi satisfies 8 Norm Recall that the relevant types and predicates are relative to the formula 1(N,x) with x of sort V1. The present Norm use of the subindex k in k should cause no confusion. 9See Remark 9 10 T ⋅ T ⋅ T In , the interpretation of the master norm is the Tsirelson norm T , but this need not hold in the model . (See Remark 11). 11 T T However, the external Tsirelsonððð ððð norm yi T need not equal xi = xi‖T‖ =1. óóó óóó óóó óóó ô ô óóó óóó óóó óóó ô ô THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 14 ∙ y ≤ 1∕3, if j ≤ i, i kj ∙ y ≥ 2∕3, if j>i, and i kj ô ô ∙ ô yiô =1, ô ô since theô correspondingô properties of xi are translated to an equality (or inequality) satisfied by the Norm∼ ⋅ ⋅ dual typeóóó óóó ∈ q at the points N = (or N = ), and y realizes q. Th ensures that, óóó óóó xi kj i ⋅ ≤ ⋅ ≤ ⋅ in models thereof, the interpretations of the norms satisfy k l pointwise for ≤ ⋅ ððð ððð ‖ ‖ k l. In particular, T is (finite) and everywhere defined on V . In fact ≤ ‖ ‖ ‖ ‖ ððð ððð yi T = lim yi k = sup yi k yi =1. ‖ ‖ j→∞ j k On the one hand, for all ôi ∈ôℕ, we haveô ô ô ô óóó óóó ô ô ô ô ô ô óóó óóó y ≥ y ≥ 2∕3, i T i ki+1 hence ô ô ô ô lim limô ôyi =ô limô yi ≥ 2∕3. i→∞ j→∞ kj i→∞ T ∞ Furthermore, for fixed j ∈ ℕ, the sequence { yi } takes values ≤ 1∕3 for i ≥ j; thus, ô ô ôkj ôi=1 ≤ ô ô ô ô limi→∞ yi k 1∕3, and hence j ô ô lim lim yôi kô≤ 1∕3. ô ô j→∞ i→∞ j This showsô ô that the formula Norm is not stable in , and we conclude that the Tsirelson norm is ô ô not definable over , by Theorem 12. ô ô 3. SCHLUMPRECHT’S SPACE 3.1. Construction of the Schlumprecht space. The construction of Schlumprecht’s space is sim- ilar to that of Tsirelson’s. We use the same notation as in section 1, except for the requirement that admissible families of finitely many nonempty finite subsets, say Ei (1 ≤ i ≤ m) of ℕ need ⋯ only satisfy E1 < E2 < < Em (and not necessarily the requirement E1 ≥ m as in Tsirelson’s L construction). Let L(t) = log2(t + 1) and (t) = t∕L(t) for t > 0. It is trivial to verify that both L and L are unbounded and strictly increasing. ⋅ ∞ Define the sequence { k=0 of norms on c00 (the Schlumprecht approximants) recursively as follows. For x = n anen ∈ c00, ‖ ‖ x 0 ≔ x l = max an , ∑ ∞ n and, for k ≥ 1, ‖ ‖ ‖ ‖ ð ð m 1 x = max E x ∶ E < E < ⋯ < E , k L(m) i k−1 1 2 m T i=1 U É where the maximum‖ is‖ taken over the set of allô admissibleô collections of any size m ≥ 1. (The reuse ⋅ ô ô of the notation k is convenient, but unrelated to the Tsirelson norms introduced in section 1.1.) Since L(1) = 1, we see that the inequality x k ≥ x k−1 holds for all x ∈ c00 and k ≥ 1, as witnessed by‖ the‖ singleton collection {E1}, where E1 = supp(x). The Schlumprecht norm of x ∈ c00 is defined by ‖ ‖ ‖ ‖ x = lim x k (= sup x k), k→∞ k∈ℕ ‖ ‖ ‖ ‖ ‖ ‖ THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 15 ⋅ and Schlumprecht’s space is the norm-completion of (c00, S ). The induced norm on is still denoted ⋅ by an abuse of notation. Routine induction shows that x ≤ a = x l for S k n n 1 x ∈ c00 and k ∈ ℕ. Thus, inequality x S ≤ x l also holds for all x ∈ c00 a priori and, a 1 ‖ ‖ ∑ posteriori‖,‖ for all x ∈ . ‖ ‖ ð ð ‖ ‖ Evidently, the Schlumprecht approximants‖ ‖ and‖ the‖ Schlumprecht norms depend only on the ab- solute values of the coefficients, and monotonically so. In contrast to Tsirelson’s construction, the admissibilityconditionfor families {Ei} in Schlumprecht’s case is invariant under the shift-by-j transformation i aiei ↦ i aiei+j for any fixed j ≥ 0. Since ⋅ = ⋅ l is also shift-invariant, it follows inductively that all approximants ⋅ are shift- 0 ∞ k ⋅ ∑12 ∑ invariant, and so is the Schlumprecht norm S itself. 3.2.‖ ‖ Non-uniform‖ ‖ convergence of the Schlumprecht approximants. ‖ ‖ ‖ ‖ 3.2.1. Auxiliary sequences of rapid growth. 15. Definition. Let L∗, L∗ be the integer-valued quasi-inverses of L, L, namely, for n ∈ ℕ, L∗ L−1 (n) = (n) = min{m ∈ ℕ ∶ m∕log2(m + 1) ≥ n}; ∗ −1 n L (n) = L (n)=2 −1 (=min{m ∈ ℕ ∶ log2(m + 1) ≥ n}). ⌈ ⌉ 16. Proposition. Given any function M ∶ ℕ → ℕ, there exist unique functions Λ ∶ (k, n) ↦ Λk(n) (k, n ≥ 1) ↦ i ≥ ∶ (k, n, i) k(n) (k, n, i 1) ↦ i ≥ ≥ ∶ (k, n, i) k(n) (k 2& n,i 1) taking values in ℕ such that the following identities hold: 1 (1) 1(n)=1; i+1 ∗ i i 2 (2) 1 (n) = L 2L[1(n)] = [1 + 1(n)] −1; i i i ≥ (3) k(n)=Λk−1[2 k(n)] (k 2); M (n) i (4) Λk(n) = i=1 k(n); 1 ≥ (5) k(n)=2 (k 2); i+1 ∑ ∗ i i i 2i ≥ (6) k (n) = L 2 L[k(n)] = [1 + k(n)] −1 (k 2). If M is strictly increasing then each of the functions , Λ, is increasing in each variable separately 1 1 ≥ (strictly increasing indeed—apart from the equalities 1(n)=1 and k(n)=2 for all k 2 and n ≥ 1). Proof. The existence and uniqueness of Λ, and are immediate, since the given conditions amount to a jointly recursive definition thereof. The asserted monotonicity of , Λ, conditional on M being strictly increasing is trivially verified. For k, n, i ≥ 1, define ⩽0 k (n)=0, and i ⩽i j k (n) = k(n). j=1 É 12 More generally, the Schlumprecht norm is invariant under monotone re-indexing transformations i aiei ↦ ⋯ aiej via any fixed choice of strictly increasing indexes 1 ≤ j1 < j2 < < ji < … . i i ∑ ∑ THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 16 ⩽M(n) ≥ (In particular, Λk(n) = k (n) for k 1.) 3.2.2. The main proposition. i ≥ 17. Proposition. There exist unique vectors Z(l), Xk(n), Yk(n) and xk(n) (k, l, n, i 1) in c00 such that (1) Z(l) = 1 l e ; L(l) i=1 i (2) X (n) = Y (n)∕ Y (n) ; k k∑ k k+1 M(n) i (3) Yk(n)=D i=1 xk(n); (4) xi (n) = Z i (n)ô ; andô 1 ∑1 ô ô (5) xi (n) = X 2ii (n) (k ≥ 2); k k −1 k (6) The support of Y (n) is I (n) ≔ 1, Λ (n) ; k k k (7) The support of xi (n) is 1,i (n) , and the coefficients of xi (n) in the direct sum (3) fill the k k k interval I i (n) ≔ ⩽i−1(n),⩽i(n) in Y (n), and also in X (n); k k k k k (8) xi (n) =1; k k (9) Yk(n) k ≤ 3; and (10) ôY (n)ô ≥ 3n. ô k ôk+1 ô ô ô ô To beginô theô proof of Proposition 17, note first that conditions (1)–(5) amount to a (unique) recursiveô definitionô of all the required vectors. Properties (6) and (7) are proved simultaneously by induction on k (and, for k fixed, by induction on i). The details are trivial and omitted. We prove that (8), (9) and (10) hold, for all n ≥ 2 and i ≥ 1, by induction on k ≥ 1. Throughout the proof, we write M for M(n). 3.2.3. The base of the induction. The proof for k =1 is as follows. We will presently write Y (n), j j j Ij, xj, j for Y1(n), I1 (n), x1(n), 1(n) (the last three expressions are constant in n). We remark l that, in the definition of ei , the admissible family achieving the maximum value, equal to i=1 1 l∕L(l) = L(l), is E ={i} (1 ≤ i ≤ L); therefore, Z(l) = 1 = xi for l,i ≥ 1 (by (1) and (4)), i ô∑ ô 1 1 proving (8) for k =1. ô ô i Mô ô ⋅ i Using the family {I } =1 in the definition of 2, we have EiY (ôn) ô1 = x 1 =1 (by translation ⋅ i ‖ ‖ ô ô invariance of 1), hence the lower bound ‖ ‖ ô ô ô ô M ô ô ô ô ‖ ‖ 1 M L Y (n) 2 ≥ 1 = = (M) ≥ 3n, L(M) 1 L(M) É ‖ ‖ since M = L∗(3n). This proves (10). To find an upper bound for Y (n) 1, note first that the coefficients of the basis elements ej in L i ⋅ Y (n) decrease with j (since the sequence 1∕ decreases with i). By the definitions of 0 and ⋅ 1, a moment’s reflection shows‖ that‖ a family realizing the maximum that defines Y (n) 1 must ≤ ≤ ≤ ≤ consist of, say, l singletons Ei = {i} for some l Λ(n) and 1 i l. Choose m M‖ so‖ that ‖ ‖ ‖ ‖ THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 17 l ∈ (⩽m−1,⩽m], and let l¨ = l − (⩽m−1) ≤ m. Then, we have m−1 i l¨ m−1 1 1 i l¨ Y (n) = xi[j] + xm[j] = + 1 ( ) ( ) L( i) L( m) L l H i=1 j=1 j=1 I L l H i=1 I m−1 É É É m−1 É ‖ ‖ L(i) l¨ L(i) L(l¨) = + ≤ + (since i,l¨ ≤ l) (3.1) L m m−1 L m i=1 L(l) L(l) ( ) i=1 L( ) ( ) Ém−1 É 1 ≤ +1 ≤ 2+1=3 (since l¨ ≤ m and L(i+1) ≥ 2L(i)). m−i−1 i=1 2 É This proves (9) and completes the proof of the base case k =1 of the induction. 3.2.4. The inductive step. Now we carry out the inductive step of the proof of (8), (9) and (10). ≥ i i i i Assume they hold for some k 1. Denote k+1(n), k+1(n), xk+1(n), Xk+1(n), Yk+1(n) and Ik+1(n) by ⋅ xi, X, Y , Ii, i and i. First, observe that Xk( ) k+1 =1 follows from the inductive hypothesis (2); therefore, xi k+1 =1 follows from (5), proving (8) for k +1. The intervals I (1 ≤ i ≤ M) satisfyô I Y ô = x = 1. Using these intervals in the i ô i ôk+2 i k+2 definitionô of ô⋅ we obtain ô ô k+2 ô ô ô ô Mô ô ô ô 1 i M L ‖ ‖ Yk+1(n) ≥ x (n) = = (M) ≥ 3n. k+2 k+1 k+1 L(M) i=1 L(M) É This proves (10)ô for k +1ô. ô ô ô ô ô ô To prove (9) for k +1, we start as in theô proofô of the base case. Since the coefficients of Y are decreasing, a moment’s reflection shows that, an admissible family achieving the maximum ⋯ that defines Yk+1(n) k+1 may be taken to consist of intervals E1 < E2 < < Em. By the ⋅ monotonicity of k, the maximizing Ej’s may be taken to be the back-to-back intervals such that m E = [1ô, Λ] = ôM I . i=1 i ô ôj=1 j A pair (j,i) with‖ ‖1 ≤ j ≤ m and 1 ≤ i ≤ M will be called relevant if Ej ∩ Ii is nonempty. ⋃We split the intervals⋃ [1,m] into two disjoint subclasses: (⊆) This class consists of those j such that Ej ⊆Ii for some i. (⊈) This is the complementary class consisting of those relevant j such that for no i is Ej a subset of Ii. Wewillusethesymbols“⊆”, “⊈” as nomenclaturefor thecorrespondingclass. Abusingthenomen- clature, we will say that i ∈ [1, M] isofclass ⊆ if (j,i) isofclass ⊆ for some j. Withthesenotations, we have ⊆ ⊈ Y k+1 = + , (3.2) where É É ⊆ ⊆ ‖ ‖ ⊈ ⊈ EjY EjY ≔ k , ≔ k . L(m) L(m) j ô ô j ô ô É É ô ô É É ô ô We will find an upper bound forô eachô of the two sums above. ô ô Case ⊆: Consider any fixed i ≤ M of class ⊆. Let mi be the number of indexes j of class ⊆ for i; they form an interval, say [ri,ri + mi − 1]. Since such Ej are disjoint nonempty subsets of Ii, THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 18 and #Ij = j, we have mi ≤ min{M,i}. Now, we have ⊆ ⊆ ⊆ r +m −1 Ej Y i i EjY = k = k . (3.3) L(m) L(m) j ô ô i j=ri ô ô ô ô ô ô É É ô ô É É ô ô as seen by using the family {E }si+mi−1 in the recursive definition of I Y , which is admissible j j=si 1 k+1 precisely because such (i,j) are of type ⊆. If there exists i ∈ [1, M] of class ⊆ such that m > , let p denoteô the largestô such j; otherwise, i i ô ô let p =0. Thus, every i>p of class ⊆ satisfies mj ≤ j. Continuing from (3.3), we write ⊆ ⊆ ⊆ = + , (3.4) ⩽p >p É É É ⊆ ⊆ ≤ where ⩽p, >p are the sums over (class-⊆) indices i p, i>p, respectively, on the right-hand side of (3.3). If p >∑0, we∑ have ⊆ ⊆ r +m −1 ⊆ i i EjY L(m ) I Y ≔ k ≤ i i k+1 L(m) L(m) ⩽p i≤p j=ri ô ô i≤p É É É ô ô É ô ô (by definitionô ofô ⋅ , since {ôE }rôi+mi−1 is an admissible family of subsets of I ) k+1 j j=ri i ⊆ ⊆ L(mi) xi k+1 L(mi) = =‖ ‖ (by property (8)—already proved for k +1) L(m) L(m) i≤p i≤p (3.5) É ô ô É p−1 ô ô L(i) ≤ 1 + (since mi ≤ i and m ≥ mp > p, by choice of p) i=1 L(p) Ép−1 p−1 p−1 L( ) L( ) 1 ≤ 1 + i = 1 + i = 1 + i i=1 L(i+1) i=1 L(i+1) i=1 2 É É Éi (since i < p for i ⊆ We have ⩽p = 0 if p = 0, i.e., if there are no class-⊆ indexes i such that mi > i, so the upper bound (equal to 1) on the right-hand side of inequality (3.5) remains valid in this case as well. ∑ ⊆ ⊆ ⊆ We now estimate >p. Just as in the case of ≤p above, have >p =0 if there is no index i>p (perhaps no index i at all) of class ⊆. Letting n =2i , we have ∑ i∑ i ∑ Yk(ni) 3 1 x = X (n ) = k ≤ = . (3.6) i k k i k 3n 2i Yk(ni) k+1 i i ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 19 We get, ⊆ ⊆ r +m −1 ⊆ r +m −1 ⊆ i i EjY i i I Y m x ≔ k = i k = i i k L(m) L(m) L(m) >p i>p j=ri ô ô i>p j=ri i>p É É É ô ô É⋅ É ô ô É ô ô (by monotonicityô ô of k, sinceôEj ⊆Iô i; furthermore,ô ô IiY is a shift of xi) (3.7) ⊆ ⊆ ∞ −i i xi 2 1 ≤ k ≤ ≤ (by (3.6), since m ≤ for i>p of class ⊆). ‖ ‖ i i i i>p L(m) i>p L(m) i=p+1 2 ô ô É ô ô É É Combining estimates (3.5) and (3.7) we obtain ⊆ ≤ 2. (3.8) É Case ⊈: For j of class ⊈, the relevant indexes i fill an interval Jj = [sj,sj + nj − 1] of length nj ≥ 2. For each i ∈ [1, M], let i be the number of j of class ⊈ such that (j,i) is relevant. Clearly, 0 ≤ i ≤ 2. Define Fji = Ej ∩ Ii. By the triangle inequality and monotonicity, ⊈ ⊈ ⊈ sj +nj −1 ⊈ sj +nj −1 EjY FjiY I Y ≔ k ≤ k ≤ i k L(m) L(m) L(m) j ô ô j i=sj ô ô j i=sj É É ô ô É É ô ô É É ô ô (3.9) ô ô ô ô M ô ô IiY xi 2 1 2 = k =2 k ≤ ≤ =1. i i i L(m) i L(m) L(m) i=1 2 i 1 ô ô ô ô É ô ô É ô ô É By (3.2), (3.8) and (3.9) we have proved (9) for k +1, completing the inductive step and the proof of Proposition 17. 3.3. Non-Definability of the Schlumprecht Norm. ⋅ 18. Theorem. Lemma5, Propositions4and6, andTheorem12holdfortheSchlumprecht norm S ⋅ ⋅ ⋅ and its approximants k in place of T and k therein. Proof. With the indicated one-for-one replacements, the proofs go through verbatim provided‖ ‖ at ‖ ‖ ‖ ‖ ‖ ‖ the beginning of the proof of the revised Lemma 5 we take y = Yk(3)∕3 as per Proposition 17 (with n =3), whose Schlumprecht approximants satisfy y k ≤ 1 and y k+1 ≥ 3. The classical Schlumprecht structure S is the Nseq structure obtained from Schlumprecht space ‖ ‖ ‖ ‖ by interpreting ∙ k as theSchlumprecht approximants, and the masternorm ∙ as the Schlumprecht S norm on . ThS is the LNseq-theory of . A Schlumprecht structure is any model of ThS . The following‖ ‖ analogue of Theorem 14 holds. Since the analogue ofððð Theoremððð 12 (for the Schlumprecht norm) holds, the proof is mutatis mutandis the same, and omitted. 19. Theorem (First-Order Non-Definability of the Schlumprecht Norm). The Schlumprecht norm ⋅ T is not definable over LNseq-structures that satisfy either of the following properties: ∙ is an elementary extension of the classical Schlumprecht space S , or ‖ ‖ ∙ is a countably saturated model of ThS . THE NON-DEFINABILITY OF THE SPACES OF TSIRELSON AND SCHLUMPRECHT 20 REFERENCES [Bel84] Steven F. Bellenot. The Banach space T and thefast growinghierarchyfromlogic. Israel J. Math., 47(4):305– 313, 1984. [BY14] Itaï Ben Yaacov. 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