Isomorphic universality and the number of pairwise non-isomorphic models in the class of Banach spaces

We study isomorphic universality of Banach spaces of a given density and a number of pairwise non-isomorphic models in the same class. We show that in the Cohen model the isomorphic universality number for Banach spaces of density $\aleph_1$ is $\aleph_2$, and analogous results are true for other cardinals (Theorem 1.2(1)) and that adding just one Cohen real to any model destroys the old universality of Banach spaces of density $\aleph_1$ (Theorem 1.5). Moreover, adding one Cohen real adds a weakly compactly generated Banach space of density $\aleph_1$ which does not embed into any Banach space of density $\aleph_1$ in the ground model. We develop the framework of natural spaces to study isomorphic embeddings of Banach spaces and use it to show that a sufficient failure of the generalized continuum hypothesis implies that the universality number of Banach spaces of a given density under a certain kind of positive embeddings (very positive embeddings), is high (Theorem 4.8(1)), and similarly for the number of pairwise non-isomorphic models (Theorem 4.8(2)).


Introduction
We shall be concerned with the isomorphic embeddings of Banach spaces, and in particular with the universality number of this class. For a quasi-ordered class (M, ≤), the universality number is defined as the smallest size of N ⊆ M such that for every 1 The author thanks EPSRC for the grants EP/G068720 and EP/I00498 which supported this research and the University of Wroc law in Poland for their invitation in October 2010 when some of the preliminary results were presented. I especially thank Grzegorz Plebanek for the many productive conversations during the development of this paper.
M ∈ M there is N ∈ N such that M ≤ N. In Banach space theory we find many examples of classes whose universality numbers have been studied, with respect to isomorphic, isometric and other kinds of embeddings. Some specific examples of this are given in sections §2 and §4. Indeed, some of the most classical results in Banach space theory come from these considerations, such as the result by Banach and Mazur [1](p.185) that C([0, 1]) is isometrically universal for all separable Banach spaces. The non-separable case is more complex. We shall note below that the question is only interesting in the context of the failure of the generalized continuum hypothesis, since GCH automatically gives one universal model for each uncountable density. The question has received considerable attention over the years, including some recent work such as that of Brech and Koszmider [3] who considered Banach spaces of density the continuum and proved that in the Cohen model for ℵ 2 many Cohen reals there is an isomorphically universal Banach space of density c = ℵ 2 . In this paper we shall also consider Cohen and Cohen-like models, and we shall prove results complementary to those of Brech and Koszmider. Specifically, in the model they considered, we shall see that the isomorphic universality number for Banach spaces of density ℵ 1 is ℵ 2 , and we shall also show that analogous results are true for other cardinals (Theorem 1.15 (1)). We shall also see that adding just one Cohen real to any model adds a Banach space of density ℵ 1 which does not embed into any such space in the ground model (Theorem 1.5). We note that in [4] it is stated (pg. 1268), without proof, that Koszmider and Thompson noted that a version of the proof from [3] gives a model where there is no isomorphically universal Banach space of density ℵ 1 . A further relation with the work of Brech and Koszmider is that in [4] they gave a model where 2 ℵ 0 = ℵ 2 and the universality number for the class of weakly compactly generated Banach spaces of density ℵ 1 is ℵ 2 , while we show that the same is true in the classical Cohen model for ℵ 2 Cohen reals (Thereom 1. 16).
A question related to that of universality is that of the number of pairwise nonisomorphic models. This is a well studied question in model theory (see Shelah's [15]) and it has received considerable attention in Banach space theory of separable Banach spaces, see Rosendal [14] for an excellent survey of this and related problems. In the non-separable context, there do not seem to be many results available. As a consequence of our non-universality results we show that in the Cohen model described above there is a family of 2 ℵ 1 many Banach spaces of density ℵ 1 which are pairwise non-isomorphic, and the same is true for other cardinals (Theorem 1.15 (2)) and wcg spaces (Thereom 1.16).
Moving away from forcing results, which give results valid in specific models of set theory, we would ideally like to establish results in the form of implications from cardinal arithmetic. An example is the above mentioned result that under GCH there is an isomorphically universal Banach spaces for every density. In the absence of GCH results of this type may be obtained by using the pcf theory. We explore this direction in §3 - §6, where we prove that for a special kind of embedding "very positive" (see Definition 4.4) a sufficient failure of GCH implies the nonexistence of universal Banach spaces in the class of spaces C(K) for K 0-dimensional, (Theorem 4.8 (1)) as well as the existence of a large number of pairwise non-isomorphic models (Theorem 4.8 (2)). While the use of forcing methods in the isomorphic Banach space theory is not new, as explained above, to our knowledge no pcf results have previously been obtained in this context. Our result required a development of a new framework, which we call natural spaces. These are model-theoretic structures which we use to talk about the spaces of the form C(K) indirectly. Model theory connected to pcf theory was used by Shelah and Usvyatsov [17] to study the isometric theory of Banach spaces and prove negative universality results about them. We review these results below. They can also serve as a motivation to a question that has interested us throughout, which is to what extent the universality number of the class of Banach spaces of given density depends on the kind of embedding considered. Let us now pass to some background results and notation. Throughout, κ stands for an infinite cardinal.
By combining the Stone duality theorem, the fact that any Banach space X is isometric to a subspace of C(B X * ) and that B X * has a totally disconnected continuous preimage, Brech and Koszmider proved the following: Fact 0.1 [Fact 1.1, [3]] (1) The universality number of the class of Boolean algebras of size κ is greater or equal to the universality number of the class of Banach spaces of density κ with isometric embeddings, which is greater or equal than the universality number of the class of Banach spaces of density κ with isomorphic embeddings. (2) The class of spaces of the form C(St(A)) for A a Boolean algebra of size κ is isometrically universal for the class of Banach spaces of density κ, and in particular its universality number with either isometric or isomorphic embeddings is the same as the universality number of the whole class of Banach spaces of density κ.
Fact 0.1 is only interesting in the context of uncountable κ, since for κ = ℵ 0 we have a universal Boolean algebra as well as an isometrically universal Banach space, as explained above. On the other hand, it is known from the classical model theory (see [5] for saturated and special models) that in the presence of GCH there is a universal Boolean algebra at every uncountable cardinal, so the questions of universality for the above classes are interesting in the context of the failure of the relevant instances of GCH. Negative universality results for Boolean algebras are known to hold when GCH fails sufficiently by the work of Kojman and Shelah [12], and in Cohen-like extensions by the work of Shelah (see [12] for a proof). Shelah and Usvyatsov proved in [17] that in the models where the negative universality results that were obtained for Boolean algebras in [12] hold, the same negative universality results hold for Banach spaces under isometric embeddings. The smallest cardinal at which these results can apply is ℵ 2 . For example, if λ is a regular cardinal greater than ℵ 1 but smaller than 2 ℵ 0 (so 2 ℵ 0 ≥ ℵ 3 ) there is no universal under isometries Banach space of density λ.
On the basis of what is known in the literature and what we obtain here, it is interesting to note that no known result differentiates between the universality number of Banach spaces of a given density under isometries or under isomorphisms. Furthermore, it is not known how to differentiate them from the universality number of Boolean algebras.
Conjecture 0.2 The universality number of the class of Banach spaces of density κ with isomorphic embeddings is the same as the universality number of the class of Boolean algebras of size κ.
It follows from the above discussion that Conjecture 0.2 would improve Fact 0.1 (1) and it would imply the negative universality results of Shelah and Usvyatsov. For all we know at this point, Conjecture 0.2 could be a theorem of ZFC, that is, it is not known to fail at any κ even consistently. A particular case of Conjecture 0.2 is the following Conjecture 0.3, which summarizes the most interesting case from the point of view of Banach space theory: Conjecture 0.3 The universality number of the class of Banach spaces of density κ with isomorphic embeddings is the same as the universality number of the class of Banach spaces of density κ with isometric embeddings.
In Banach space theory one studies other kinds of embeddings but isomorphisms and isometries, an example is described in §2. It would be interesting to refine the above conjectures in other contexts, for example in the context of embeddings with a fixed Banach-Mazur diameter, see §2 for a discussion. We now finish the introduction by giving some background information for the readers less familiar with Banach space theory.
Definition 0.4 A Banach space is a normed vector space complete in the metric induced by the norm. A linear embedding T : X → Y between Banach spaces is an isometry if for every x ∈ X we have ||x|| = ||T x||, where we use T x to denote T (x). A linear embedding T : X → Y between Banach spaces is an isomorphism if there is a constant D > 0 such that for every x ∈ X we have 1 D ||x|| ≤ ||T x|| ≤ D||x||.
Remark 0.5 Every isometry is an isomorphism. An isomorphism is in particular an injective continuous function and in fact, a linear map T is an isomorphism iff both T and T −1 are linear and continuous.
Throughout the paper letters A and B will be used for Boolean algebras, κ, λ for infinite cardinals and K and L for compact spaces. We shall write St(A) for the Stone space of a Boolean algebra A. Let us note that Fact 0.1 implies Observation 0.6 The universality number of Banach spaces of density κ, under any kind of embeddings, is either 1 or ≥ κ + . This is so because if for any α * ∈ [1, κ + ) we had that {X α : α < α * } were a universal family of Banach spaces of density κ, then we could assume that each X α = C(St(A α )) for some Boolean algebras A α of size κ. Therefore we could find a single algebra A of size κ such that all A α embed into it 2 and hence then C(St(A)) would be a single universal Banach space of density κ. 2 simply by freely generating an algebra by a disjoint union of all A α

Universals in Cohen-like extensions
It is a well known theorem of Shelah (see [12], Appendix, for a proof) that in the extension obtained by adding a regular κ ≥ λ ++ number of Cohen subsets to a regular λ over a model of GCH, the universality number for models of size λ + for any unstable complete first order theory is κ = 2 λ , so the maximal possible. This in particular applies to Boolean algebras. We explore to what extent this can be adapted to the class of Banach spaces of density λ by looking at variants of Cohen forcing. Our results show that Conjecture 0.2 holds in this special case, that is, in these circumstances we obtain the strongest possible negative universality result.
Let λ = λ <λ . Fix a set A = {a i : i < λ + } of indices which we shall assume forms an unbounded co-unbounded subset of λ + . Definition 1.1 The forcing P(λ) consists of Boolean algebras p generated on a subset of λ + by some subset w p of A of size < λ satisfying p ∩ A = w p . The ordering on P is given by p ≤ q if p is embeddable as a subalgebra of q and the embedding fixes w p , where in our notation q is the stronger condition.
Let us check some basic properties of the forcing P(λ), reminding the reader of the following notions: Definition 1.2 (1) A forcing notion P is said to be λ + -stationary cc if for every set {p i : i < λ + } of conditions in P, there is a club C and a regressive function f on C satisfying that for every i, j < λ + of cofinality λ satisfying f (i) = f (j), the conditions p i and p j are compatible.
(2) A subset Q of a partial order P is directed if every two elements of Q have an upper bound in P . A forcing notion P is (< λ)-directed closed if every directed Q ⊆ P of size < λ, has an upper bound in P. Lemma 1.3 (1) P(λ) satisfies the λ + -stationary cc and is (< λ)-directed closed.
Proof. For part (1), suppose that {p i : i < λ + } are conditions in P(λ). Let us denote by w i the set w p i and let us consider it in its increasing enumeration. Let the isomorphism type of w i be determined by the order type of w i and the Boolean algebra equations satisfied between the elements of w i , denote this by t(w i ). Note that by λ <λ = λ the cardinality of the set I of isomorphism types is exactly λ, so let us fix a bijection g : I × [λ + ] <λ → λ + . Note that there is a club C of λ + \ 1 such that for every point γ in C of cofinality λ we have i < γ iff w i ⊆ γ and g"( I × γ) ⊆ γ. Define f on C by letting f (j) = g(t(w j ), w j ∩ j) for j of cofinality λ and 0 otherwise. Hence f is regressive on C and if i < j both in C have cofinality λ and satisfy f (i) = f (j) then we have that w i ∩ i = w j ∩ j = w i ∩ w j and that p i and p j satisfy the same equations on this intersection. Hence p i and p j are compatible.
For the closure, note that a family of conditions in P(λ) being (< λ)-directed in particular implies that the conditions in any finite subfamily agree on the equations involving any common elements. Therefore the union of the family generates a Boolean algebra which is a common upper bound for the entire family.
For part (2), note that if p ∈ P and a i is not in w p , then we can extend p to q which is freely generated by w p ∪ {a i }, except for the equations present in p. Hence the set D i = {p : a i ∈ w p } is dense. Note that if p, q ∈ G satisfy p ∩ A = q ∩ A, then p and q are isomorphic over w p . Let H be a subset of G which is obtained by taking one representative of each isomorphism class with the induced ordering, hence H is still a filter in P and it intersects every D i . By the downward closure of G it follows that for every F a subset of A of size < λ, there is exactly one element p F of H with w p F = F . (Note that no new bounded subsets of λ are added by P, since (1) holds). Now note that {p F : F ∈ [A] <λ } with the relation of embeddability over A form an inverse system of Boolean algebras directed by ([A] <λ , ⊆). Let B be the inverse limit of this system (see [9] for an explicit construction of this object), so B is generated by {a α : α < λ} and it embeds every element of H. (We shall call this algebra "the" generic algebra. It is unique up to isomorphism). ⋆ 1.3 Note that in the case of λ = ℵ 0 , we in particular have that the forcing is ccc.

Theorem 1.4 If
A is the generic Boolean algebra for some P(λ), then there is no Banach space X * in the ground model such that C(St(A)) isomorphically embeds into X * in the extension by P(λ).
Proof. Let us fix a λ. Suppose for a contradiction that there is an X * and embedding T in the extension contradicting the statement of the theorem. Let p * force that T is an isomorphic embedding and, without loss of generality p * decides a positive constant c such that 1/c · ||x|| ≤ ||T x|| ≤ c · ||x|| for every x. Let n * be a positive integer such that c < n * . For each i < λ + let p i ≥ p * be such that a i ∈ w p i , and without loss of generality p i decides the value x i = T (χ [a i ] ). Now let us perform a similar argument as in the proof of the chain condition, and in particular by passing to a subfamily of the same cardinality we can assume that the sets w p i form a ∆-system with a root w * . Since a i ∈ w p i , we can assume that w i = w * , and in particular that a i ∈ w i \ w * . Let us take distinct i 0 , i 1 , . . . i n 2 * . Since there are no equations in p i or p j that connect a i and a j for i = j, we can find two conditions q ′ and q ′′ which both extend p i 0 , . . . p n 2 * and such that in q ′ we have a i 0 ≤ . . . ≤ a i n 2 * and in q ′ we have that a i 0 , . . . a i n 2 * are pairwise disjoint. Hence q ′ forces that χ [a i 0 ] + . . . χ [a i n 2 * ] has norm n 2 * + 1 and q ′′ forces that it has norm 1. Therefore the vector x i 0 + . . . x i n 2 by ℵ 2 Cohen reals, and the earlier proof by Džamonja and Shelah ([7], Theorem 3.4) that in the model considered there there is no universal normed vector space over Q of size ℵ 1 under isomorphic vector space embeddings. In fact in the case of λ = ℵ 0 we can expand on these ideas to prove the following Theorem 1.5. We shall use a simplified (ω, 1)-morass constructed in ZFC by Velleman in [18] 3 . Theorem 1.5 Forcing with one Cohen real adds a Boolean algebra A of size ℵ 1 such that in the extension the Banach space C(St(A)) does not isomorphically embed into any space X * in the ground model of density ℵ 1 .
For future use in the proof we shall need the following: The sequence θ α α<ω is a non-decreasing sequence diverging to ∞.
To see this, notice that the fact that id θα ∈ F α,α+1 guarantees that θ α ≤ θ α+1 . For the rest, suppose that θ α α<ω is bounded by some k < ω. This implies that there is α < ω such that for all γ ≥ α we have θ γ+1 = θ γ ≤ k and F γ,γ+1 = {id γ }. This implies by (4) that F α,ω = {id α }, in contradiction with (3). Fixing a simplified (ω, 1)-morass as in the above definitions, we shall define a Boolean algebra A as follows. By induction on α ≤ ω we define a Boolean algebra A α on a subset of θ α ∪ (ω 1 × {0}) generated by {i : i < θ α }, and at the end we shall have A = A ω . The first requirement of the induction will be: (i) if β < α and f ∈ F β,α then f gives rise to a Boolean algebra embedding.
This requirement guarantees that the final object A is indeed a well defined Boolean algebra. This is so because {A α : α ≤ ω} form an inverse system of Boolean algebras directed by the ordinal ω + 1, see [9] for the details. For the main part of the proof, suppose that we have Banach space X * in the ground model V with density ℵ 1 and a fixed dense set {z i : i < ω 1 } of X * in V. We shall guarantee that for all natural numbers n * ≥ 3 and for all j : To see that our algebra, once constructed, has the required properties, suppose that T is an isomorphic embedding of C(St(A)) into some X * as above and that ( * ) holds for a dense set case is similar. Now we claim that to guarantee the condition ( * ) for any fixed X * and {z i : i < ω 1 }, it suffices to assure that for all ( * * ) For this, let us recall that for every j : [Namely, if j is a name for a function j : ω 1 → ω 1 then for every α < ω 1 there is p ∈ G deciding the value j(α). Since the forcing notion is countable, there is p * in G such that the set A of all α for which p decides the value of α is uncountable, and then it suffices to define j 0 on A by j 0 (α) = β iff p forces j(α) = β.] In order to guarantee this, we first define by induction on n an increasing sequence of natural numbers α n such that α 0 = 0 and θ α n+1 > 2θ αn + n. In order to formulate the second requirement of the inductive construction we need a lemma. Lemma 1.7 Let n < ω. Then there is a set A n ⊆ θ α n+1 of size n + 1 such that for any β ≤ α n and f ∈ F β,α n+1 , the intersection A n ∩ f "θ β is empty.
Proof of the Lemma. Requirement (4) in the definition of the morass shows that it suffices to work with β = α n as for β < α n we have {h"θ β : h ∈ F β,α n+1 } ⊆ {f "θ αn : f ∈ F αn,α n+1 }. The proof proceeds on a backward induction on the size k of α n+1 − α n ≥ 1. In the case k = 1 we have that |F αn,α n+1 | ≤ 2 and for each f ∈ F αn,α n+1 we have |f "θ αn | ≤ θ αn and therefore θ α n+1 \ | f ∈Fα n,αn+1 f "θ αn | ≥ θ α n+1 − 2θ αn > n, so we can take A = θ α n+1 \ f ∈Fα n,αn+1 f "θ αn . For the step k + 1, let β = α n + 1 and then it suffices to again apply (4) from the definition of a morass. ⋆ 1.7 Our second requirement of the induction will be as follows, where r is the generic Cohen real, viewed as a function from ω to 2 : (ii) Let n < ω and suppose that A αn has been defined. Then A α n+1 is generated freely over A αn except for the equations induced by requirement (i) and the requirement that Given Lemma 1.7, it is clear that conditions (i) and (ii) can be met in a simple inductive construction of A a , as there will be no possible contradiction between (i) and the requirements that (ii) puts on the elements of A n . Let us now show that the resulting algebra is as required. Hence let us fix n * , X * , {z i : i < ω 1 }, A and j 0 as in ( * * ). Is it is then sufficient to observe that the following set is dense in the Cohen forcing (note that the set is in the ground model): p : (∃n ∈ dom(p))(n ≥ n 2 * and p(n where {i 0 , i 1 , . . . , i n 2 * } is the increasing enumeration of the first n 2 * + 1 elements of A n . ⋆ 1.5 We note in the following Theorem 1.9 that the construction from Theorem 1.5 can be refined so that the resulting Banach space is weakly compactly generated. This give rise to Theorem 1.16 which shows that in the Cohen model for ℵ 2 Cohen reals the universality number for wcg Banach spaces of density ℵ 1 is ℵ 2 . A model for this was obtained by Brech and Koszmider in [4] using a ready-made forcing. Theorem 1.16 answers a question raised by Koszmider (private communication). The reader not interested in wcg spaces can jump directly to after the proof of Theorem 1.9, those who proceed to read Theorem 1.9 will be assumed to know what wcg spaces are. Let us however recall one definition: Definition 1.8 A subset C of a Boolean algebra A has the nice property if for no finite F ⊆ C do we have F = 1. A Boolean algebra A is a c-algebra iff there is a family {B n : n < ω} of pairwise disjoint antichains of A whose union has the nice property and generates A.
Bell showed in [?] that the Stone space of a c-algebra A is a uniform Eberlein compact and hence C(St(A)) is a wcg space, see [4] for details. We prove: Theorem 1.9 Forcing with one Cohen real adds a c-algebra A of size ℵ 1 such that in the extension the Banach space C(St(A)) does not isomorphically embed into any space C(St(B)), where B is any Boolean algebra in the ground model of size ℵ 1 .
Proof. Fixing a simplified (ω, 1)-morass as in the proof of Theorem1.5, we shall define a Boolean algebra A as follows. By induction on α ≤ ω we define a Boolean algebra A α on a subset of θ α ∪ (ω 1 × {0}) generated by {i : i < θ α } freely except for the requirements in the next sentence, and at the end we shall have A = A ω . At the same time we shall define pairwise disjoint antichains B n α : n < n α such that n<nα B n α = {i : i < θ α } and this set has the nice property. The basic requirements of the induction will be: (i) if β < α and f ∈ F β,α then f gives rise to a Boolean algebra embedding.
Requirement (i) guarantees that the final object A is indeed a well defined Boolean algebra. The second requirement guarantees that the algebra obtained is a c-algebra, as will become clear later. As in the proof of Theorem 1.5 and using the same notation as there, we shall guarantee the requirement ( * * ).
Our final requirement of the induction will be as follows, where r is the generic Cohen real, viewed as a function from ω to 2 : (iii) Let n < ω and suppose that A αn has been defined. Then A α n+1 is generated freely over A αn except for the equations induced by requirements (i) and (ii) and the re- . . , i n } is the increasing enumeration of the first n + 1 elements of A n . For β ∈ (α n , α n+1 ) we let A β be the Boolean span in A α n+1 of θ β .
Given Lemma 1.7, it is clear that conditions (i) and (ii) can be met in a simple inductive construction of A a , as there will be no possible contradiction between (i) and the requirements that (ii) puts on the elements of A n . Let us comment on how to meet the requirement (iii), say at stage n + 1. We have chosen A n so that it elements are not in the image of any B m β for any β < α n+1 and any m < n β . Therefore we are free to either set n α n+1 = n α + 1 and put all elements of A n into one antichain, if r(n) = 0, or set n α n+1 = n α + |A n | and put each element of A n into a new antichain, requiring There are no problems in generating A α+1 freely except for these requirements and those dictated by (i)-(iii) as the requirements are not contradictory. Lemma 1.10 Suppose that the requirements (i)-(iii) from above are satisfied. Then letting for n < ω, B n = α≤ω,n<n α B n α , we obtain disjoint antichains which have the nice property in A and A is generated by their union.
Proof. (of the Lemma) As in the proof of Theorem 1.5, (i) guarantees that A is a Boolean algebra. Requirement (ii) guarantees that B n s are disjoint antichains. The set B = n<ω B n has the nice property since this property is verified by finite subsets of B n . In other words, if B does not have the nice property, then there is a finite F ⊆ B with ∨F = 1. There must be α < ω such that F ⊆ n<nα B n α , so ∨F = 1 already holds in A α . However, this is in contradictory to the fact that A α is generated freely except for the equations required in (i)-(iii). It is also clear B generates A. ⋆ 1.10 The proof that ( * * ) holds is exactly the same as in the proof of Theorem 1.5. ⋆ 1.9 We would now like to extend Theorems 1.4 and 1.5 by using "classical tricks" with iterations to obtain the non-existence of an isomorphically universal Banach space of density λ + in an iterated extension. It turns out that the cases λ = ℵ 0 and larger λ are different. Let us consider the former case first.
Scenario. Let P denote P(ω 1 ) and let Q be the iteration of ω 2 steps of P, say over a model of GCH, or simply the classical extension to add ℵ 2 Cohen reals over a model of GCH. Then in the extension by Q we have that 2 ℵ 0 = ℵ 2 and the new reals are added throughout the iteration. We try to argue that the universality number for the class of Banach spaces of density ℵ 1 is ℵ 2 , so we try for a contradiction. Suppose that the number is ≤ ℵ 1 , then by Observation 0.6 there is actually a single universal element, say X * . We can by Fact 0.1 assume that X * = C(St(A * )) for some Boolean algebra A * of size ℵ 1 . By standard arguments, we can find an intermediate universe in the iteration, call it V, which contains A * . If we could say that C(St(A * )) is also in V, then we could apply Theorem 1.4 to conclude that for the generic Boolean algebra B added to V, C(St(B)) does not embed into C(St(A * )) and we would be done. The problem is that since we keep adding reals, C(St(A * )) keeps changing from V to the final universe and hence there is no contradiction.We resolve this difficulty by a use of simple functions with rational coefficients. 4 . Theorem 1.11 Let κ be a cardinal with cf(κ) ≥ ℵ 2 and let G be a generic for the iteration of length κ with finite supports of the forcing to add the generic Boolean algebra of size ℵ 1 by finite conditions over a model V of GCH, or the forcing to add κ many Cohen reals over a model V of GCH. Then in V[G] the universality number of the class of Banach spaces of density ℵ 1 under isomorphisms is κ.
Proof. For simplicity, let us show that there is no single universal element, the proof in the case of < κ universals being very similar. Suppose that C(St(A)) is a universal Banach space of density ℵ 1 and without loss of generality, assume that A is in the ground model. Let p * forceṪ to be an isomorphic embedding from the C of the Stone space of the first generic algebra to C(St(A)) with ||T || ≤ c < n * . Let ε > 0 be small enough, We proceed similarly to the proof of Theorem 1.4, and without loss of generality assume that a i ∈ dom(p i (0)), for every i. Now do several ∆-system arguments as standard, and take "clean" i 0 < i 1 < . . . i n 2 * , hence n 2 * k=0 p i k does not decide any relation between a i k s. If ||Σ n 2 * k=0 h i k || ≥ n * , it follows by the choice of ε that there cannot be an extension of n 2 * k=0 p i k forcing a i k s and to be disjoint, a contradiction, and if ||Σ n 2 * k=0 h i k || < n * then it follows that there cannot be an extension of n 2 * k=0 p i k forcing a i k s to be increasing with k, again a contradiction. ⋆ 1.11 In the situation in which we add a generic Boolean algebra of cardinality λ + ≥ ℵ 2 with conditions of size < λ, the forcing is countably closed and therefore it does not add any new reals. Hence the attempted argument of proving the analogue of Theorem 1.11 for isomorphic embeddings works in this case. However, to make this into a theorem we need an iteration theorem for the forcing, in order to make sure that the cardinals are preserved. Paper [6] reviews various known forcing axioms and building up on them proves the following Theorem 1.12, which is perfectly suitable for our purposes. Theorem 1.12 (Cummings, Džamonja, Magidor, Morgan and Shelah) Let λ = λ <λ . Then, the iterations with (< λ)-supports of (< λ)-closed stationary λ + -cc forcing which is countably parallel-closed, are (< λ)-closed stationary λ + -cc.
Here, the property of countable parallel-closure is defined as follows: Definition 1.13 Two increasing sequences p i : i < ω and q i : i < ω of conditions in a forcing P are said to be pointwise compatible if for each i < ω the conditions p i , q i are compatible. The forcing P is said to be countably parallel-closed if for every two ω-sequences of pointwise compatible conditions as above, there is a common upper bound to {p i , q i : i < ω} in P.
Proof. Suppose that p i : i < ω and q i : i < ω are pairwise compatible increasing sequences. In particular this means that, on the one hand, each p i , q i agree on their intersections, and on the other hand that i<ω p i and i<ω q i each form a Boolean algebra. Furthermore i<ω p i and i<ω q i agree on their intersection, and hence their union can be used to generate a Boolean algebra, which will then be a common upper bound to the two sequences. ⋆ 1.14 Theorem 1.15 Let λ = λ <λ , let cf(κ) ≥ λ ++ and let G be a generic for the iteration with (< λ)-supports of length κ of the forcing to add the generic Boolean algebra of size λ + by conditions of size < λ over a model of GCH, or in the case of λ = ℵ 0 the forcing to add κ many Cohen reals. Then in V[G]: (1) the universality number of the class of Banach spaces of density λ + under isomorphisms is κ = 2 λ + .
(1) For the case of λ = ℵ 0 we use Theorem 1.11. In the case of uncountable λ, it follows from Lemma 1.3, Lemma 1.14 and Theorem 1.12 that the iteration of the forcing P(λ) described in the statement of the Theorem, preserves cardinals. By the countable closure of the forcing no reals are added, and hence the argument described in the Scenario, with λ + in place of ℵ 1 , works out to give the desired conclusion.
(2) The claim is that the Banach spaces C (St(A)) added at the individual steps of the iteration are not pairwise isomorphic. To prove this, we only need to notice that an embedding of a Banach space into another is determined by the restriction of an embedding onto a dense set, hence in our case a set of size λ + , and then to argue as in (1). ⋆ 1.15 Using the same reasoning as in Theorem 1.15 with λ = ℵ 0 and relating this to Theorem 1.9 we obtain Theorem 1.16 In the Cohen model for 2 ℵ 0 = κ, where cf(κ) ≥ ℵ 2 , the universality number of the wcg Banach spaces is at least cfκ and there are cf(κ) pairwise nonisomorphic wcg Banach spaces.

Isometries and isomorphisms of small Banach-Mazur diameter
As we have seen in §1, the Cohen-like extensions do not distinguish the universality number of Banach spaces of a fixed density under isomorphisms or under isometries, the subject of Conjecture 0.3. This short section gives evidence that some forms of that conjecture are just true in ZFC and serves as an interlude to motivate study of other kinds of embedding but isometries and isomorphisms. The results will be easy to a Banach space theorist as they use a well known theorem in the subject, due to Jarosz in [10].
is an isomorphic embedding such that ||T || · ||T −1 || < 2. Then there is a closed subspace L 1 ⊆ L which can be continuously mapped onto K.
We can use Jarosz's theorem to reduce the question of the existence of isometrically universal Banach spaces or more generally, universals under isomorphisms of small Banach-Mazur diameter to that of Boolean algebras and their ideals. Let us recall the following Observation 2.2 Suppose that L maps onto K, then C(K) isometrically embeds into C(L).
Proof. Let F be a continuos mapping from L that maps onto K. Defining for f ∈ C(K) T (f ) by T f (x) = f (F (x)) for x ∈ L gives the desired isometry. ⋆ 2. 2 We shall also use the following Proof. Let κ be fixed. Suppose that {X α : α < α * } are universal under isomorphisms of the Banach-Mazur diameter < 2 in the class of Banach spaces of density κ. By the introductory remarks, it follows that we can assume that for every α there is a Boolean algebra A α of size κ such that X α = C(St(A α )). The closed subspaces of St(A α ) are of the form St(A α /I) where I is an ideal in A α . Let A be a Boolean algebra of size κ and let K be it Stone space. Hence C(K) embeds with a Banach-Mazur distance < 2 in some X α . By Jarosz's theorem, there is a closed subspace L of St(A α ) such that L maps continuously onto K. Let I be an ideal on A α such that L = St(A α /I). It follows that A is a subalgebra of A α /I. Hence every Boolean algebra of size κ embeds in a factor of some {A α : α < α * }.
On the other hand suppose that {A α : α < α * } is a family of Boolean algebras of size κ such that every Boolean algebra of size κ embeds in a factor of some A α . Let X α = C(St(A α )) and we claim that {X α : α < α * } are universal under isometries in the class of Banach spaces of density κ. It suffices to show that for every Boolean algebra A of size κ, C(St(A)) isometrically embeds into some X α . Given such A, let α < α * be such that for some ideal I on A we have that A embeds into A α /I. Hence St(A) is a continuous image of the St(A α /I), which is a closed subspace L of X α . By Observation 2.2, C(St(A)) embeds isometrically into C(L). Let B a basis of C(L) as a vector space. By Tietze's extension theorem, every function f ∈ B can be extended to a function g ∈ X α = C(St(A α )) of the same norm. Let this association be called T . Now extend T to a linear embedding from C(L) 5 , which finally shows that C(L) and hence C(K) embeds isomorphically into X α . As the universality number of the class of Banach spaces of density κ under isometries is clearly ≥ the universality number of the class of Banach spaces of density κ with isomorphic embeddings of the Banach-Mazur diameter < 2, we obtain the desired equality. ⋆ 2.4 As we have seen in §2, there is a good motivation to study other kinds of embeddings of Banach spaces but isometries and isomorphisms. Sticking to the spaces of the form C(K), among the classically studied isomorphic embeddings are those that preserve multiplication, or the ones that preserve the pointwise order of functions. It is known for either one of them (Gelfand and Kolmogorov [8] for the former and Kaplansky [11] for the latter) that if they are onto they actually characterize the topological structure of the space, that is if T : C(K) → C(L) is an onto embedding which either preserves multiplication or the pointwise order, then K and L are homeomorphic. We shall show that in moving from the order preserving onto assumption just a small bit, we no longer have the preservation of the homeomorphic structure, but under the assumption that GCH fails sufficiently, we do have a large number of pairwise nonisomorphic spaces and a large universality number. The same methods can be applied to study embeddings with some amount of preservation of multiplication, which we shall not do here. Our methods will involve a combination of model theory, set theory and Banach space theory. In this section we introduce a simple model-theoretic structure which will be used to achieve that mixture of methods.
Suppose that A is a Boolean algebra. We shall associate to it a simple structure whose role is to represent the space C(K), where K is the Stone space of A, K = St(A). The idea is as follows. We are interested in the set of all simple functions with rational coefficients defined on K, so functions of the type Σ i≤n q i χ [a i ] , where each q i is rational, a i ∈ A and [a i ] denotes the basic clopen set in K determined by a i . Every element of C(K) is a limit of a sequence of such functions, since the limits of such sequences form exactly the class of Lebesgue integrable functions, which of course includes C(K). Let us then consider the vector space freely generated by A over Q, call it V = V (A). 6 Hence every simple function on K with rational coefficients corresponds uniquely to an element of V , via an identification of each a ∈ A with χ [a] . Using coordinatwise addition and scalar multiplication the product W = V ω becomes a vector space. Any function f in C(K) can be identified with an element of this vector space, namely a sequence of simple rational functions whose limit is f , and hence C(K) can be identified with a subset of W .
To encapsulate this discussion we shall work with vector spaces with rational coefficients and with two distinguished unary predicates C, C 0 satisfying C 0 ⊆ C. With our motivation in mind, we shall call them function spaces. If such a space (V, C, C 0 ) is the space of sequences of simple rational functions over a Stone space K = St(A) and C, C 0 correspond respectively to the set of such sequences which converge or converge to 0, then we call (V, C, C 0 ) a natural space and we denote it by N(A). In spaces of the form N(A) for an elementf of C N (A) we define ||f || as the norm in C(St(A)) of the limit f of f . If φ is an embedding between N(A) and N(B) we shall say that D > 0 is a constant of the embedding if for everyf of C N (A) we have that 1 D · ||f || ≤ ||φ(f )|| ≤ D · ||f ||. Not every embedding has such a constant, but we shall only work with the ones which do. We shall mostly be interested in a specific case of the representation of continuous functions as limits of simple functions, given by the following observation:

Lemma 3.1 Suppose that K = St(A) is the Stone space of a Boolean algebra
A and let f ≥ 0 be a function in C(K) with ||f || ≤ D * for some D * > 0. Then there is a sequence f n : n < ω of simple functions, where each f n is of the form Σ i≤n q i χ [a i ] , with each q i rational in (0, D * ] and a i ∈ A, such that f = lim f n . Proof. By multiplying by a constant if necessary, we can assume that D * = 1. Functions of the form Σ i≤n r i χ [a i ] , with each r i is real, contain the constant function 1, form an algebra and separate the points of K, hence by the Stone-Weierstrass theorem they form a dense subset of C(K). Notice that every function Σ i≤n r i χ [a i ] can be, by changing the coefficients and the sets a i if necessary, represented in the form where all [a i ]s are pairwise disjoint, so we can without loss of generality work only with such functions. Given ε > 0 and Σ i≤n r i χ [a i ] with a i s disjoint, we can find for i ≤ n rational numbers q i with |q i − r i | < ε, hence the function Σ i≤n r showing that also functions with rational coefficients and sijoint a i s are dense. Now given f ≥ 0 a function in C(K) with ||f || ≤ 1 and ε > 0, let Σ i≤n q i χ [a i ] be a function with rational coefficients and disjoint a i s satisfying ||f − Σ i≤n q i χ [a i ] || < ε, recalling that the || || in C(K) is the supremum norm. Define now: and consider the function Σ i≤n s i χ [a i ] . We claim that ||f − Σ i≤n s i χ [a i ] || < ε. By the assumptions that a i s are disjoint, for any x there is at most one Definition 3.2 Suppose thatf = f n : n > ω is a sequence in N(A) and suppose that f ′ = f ′ n : n > ω was obtained by first replacing each f n with an equivalent function Σ i≤n q i χ [a i ] with disjoint a i s and then replacing the coefficients q i by s i using the procedure described in the proof of Lemma 3.1. We say thatf ′ is a top-up off . Proof. Let f = lim n f n , hence f can be written as f = f + − f − where f + = max{f, 0} and f − = min{f, 0} are both continuous and positive. Therefore, by the closure of A under linear combinations it suffices to prove the Corollary in the case of f ≥ 0. Let D * = ||f ||, and we now apply Lemma 3.1. ⋆ 3.3 The point of these definitions is the connection between the embeddability in the class of spaces of the form C (St(A)) and the class of function spaces. Namely, we have the following Proof. Let T : C(St(A)) → C(St(B)) be an isomorphic embedding, so ||T || < ∞. We intend to define an isomorphic embedding φ from A to N(B). By linearity it is sufficient to work with the basis of A, so denote by A ′ the sequences of simple rational functions whose coefficients are in [0, 1]. Let us use the notation π n for the projection on the n-th coordinate. First we define the action of φ on those f ∈ A ′ which have the property that there is at most one n such that π n (f ) is not the identity zero function, and then π n (f ) is a function of the form χ [a] for some a ∈ A. If there is no such n with a = 0 then we let φ(f ) be the element of N(B) whose all projections π n are zero. Otherwise, let n be such that π n (f ) = 0 and consider T (π n (f )), which is well defined. We have no reason to believe that T (π n (f )) is a simple function with rational coefficients. However, there is a function F ((π n (f ))) which is a simple function with rational coefficients and whose distance to T (π n (f )) in C(St(B)) is less than 1 2 n+1 . We define φ(f ) to be the unique element g of N(B) such that the only π m (g) which is not identically zero is π n (g) and π n (g) = F ((π n (f ))). Now suppose thatf ∈ A ′ is such that for exactly one n, π n (f ) is not identity zero and π n (f ) = m i=0 q i χ [a i ] for some a 0 , . . . a m ∈ A and some rational q 0 , . . . q m in [0, 1]. For i ≤ m letf i be the element of N(A) whose n-th projection is χ [a i ] and all other projections are identity zero. Hence we have already defined φ(f i ) and we let φ(f ) = i≤m q i φ(f i ).
Finally suppose that f = f n : n < ω is any element of A ′ . Therefore for every n we have already defined g n = φ( 0, . . . , f n , 0 . . . )), where f n is on the n-th coordinate.
Let φ(f ) = g n : n < ω . Hence we have defined a linear embedding of A ′ to N(B). We extend this embedding to A by linearity. We need to check that this embedding preserves C and C 0 . So suppose thatf = f n : n < ω is in C N (A) ∩ A and letḡ = g n : n < ω be its image under φ. We shall show thatḡ ∈ C N (B) by showing that it is a Cauchy sequence. Let n, m < ω, we shall consider ||g n − g m ||. Clearly it suffices to assume that f ∈ A ′ . Let f n = i≤k q i · χ [a i ] and f m = j≤l r j · χ [b j ] . We have , which goes to 0 as n, m → ∞.

Invariants for the natural spaces and very positive embeddings
We shall now adapt the Kojman-Shelah method of invariants [12], to the natural spaces and a specific kind of isomorphic embeddings between Banach spaces, which we call very positive embeddings (see Definition 4.4). From this point on we assume that λ is a regular uncountable cardinal. (2) For a regular cardinal θ < λ we use the notation S λ θ for {α < λ : cf(α) = θ}. (3) A club guessing sequence on S λ θ is a sequence C δ : δ ∈ S λ θ such that each C δ is a club in δ, and for every club E ⊆ λ there is δ such that C δ ⊆ E.
Observation 4.2 Suppose that θ > ℵ 1 and there is a club guessing sequence C δ : δ ∈ S λ θ . Then there is a club guessing sequence D δ : δ ∈ S λ θ such that for all i < θ where α δ i : i < θ is the increasing enumeration of D δ , for each δ.
Proof. First of all notice that by passing to subsets if necessary we can without loss of generality assume that each C δ has order type θ. Given δ, let C ′ δ consists of the points of C δ of cofinality > ω and let D δ be the closure of C ′ δ in δ. Since θ > ℵ 1 we have that C ′ δ is unbounded in δ, so it is clear that D δ is a club of δ and since we have D δ ⊆ C δ , we obtain that the resulting sequence is a club guessing sequence on S λ θ . It also follows that otp(D δ ) = θ, so the increasing enumeration as claimed exists. The main definition we need is the definition of the invariant. Let us suppose that θ > ℵ 1 is regular and that D δ : δ ∈ S λ θ is a club guessing sequence with an increasing enumeration α δ i : i < θ of D δ , for each δ and satisfying the requirement (1). This sequence will be fixed throughout. The existence of such a sequence will be discussed at the end of the section but for the moment let us say that Shelah (see Theorem 4.7) proved that such a sequence exist in many circumstances, notably for any λ regular ≥ θ ++ .

Definition 4.3
Suppose that A is a Boolean algebra of size λ,Ā = A α : α < λ a filtration of A, that δ ∈ S λ θ and thatf We shall be interested in the kind of embeddings between Banach spaces which will allow us to define appropriate φ which preserve the invariants, see the Preservation Lemma 4.5. We have succeeded to do this in the case of a special kind of positive embeddings, as defined in the following definition.

Definition 4.4
We say that an isomorphic embedding T : C(K) → C(L) is very positive if the following requirements hold: We do not know if very positive embeddings were studied in the literature but clearly, one kind of embedding that is very positive, is an order preserving onto embedding. In this case we have Kaplansky's theorem [11] mentioned above, which shows that in the presence of such an embedding from C(K) to C(L) we have that K and L are homeomorphic. We show in the example in §5 that the analogue is not true for very positive embeddings, not even when they are assumed to be onto. In particular the question of the number of pairwise nonisomorphic by very positive embeddings spaces of the form C(St(A)) does not reduce to the well studied and understood question of the number of pairwise nonisomorphic Boolean algebras of a given cardinality (which for any infinite κ is always equal to 2 κ , see Shelah's [15]).
Let us now make a further assumption on λ: If φ : A → N(B) is an isomorphic embedding satisfying that lim φ(f ) = T (lim(f )) for everyf ∈ A, then there is a club E of λ such that for every δ with D δ ⊆ E and for everyf ∈ A ′ \ N(A δ ) with 0 ≤ lim n f n and || lim n f n || = 1 we have that invĀ ,δ (f ) = invB ,δ (φ(f )).
Proof. We may assume that the underlying set of A and B is the ordinal λ. Let us define a model M with the universe two disjoint copies of the ω-sequences of the simple functions on λ with rational coefficients, interpreted as the elements of N (A) and N(B), all the symbols of N(A) and N(B) with interpretations induced from these models, and the symbols A, A ′ and φ. By the assumption (3), there is a club E of λ such that for every δ ∈ E of cofinality not ω we have that M restricted to the sequences whose ordinal coefficients are < δ is an elementary submodel of M and that is has universe corresponding to N(A δ ) ∪ N(B δ ). Let us denote the latter model by M ↾ δ.
. Notice that the requirement (2) will hold if we replacef ′ by any top-upf ′′ (see Definition 3.2) as 0 ≤ f ≤ 1 and hence for all Since the topping up procedure is definable in M ↾ α δ i+1 , we may assume that 0 ≤ f ′ n ≤ 1 for all n and 0 ≤ f ′ ≤ 1. By Lemma 3.1 applied within M α δ i+1 we can assume thatf ′ ∈ A. By the choice of φ we have that lim φ(f ′ ) = T (f ′ ) and similarly 0 ≤ lim φ(f ′ ). By the fact that D δ ⊆ E and since φ is an isomorphism we have ) . We would like to use φ(f ′ ) to witness that i ∈ invB ,δ (φ(f )), so let us try. By the choice of φ we have that lim φ(f ′ ) = T (f ′ ) and similarly 0 ≤ lim φ(f ′ ). It remains to check the property (2) of φ(f ′ ).
Suppose for a contradiction that there isḡ ∈ N(B α δ i ) such that 0 ≤ g def = lim n g n ≤ |T f − T f ′ | but thatḡ / ∈ C 0 . Applying (ii) we can find h ≥ 0 with 0 ≤ T h ≤ g and h = 0. By Corollary 3.3 we can assume that there ish ∈ A with h = lim n h n and hence T h = lim n φ(h). Translating (ii) into the terms of φ and applying the elementarity of M ↾ α δ i we can assume thath ∈ N(A α δ i ). Now we apply (iii) to find s ≥ 0, s = 0 definable from h and satisfying s ≤ |f − f ′ |. Being definable from h, s has an approximations withs definable fromh, hences ∈ N(A α δ i ). By topping up if necessary as in Lemma 3.1 and in the above paragraph, we can assume that every element s n ins satisfies s n ≥ 0, therefores contradicts the choice off ′ . Now let us prove the other direction of the desired equality. Let i ∈ invB ,δ (φ(f )) as exemplified by someḡ ∈ N(B α δ i+1 ). As in the previous paragraphs, we can assume that 0 ≤ g def = lim n g n and hence by (ii) we can assume that for some f ′ ≥ 0 we have 0 ≤ T f ′ ≤ g and by the same argument as above we can assume that there isf ′ = f ′ n : n < ω ∈ N(A α δ i+1 ) ∩ A such that lim n f ′ n = f ′ . Now we claim thatf ′ exemplifies that i ∈ invĀ ,δ (f ). Suppose for a contradiction that h n : n ∈ ω ∈ N(A α δ i ) \ C 0 and 0 ≤ lim h n ≤ | lim f n − lim f ′ n |. Let h = lim h n . As before, we can assume thath ∈ A and each h n ≥ 0. So by the positivity we have 0 The next task is to construct lots of Boolean algebras A with different invariants for N(A) and then to us the Preservation Lemma to show that no fixed N(B) can embed them all. Lemma 4.6 (Construction Lemma) Suppose that θ + < λ. Then the club guessing sequence D δ : δ ∈ S λ θ can be chosen so that for any A ⊆ θ which is a closed set of limit ordinals, there is a Boolean algebra A = A[A], a filtrationĀ of A and a club E of λ such that for every δ ∈ E there isf ∈ A ′ \ N(A δ ) with invĀ ,δ (f ) = A ∩ S θ =ω and || lim f n || = 1.
The proof of this lemma is presented in the §6. The following theorem of Shelah will be used in the proof of the Construction Lemma as well as in the the proof of Theorem 4.8.
Proof. Fix sequences c δ : δ ∈ S ⊆ S λ θ and P α : α < λ as guaranteed by Theorem 4.7. Notice that c δ : δ ∈ S satisfies that with α δ i : i < θ being the increasing enumeration of c δ , we have that cf(i) = ω =⇒ cf(α δ i ) = ω. Hence letting D δ = c δ for δ ∈ S and D δ an arbitrary club of δ of order type θ satisfying cf(i) = ω =⇒ cf(α δ i ) = ω for δ ∈ S λ θ \ S, the sequence can be used in the context of the Preservation Lemma 4.5. It can also be used in the context of the Construction Lemma 4.6. Let us therefore find the Boolean algebras A[A] as described in the statement of the Construction Lemma. Notice that there are 2 θ many different choices for A ∩ S θ =ω . 1. Suppose for a contradiction that there is a family {C(St(A α )) : α < α * } for some α * < 2 θ for some algebras A α of size λ which is very positively universal for all C(St(A)) for Boolean algebras A of size λ. Notice that the assumptions we have made on λ imply that λ ℵ 0 = λ so the size of each N(A α ) is λ. Let F be the family of all subsets of θ that appear as invariants of elements of α<α * N(A α ), hence the size of F is < 2 θ , and in particular there is A ⊆ θ a closed set of limit ordinals such that An example of circumstances when Theorem 4.8 applies is when

Example
We give an example of two 0-dimensional spaces K and L which are not homeomorphic yet they admit a very-positive isomorphism onto. The example itself was constructed by Grzegorz Plebanek in [13], Example 5.3 when considering positive onto isomorphisms.
Let K consist of two disjoint convergent sequences x n : n < ω with lim n x n = x and y n : n < ω with lim n y n = y = x and let L consist of a single convergent sequence z n : n < ω with lim n z n = z. Define T : C(K) → C(L) by letting for all n T f (z 0 ) = f (y), Plebanek shows that T is a positive isomorphism onto C(L) and moreover he calculates the inverse S = T −1 which is given by We shall show that T is a very positive embedding. Considering property (ii) of Definition 4.4, suppose that g ∈ C(L) \ {0} with 0 ≤ g, we need to find h ≥ 0 with 0 ≤ T h ≤ g and h = 0. Let t in C(L) be such that t(z 0 ) = 0 and for n ≥ 1 the sequence {t(z n )} n is non-negative and not identically 0, converges to 0 and is bounded by the sequence {g(z n )} n . It is possible to find such a sequence since g ≥ 0 and g = 0. Let t(z) = 0, hence t ∈ C(L). Let h = St, and we note from the definition of S that h ≥ 0 and we have T h = t ≤ g.
For the property (iii), we shall have an existential proof of the existence of the s as required. Let S be the family of all non-negative functions in C(K) for which there is exactly one point with non-zero value, and on that point the value is equal to that of h. Each element of S is clearly definable from h. We claim that some s ∈ S can be chosen to demonstrate (iii). Namely, since we do not have T f ≤ T h, we cannot have f ≤ h by positivity. Hence there is some value w with h(w) < f (w), and by the continuity of the functions h and f , there must be some such w ∈ {x n , y n : n ∈ ω}. Then letting s(w) = h(w) and s(v) = 0 for v = w gives a function in S and we have 0 ≤ s ≤ f and 0 = s.

Proof of the Construction Lemma
We present a proof of Lemma 4.6. Let S ⊆ S λ θ and sequences c δ : δ ∈ S , P α : α < λ be as in the statement of Theorem 4.7, while D δ : δ ∈ S λ θ is such that D δ = c δ for δ ∈ S. For all the definitions of invariants we use here, the value of the invariant is the same with respect to c δ : δ ∈ S as it is with respect to D δ : δ ∈ S λ θ so we shall not make a difference between the two. We start with a construction lemma for a certain family of linear orders, as obtained by Kojman and Shelah in [12]. Let us give their definition of the invariants of linear orders: Definition 6.1 Suppose that L is a linear order with the universe λ and L = L δ : δ < λ is a filtration of L. Then for every δ ∈ S such that the universe of L δ is δ we define [12] proves that under the assumptions we have stated, for every closed set A of limit ordinals in θ there is a linear order L[A] with universe λ and a filtration The idea of our proof is to transfrom the Kojman-Shelah construction first into a construction of a family of Boolean algebras of size λ and then to use these Boolean algebras to define natural spaces of functions with appropriate invariants. Definition 6.2 Suppose that A is a Boolean algebra with the set of generators {a α : α < λ} andĀ = A δ : δ < λ is a filtration of A, while δ ∈ S is such that A δ is generated by {a α : α < δ}. We define i means that for any element w of A α δ i we have w ≤ a iff w ≤ b. Definition 6.3 Suppose that L is a linear order with universe λ. We define a Boolean algebra A[L] as being generated by {a α : α < λ} freely except for the equations Since the equations in (4) are finitely consistent with the axioms of a Boolean algebra it follows from the compactness theorem that the algebra A[L] is well defined. Now we shall see a translation between the calculation of the invariants of the linear orders and the associated Boolean algebras. Sublemma 6.4 Let L be a linear order on λ and A[L] the algebra associated to L as per Definition 6.3. LetL andĀ be any filtrations of L and A[L] respectively. Then there is a club E such that for every δ ∈ S ∩ E we have invL ,δ (δ) = invĀ ,δ (a δ ) and moreover, for any i ∈ invL ,δ (δ), this is exemplified by δ ′ iff i ∈ invĀ ,δ (a δ ) is exemplified by a δ ′ .
Proof. Let E be the club of δ such that the universe of L δ is δ, L δ is an elementary submodel of L, A[L] δ is generated by {a α : α < δ} and is an elementary submodel of A. Suppose that δ ∈ E ∩ S.
Suppose that z > 0 is in A α δ i and satisfies z ≤ a c α ∩ a δ . By the Disjunctive Normal Form for Boolean algebras, we can assume that z = i≤n j≤k i a l(i,j) β(i,j) for some l(i, j) ∈ {0, 1} and β(i, j) < α δ i . It suffices to prove that for every i we have j≤k i a l(i,j) β(i,j) ≤ a c α ∩a ′ δ . Fix an i and without loss of generality assume that j≤k i a l(i,j) β(i,j) > 0, as otherwise the conclusion is trivial.
Suppose now that δ < L β 1 . Therefore a β 1 ∩a c δ = 0. On the other hand, a c β 0 ∩a β 1 ∩a c δ ≤ a δ ∩ a α c ∩ a δ c = 0 and hence we must have a β 0 ∩ a β 1 ∩ a c δ > 0, which, taking into account β 0 < L β 1 gives that a β 0 ∩ a c δ > 0 and hence δ < L β 0 , a contradiction. Hence we have β 1 < L δ. By the choice of δ ′ we have β 1 < L δ ′ and hence a β 1 ≤ a δ ′ and in particular a c β 0 ∩ a β 1 ≤ a δ ′ , as required. Since the roles of δ and δ ′ in this proof were symmetric we can prove in the same way that for any z > 0 is in A α δ i which satisfies z ≤ a c α ∩ a ′ δ we also have z ≤ a c α ∩ a δ . Case 2. α > L δ, so α > L δ ′ by the choice of δ ′ . We have a c δ ≥ a c α , so a c α ∩ a δ = 0 and similarly a c α ∩ a ′ δ = 0. We also have a α ∩ a δ = a δ and similarly for δ ′ , hence we need to prove that a δ = a δ ′ mod A α δ i . As in Case 1, it suffices to show that for every β 0 , β 1 < α δ i with 0 < a c β 0 ∩ a β 1 < a δ , we have a c β 0 ∩ a β 1 ≤ a ′ δ (the equality cannot occur), and vice versa. Let us start with the forward direction. As before, from 0 < a c β 0 ∩ a β 1 we conclude β 0 < L β 1 . Also, if β 0 > L δ then we have a c β 0 ≤ a c δ , contradicting that a c β 0 ∩ a δ > 0. Hence β 0 < L δ.
If β 1 < L δ then β 1 < L δ ′ so a β 1 ≤ a ′ δ and hence a c β 0 ∩ a β 1 ≤ a ′ δ , as required. So assume that δ < L β 1 . Hence a β 1 > a δ and so a c β 0 ∩ a β 1 > a c β 0 ∩ a δ ≥ a c β 0 ∩ a β 1 , a contradiction. This finishes the proof of the forward direction, and the other direction follows from the symmetry of the roles of δ and δ ′ in the proof. Now suppose that i ∈ inv A,δ (a δ ) as exemplified by a ′ δ . Let α < α δ i , we need to prove α < L δ ⇐⇒ α < L δ ′ . If α < L δ then a α < a δ hence a α < a δ ′ by the assumption, and hence α < L δ ′ by the definition of A[L]. The other direction follows by symmetry. ⋆ 6.4 Sublemma 6.5 Let A[L] be one of the algebras described in the above andĀ its filtration. Then there is a club E of λ such that for every δ ∈ S with D δ ⊆ E, we have that invĀ ,δ (χ [a δ ] ) = invĀ ,δ (a δ ) and moreover, for any i ∈ invĀ ,δ (χ [ a δ ]), this is exemplified by χ [a ′ δ ] iff i ∈ invĀ ,δ (a δ ) is exemplified by a δ ′ . Here, the invariant on the left refers to the invariant in the natural space N(A) and the invariant on the right to the invariant in the algebra A. The notation χ [a] is used for the sequence χ [a] , χ [a] , χ [a] , . . . in C N (A) .
Proof. Let M * be a model consisting of L, A, two disjoint copies of the ω-sequences of the simple functions on λ with rational coefficients, interpreted as the elements of N(A) and all the symbols of N(A) with induced interpretations induced from these models. Recall the assumption that ∀κ < λ we have κ ℵ 0 < λ and notice that it implies that there is a club E 0 of λ such that for every δ ∈ E 0 of cofinality > ℵ 0 , the model M * ↾ δ is ℵ 1 -saturated in M * , that is it realizes all the types with countably many parameters in M * ↾ δ which are realized in M * . LetL be any filtration of L, let E ⊆ E 0 be a club witnessing Sublemma 6.4, and let δ ∈ S be such that D δ ⊆ E.
Suppose i ∈ invĀ ,δ (a δ ) as exemplified by a δ ′ butḡ ≥ 0 withḡ ∈ C N (A α δ i ) \ C 0 and the limit g ofḡ satisfies g ≤ |χ [ a δ ] − χ [ a ′ δ ]| = χ [a δ ∆a δ ′ ] . By topping up if necessary (see Definition 3.2) we may assume that each g n ≥ 0, and by throwing away unnecessary elements ofḡ we may assume that every g n = 0. We can then assume that for each n there are pairwise disjoint {b n 0 , . . . b kn n } ∈ A α δ i and q n i (i ≤ k n ) ∈ Q + such that g n = Σ i≤kn q n i χ [b n i ] . Since ||g n − χ [a δ ∆a δ ′ ] || → 0, there has to be a [b n i ] with a non-empty intersection with [a δ ∆a δ ′ ]. By applying the Disjunctive Normal form, we can assume that Claim 6.6 Suppose that i ∈ invĀ ,δ (χ [a δ ] ) as exemplified by somef . Without loss of generality we can assume thatf = χ [a ′ δ ] for some δ ′ . Proof of the Claim. First let us notice that if f = lim n f n then for f ′ = min{f, 1} we have |χ [a δ ] − f ′ | ≤ |χ [a δ ] − f | so we can without loss of generality assume that f ≤ 1. Similarly we can assume that f ≥ 0, and then by applying a similar logic we can also assume that 0 ≤ f n ≤ 1 for all n and that f n = 0. Each f n is a simple function with rational coefficients defined on (without loss of generality) disjoint basic clopen sets of the form [a l β β ] where β < α δ i+1 and l β < 2. Let {β n : n < ω} enumerate all the relevant β. For each n and R ∈ {< L , > L } let j n R be the truth value of "a βn Ra δ ". Consider the following sentence with parametersf , α δ i and the elements of {β n : n < ω}: there is β such that • for allḡ ∈ N(A α δ i ) if 0 ≤ lim n g n ≤ | lim f n − χ [a β ] |, we haveḡ ∈ C 0 , • for all n and R ∈ {< L , > L } we have a βn Ra β iff j n R = 1.
This sentence is true as exemplified by δ, so by the choice of E 0 it is true in M * ↾ α δ i+1 , say as exemplified by δ ′ . Let us note that in L we have δ < L δ ′ or δ ′ < L δ, let us assume that δ ′ < L δ, as the other case is symmetric. We claim that χ [a ′ δ ] exemplifies that i ∈ invĀ ,δ (χ [a δ ] ). If not, we can findḡ ∈ C N (A α δ i ) \ C 0 with 0 ≤ḡ and g def = lim n g n ≤ |χ [a δ ] − χ [a ′ δ ] | = χ [a δ ∆a ′ δ ] ≤ 1. By the triangle inequality it follows that g ≤ |χ [a δ ] − f | + |f − χ [a ′ δ ] |. We have that for every x both |χ [a δ ] − f |(x) and |χ [a δ ] − f |(x) are equal to f (x) if x ∈ [a c δ ] and 1 − f (x) if x ∈ [a δ ′ ]. The possible difference is on [a δ \ a δ ′ ], where the former function is equal to 1 − f (x) and the latter to f (x). We now claim that f is constant on [a δ \ a δ ′ ].
Clearly, it suffices to show that each f n is constant on [a δ \ a δ ′ ], and by the choice of {β n : n < ω}, it suffices to show that for each l < 2 and each n, χ [a l βn ] is constant on [a δ \ a δ ′ ]. Let β = β n for some n. If β ≤ L δ ′ then a β ≤ a δ ′ so χ [a β ] is constantly 0 on [a δ \ a δ ′ ]. In addition, we have a c β ≥ a c δ ′ ≥ (a δ \ a ′ δ ) so χ [a c β ] is constantly 1 on [a δ \ a δ ′ ]. If β ≥ L δ ′ then β ≥ L δ by the choice of δ ′ so a β ≥ a δ and hence χ [a β ] is constantly 1 on [aδ \ a δ ′ ]. In addition, a c β ≤ a c δ so χ [a c β ] is constantly 0 on [a δ \ a δ ′ ], and the statement is proved.