aa r X i v : . [ m a t h . L O ] A p r Model Theory For C p -Theorists Clovis Hamel and Franklin D. Tall April 15, 2020
This paper is dedicated to Prof. A. V. Arhangel’ski˘ı, whose research hasinspired the second author for more than fifty years, and who is the founder of C p -theory as a coherent subfield of general topology. Abstract
We survey discrete and continuous model-theoretic notions which haveimportant connections to general topology. We present a self-containedexposition of several interactions between continuous logic and C p -theorywhich have applications to a classification problem involving Banach spacesnot including c or l p , following recent results obtained by P. Casazza andJ. Iovino for compact continuous logics. Using C p -theoretic results in-volving Grothendieck spaces and double limit conditions, we extend theirresults to a broader family of logics, namely those with a first countableweakly Grothendieck space of types. We pose C p -theoretic problems whichhave model-theoretic implications. Introduction
It is perhaps impossible to find an area of modern mathematical research inwhich topology does not play a relevant role. This, of course, does not imply thata particular instance of this fact has to be of interest for topologists in general.This survey paper is intended to serve as an introduction to model theory,especially to continuous logics, for those who have had little or no previouscontact with this branch of mathematical logic, showing how several model-theoretic results can be equivalently translated into topological statements andhow answering C p -theoretic questions can lead to solving problems in the contextof continuous logics. In particular, we present new results concerning Gowers’problem [16] about the definability of a pathological Banach space. Research supported by NSERC grant A-7354.2020 Mathematics Subject Classification. 03C45, 03C75, 03C95, 03C98, 54C35, 46A50, 46B99.Key words and phrases: Tsirelson’s space, Gowers’ problem, explicitly definable Banachspaces, C p -theory, model-theoretic stability, definability, double limit conditions, Grothendieckspaces. L , two topologicalspaces are of interest: the space of L -structures and the space of types. TheCompactness Theorem holds if and only if those spaces are compact.Let c denote the space of eventually zero sequences of real numbers, c the space of sequences for which the limit is zero, and l p the space of sequences( h x n : n < ω i ) of real numbers such that P n<ω | x n | p < ∞ . The main topicpresented here is a classification problem in continuous logic, which involves afair amount of C p -theory. In 1974, B. Tsirelson [41] constructed a Banach spacewhich does not include a copy of l p or c . The construction involves a processof approximation that had its inspiration in Cohen’s forcing method. What isnow called Tsirelson’s space is due to T. Figiel and W. Johnson [15] and is thedual of the original space constructed by Tsirelson: let { x n : n < ω } be thecanonical basis for c . If x = P n<ω a n x n ∈ c and E, F ∈ [ ω ] <ω , denote P n ∈ E a n x n by Ex and write E ≤ F if and only if max E ≤ min F . The norm k k T of Tsirelson’s space T is constructed by an approximation process and isthe unique norm satisfying: k x k T = max {k x k c , max { P i We first set out some elementary notions that will be used throughout. For amore detailed exposition, see [36]. A language is a set of constants, functionsymbols and relation symbols, e.g. {∈} is the language of set theory, where ∈ is a binary relation, and { ¯ e, ∗ , − } is the language of group theory, where ¯ e is aconstant, ∗ is a binary relation and − is an unary function. First-order logic is the usual logic used in mathematics: given a language L formulas are builtrecursively as certain finite strings of symbols built using the members of L ,parentheses, variables, and the logical connectives ∧ , ∨ , → , ↔ , ¬ and quantifiers ∃ , ∀ . For the rest of this section let L be a fixed language. Sometimes it will beuseful to denote a formula ϕ by ϕ and ¬ ϕ by ϕ .An L -structure M is a set M , called the universe of M together with an interpretation of elements of L , i.e. a function which assigns an element of M to each constant symbol, a function from M n to M to each n -ary functionsymbol, and a subset of M n to each n -ary relation symbol. Given an L -formula ϕ , we denote by ϕ M the interpretation of ϕ in M . A substructure N of M isjust an L -structure, the universe of which is N , a subset of M , containing allthe same interpretations of symbols from M . For instance, in the language ofgroup theory, the formula ( ∀ x )( ∃ y )( x ∗ y = ¯ e ) is interpreted as ( ∀ x ∈ Z )( ∃ y ∈ Z )( x + y = 0) in the additive group Z .A variable is said to be free in a formula if it is not under the scope of anyquantifier; a sentence is a formula without free variables. A theory is a set of sen-tences and their logical consequences; a theory is consistent if it has a model anda maximal consistent theory is said to be complete (Note consistency is often de-fined proof-theoretically: not every sentence can be proved. The Completeness3heorem for first-order logic asserts that consistency is equivalent to satisfiabil-ity). For example, the axioms and theorems of group theory constitute a theory.If T is a theory, the satisfaction relation M | = T is defined recursively to mean M is a model of T , i.e. T is true in M . As examples, G | = axioms of group theory,for any group G , and R | = ( ∀ x )( ∀ y )( x ∗ y = y ∗ x ). A submodel N of M is elementary if for any L -formula ϕ ( x , . . . , x n ) and a , . . . , a n ∈ N , we have N | = ϕ ( a , . . . , a n ) if and only if M | = ϕ ( a , . . . , a n ); in this case we write N (cid:22) M . Two models M and N are elementarily equivalent if for any L -sentence ϕ we have M | = ϕ if and only if N | = ϕ ; this is denoted by M ≡ N .If M is an L structure and A ⊆ M , we can extend the language L to L ( A )by adding a constant symbol ˙ a for each a ∈ A . In this case, an L ( A )-formula issimply an L -formula with parameters from A .Given a language L , two interesting topological spaces arise: the space oftypes and the space of L -structures. We shall now introduce them and see howthe Compactness Theorem for first-order logic is equivalent to each of thosespaces being compact. Definition 2.1 Let M be an L -structure and A ⊆ M . A complete n -type over A in M is a maximal satisfiable set of L ( A ) -formulas in the variables x , . . . , x n ; i.e.a set p ( x , . . . , x n ) of formulas ϕ ( x , . . . , x n ) such that there are a , . . . , a n ∈ M such that for every ϕ ∈ p we have M | = ϕ ( a , . . . , a n ) . The set of all n -types isdenoted by S n ( A ) . Give a formula ϕ = ϕ ( x , . . . , x n ), we define[ ϕ ] = { p ∈ S n ( A ) : ϕ ∈ p } where the superscript is omitted if it is 1.Note that all sets of the form [ ϕ ] form a basis for a topology on S n . Moreover,since the negation: ¬ is available in first-order logic, each [ ϕ ] is clopen. We willsoon be dealing with other logics in which negation is not at our disposal andthe [ ϕ ]’s will only constitute a basis for the closed sets. Most of the time, itis enough to study 1-types, and then the same proofs can be carried out for n -types. Definition 2.2 Let Str ( L ) be the set of all equivalence classes under ≡ of L -structures. For each theory T let [ T ] = { ¯ M ∈ Str ( L ) : M | = T } where ¯ M is theequivalence class of M . All the sets of the form [ T ] constitute a basis for theclosed sets of the topology on Str ( L ) known as the space of L -structures. Wewrite [ ϕ ] instead of [ { ϕ } ] . Remark 2.3 Considering equivalence classes under ≡ instead of models is a wayto go around foundational issues: the upward L¨owenheim-Skolem theorem statesthat if a theory has a model of cardinality κ then it has a model of cardinality λ for every λ ≥ κ . Consequently, dropping the modulo ≡ would make Str( L ) andMod( T ) proper classes. Considering equivalence classes is equivalent to simplyconsidering the set of all consistent theories, which is a set of cardinality ≤ | L | .A topological consequence of mod-ing out by ≡ is that we guarantee that the4opology on Str( L ) is Hausdorff. This is not necessarily the case otherwise, sincethere could be two different elementarily equivalent structures M and N whichare not topologically distinguishable.Now we state a result which is a cornerstone of first-order logic: Theorem 2.4 (The Compactness Theorem) Let T be an L -theory in first-orderlogic. If T is finitely satisfiable, i.e. for any ∆ ∈ [ T ] <ω there is an L -structure M ∆ satisfying M ∆ | = T , then T is consistent. Proof. Notice that if equivalent L -formulas are identified, we get a Booleanalgebra and then its corresponding Stone space S is compact. An element ofthe Stone space is simply a complete L -theory. We prove the contrapositive.Suppose T is inconsistent and take an enumeration T = { ϕ α : α < κ } . Since T is finitely satisfiable, O = { [ ¬ φ α ] : α < κ } , where [ ¬ ϕ α ] = { x ∈ S : ¬ ϕ α ∈ x } covers S since T { [ ϕ α ] : α < κ } = ∅ . If { [ ¬ ϕ α i ] : i < n } is a finite subcover of O ,then T { [ ¬ ϕ α i ] : i < n } = ∅ and thus { ϕ α i : i < n } is a finite inconsistent subsetof T . (cid:3) The relationship between the Compactness Theorem and the space of struc-tures and the space of n -types is fundamental. Basically, a logic satisfies theCompactness Theorem if and only if its space of structures is compact, and ifand only if its space of n -types is compact. To formulate this precisely, we wouldneed to formulate precisely what a logic is, as is done for example in [14]. Wewon’t do this here; the following two theorems will easily be seen to apply towhatever logics are considered in this paper. Theorem 2.5 Given a logic and a language L , every finitely satisfiable set of L -formulas is satisfiable if and only if the space of L -structures Str ( L ) is compact. Proof. ⇒ ) Let { [ ϕ α ] : α < κ } be a centred family (i.e. every finite subfamily hasnon-empty intersection) of basic closed sets of Str( L ). Since [ ϕ α ] ∩ [ ϕ β ] = [ ϕ α ∧ ϕ β ], T = { ϕ α : α < κ } is finitely satisfiable. The Compactness Theorem yieldsa model M of T , i.e. M ∈ T { [ ϕ α ] : α < κ } . ⇐ ) If T = { ϕ α : α < κ } is finitelysatisfiable, then { [ ϕ α ] : α < κ } is a centred family and any M ∈ T { [ ϕ α ] : α < κ } is a model of T . (cid:3) Theorem 2.6 Given a logic, a language L and a n < ω , every finitely satisfiableset of L -formulas is satisfiable if and only if the space of types S n is compact. Proof. ⇒ ) Let { [ ϕ α ] : α < κ } be a centred family of basic closed sets of S n .Then { ϕ α : α < κ } is finitely satisfiable, and so it is satisfiable, i.e. there isan L -structure M with a , . . . , a n ∈ M such that for every α < κ we have M | = ϕ α ( a , . . . , a n ). By an application of Zorn’s lemma, T can be extended toan n -type p . Clearly, p ∈ T { [ ϕ α ] : α < κ } . ⇐ ) If T = { ϕ α : α < κ } is finitelysatisfiable then { [ ϕ α ] : α < κ } is a centred family of closed sets in S n . Take p ∈ T { [ ϕ α ] : α < κ } ; then p is a type and so it is satisfied by a model M . Inparticular M | = ϕ α for each α < κ . (cid:3) Stability and Definability in First-Order Logic It is a fruitful area of research to consider logics other than first-order logic.However, one cannot expect the Compactness Theorem to hold in every case.The following sections are devoted to presenting an exposition of logics withstronger expressive powers and showing how to use different topological toolswhen the Compactness Theorem does not hold.We first examine the concept of stability which has been a keystone of modeltheory since Shelah [32] introduced it. The purpose of this section is to showthat studying stability is a natural way to approach definability. Definition 3.1 If x is a n -tuple of variables, we write l ( x ) = n . Let M be an L -structure, ϕ ( x , y ) an L -formula where l ( x ) = n and l ( y ) = m , A ⊆ M and b ∈ M n . An n - ϕ -type over A in M is a set of formulas tp ϕ ( b , A, M ) = { ϕ t ( x , a ) : a ∈ A m ∧ t < ∧ M | = ϕ t ( b , a ) } . Thus, the space of n - ϕ -types is S nϕ = { tp ϕ ( b , A, M ) : b ∈ M n } . Again, we omit the n from the notation when itis . Definition 3.2 Let T be an L -theory (i) A model M of T is stable on an infinite cardinal λ if for every A ⊆ M such that | A | ≤ λ we have |S ( A ) | ≤ λ . (ii) T is stable if all its models are stable. (iii) An L -formula ϕ ∈ T is stable in M on an infinite cardinal λ if for every A ⊆ M such that | A | ≤ λ we have that |S ϕ ( A ) | ≤ λ . Since the previous definition of stability is usually cumbersome to work with,we present several equivalent definitions. For a more detailed exposition, see[33]. Despite a superficial resemblance to the concept of stability in C p -theory,we have not been able to find a nice connection between the two. Definition 3.3 An L -formula ϕ ( x , y ) has the order property in M if there aresequences h a i : i < ω i and h b j : j < ω i of tuples from M such that, for every i, j < ω , M | = ϕ ( a i , b j ) if and only if i < j . Remark 3.4 It is a consequence of the Compactness Theorem and Ramsey’sTheorem that having the order property is symmetric in x and y , i.e. if ϕ ( x , y )has the order property, so does ϕ ( y , x ). For a detailed proof of this fact, see[33] or [36]. Theorem 3.5 Let T be an L -theory. Then the following are equivalent: (i) T is stable on some λ ≥ ℵ . (ii) T is stable on every λ ≥ ℵ . (iii) Every formula ϕ ( x , y ) in T is stable on some λ ≥ ℵ in every model of T . iv) Every formula ϕ ( x , y ) in T is stable on every λ ≥ ℵ in every model of T . (v) For every formula ϕ ( x , y ) in T and every model M of T , ϕ does not havethe order property in M . For a proof see Shelah [33].We now state the classical theorem from first-order model theory whichrelates the concepts of stability and definability. Definition 3.6 Let M be an L -structure and A, B ⊆ M (i) We say that A is B -definable if there is an L ( B ) -formula ϕ ( x ) such that A = { x ∈ M : M | = ϕ ( x ) } . (ii) An n - ϕ -type over A in M is B -definable if there is an L -formula ψ ( x , y ) and a tuple b from B so that for every a ∈ A n , ϕ ( x , a ) ∈ p if and only if M | = ψ ( a , b ) . The following is a classical theorem, the proof of which can be found in theliterature. See for instance, [33] or [36]. Theorem 3.7 (The Definability Theorem) An L -formula ϕ is stable in M if andonly if for every A ⊆ M and for every p ∈ S ϕ ( A ) , p is A -definable. As done in [8] for compact continuous logics, a reasonable scheme for approach-ing the problem of the definability of Tsirelson-like spaces consists of the fol-lowing: (i) Extend the notions of stability and definability from first-order logic tothe logic in question. (ii) Prove an analogue of the Definability Theorem. (iii) Establish a relationship between stability and double limit conditions. This is where C p -theory plays a fundamental role. (iv) Decide on the definability in that logic of Tsirelson’s space (or any otherBanach space) by using the double limit condition.Given a model M , it is convenient to also view an L -formula ϕ ( x, y ) as a function ϕ : M × M → ϕ ( a, b ) = 1 if and only if M | = ϕ ( a, b ). Then a ϕ -type q = tp ϕ ( a, A, M ) can be identified with the function a → ϕ ( a, b ) (the ϕ -type isrecovered by taking the pre-image of { } ).In the following chapters, stability will be defined in terms of double-(ultra)limitconditions. We remind the reader that if U is an ultrafilter over κ and h x α : α <κ i is a κ -sequence, we write lim α →U x α = x if for every neighbourhood U of x , { α : x α ∈ U } ∈ U . As a motivation, we see that this is the case in first-orderlogic, following Iovino [20]: 7 emma 3.8 Let ϕ : A × B → [0 , be a function, h a n : n < ω i and h b m : m < ω i be sequences in A and B respectively and U and V ultrafilters on ω . If lim n →U lim m →V ϕ ( a n , b m ) = α and lim m →V lim n →U ϕ ( a n , b m ) = β then there are subsequences h a n i : i < ω i and h b m j : j < ω i such that lim i Given ε > (i) ( ∃ U ε ∈ U )( ∀ n ∈ U ε ) | lim m →V ϕ ( a n , b m ) − α | < ε (ii) ( ∃ V ε ∈ V )( ∀ m ∈ V ε ) | lim n →U ϕ ( a n , b m ) − β | < ε (iii) ( ∀ m < ω )( ∃U mε ∈ U )( ∀ n ∈ U mε ) | ϕ ( a n , b m ) − lim n →U ϕ ( a n , b m ) | < ε . (iv) ( ∀ n < ω )( ∃V nε ∈ V )( ∀ m ∈ V nε ) | ϕ ( a n , b m ) − lim m →V ϕ ( a n , b m ) | < ε .Notice that (i) and (ii) follow from the convergence hypothesis, and (iii) and (iv) follow from the compactness of [0,1]. Construct recursively two increasingsequences h n i : i < ω i and h m j : j < ω i such that n ∈ U , m ∈ V ∩ V n and n i ∈ U /i +1 ∩ T k
Theorem 3.9 Given a formula ϕ ( x, y ) , the following are equivalent: (i) ϕ ( x, y ) is stable. (ii) For any sequences h a n : n < ω i and h b m : m < ω i in M , and any ultrafilters U and V on ω , lim n →U lim m →V ϕ ( a n , b m ) = lim m →V lim n →U ϕ ( a n , b m ) . Proof. ( ii ) → ( i ) Assume that ϕ has the order property witnessed by h a n : n < ω i and h b m : m < ω i . Then lim m →∞ ϕ ( a n , b m ) = lim m →V ϕ ( a n , b m ) = 1 for anyfixed n < ω . Similarly, lim n →∞ ϕ ( a n , b m ) = lim n →U ϕ ( a n , b m ) = 0 for any fixed m < ω . So lim n →U lim m →V ϕ ( a n , b m ) = 1 and lim m →V lim n →U ϕ ( a n , b m ) = 0.( i ) → ( ii ) By the lemma, there are subsequences h a n i : i < ω i and h b m j : j < ω i such that: lim i The logics that we will consider have their motivation in overcoming some noto-rious limitations of first-order logic, namely that the only possible truth valueslie in { , } , leaving little room for the usual ε -play and approximations whichare fundamental in analysis, and that only finite conjunctions are allowed, re-stricting certain recursive definitions. We will first look at continuous logics,particularly logics for metric structures, where the first limitation is overcomeby allowing each real in [0 , 1] as a possible truth value. We will deal with thesecond difficulty later when we introduce the continuous version of L ω ,ω , inwhich countable conjunctions are allowed. The origins of continuous logic lie inW. Henson’s approximate satisfaction [18]. A [0 , continu-ous first-order logic has been an active area of research for the last decade sinceit was introduced by I. Ben Yaacov and A. Usvyatsov [6] following ideas of C.C. Chang and H. J. Keisler [10], and Henson and Iovino [19]. We will deal withvariations and generalizations introduced by Ben Yaacov and Iovino [5] and C.J. Eagle [12], following mostly the last one.For the topologist reader who is not so interested in analysis, we emphasizethat we could just consider the usual first-order logic (and later, L ω ,ω ). Wethen do not have to worry about continuous connectives, non-trivial metricstructures, unexpected definitions of definability, etc. We will still use non-trivial theorems of C p -theory concerning C p ( X, L ω ,ω .In the discrete case, the interpretation in a model M of an n -ary relation,also called an n -ary predicate, was defined as a subset of M n and so it can alsobe regarded as a function M n → Definition 4.1 Assume for the sake of notational simplicity that all metric struc-tures have diameter 1. (i) If ( M, d ) and ( N, ρ ) are metric spaces and f : M n → N is uniformly con-tinuous, a modulus of uniform continuity of f is a function δ : (0 , ∩ Q → (0 , ∩ Q such that whenever x = ( a , ..., a n ) , y = ( b , ..., b n ) ∈ M n and ǫ ∈ (0 , ∩ Q , sup { d ( a i , b i ) : 1 ≤ i ≤ n } < δ ( ε ) implies ρ ( f ( x ) , f ( y )) < ε . (ii) A language for metric structures is a set L which consists of constants,functions with an associated arity and a modulus of uniform continuity,predicates with an associated arity and a modulus of uniform continuity,and a symbol d for a metric. (iii) An L -metric structure M is a metric space ( M, d M ) together with in-terpretations for each symbol in L : c M ∈ M for each constant c ∈ L ; f M : M n → M is an uniformly continuous function for each n -ary func-tion symbol f ∈ L ; P M : M n → [0 , is an uniformly continuous functionfor each n -ary predicate symbol P ∈ L . language instead of language for metric structures whenever thereis no room for confusion.Now we proceed to define the continuous analogue of first-order logic in thecontext of metric structures. The definition will be recursive and relies on thenotion of syntactical objects called terms . We will present the continuous casehere following [12]; for an exposition of this in the discrete case, see [36]. Definition 4.2 Suppose { x n : n < ω } is a set of variables and fix a language L .Continuous first-order logic L - terms are defined as follows: All variables andconstants are L - terms . If t , . . . , t n are L - terms and f is an n -ary functionsymbol, then f ( t , . . . , t n ) is an L -term. The interpretation of a term in a metric structure is defined recursively in thenatural way. As we discussed in Section 2, a formula ϕ can also be seen asa function such that for a structure M and a ∈ M , ϕ ( a ) = 1 if and only if M | = ϕ ( a ). For an L -sentence ϕ we write M | = ϕ if and only if ϕ ( a ) = 1 forevery a ∈ M . Definition 4.3 Let L be a language for metric structures. The continuous first-order logic L -formulas are defined recursively as follows: (i) Whenever t and t are L -terms, d ( t , t ) is an L -formula. (ii) If t , . . . , t n are L -terms and P an n -ary predicate symbol, then P ( t , . . . , t n ) is an L -formula. (iii) If ϕ , . . . , ϕ n are L -formulas and g : [0 , n → [0 , is continuous, then g ( ϕ , . . . , ϕ n ) is an L -formula. (iv) If ϕ is an L -formula and x is a variable, then sup x ϕ and inf x ϕ are L -formulas. Remark 4.4(i) sup x ϕ and inf x ϕ can be thought as the quantifiers analogous to ∀ and ∃ .However, they are not quite the same: inf x ϕ = 1 means that for every n < ω there is an x such that ϕ ( x ) > − n , which does not imply thatthere is an x such that ϕ ( x ) = 1. (ii) If M is an L -structure we have M | = min { ϕ, ψ } if and only if M | = ϕ and M | = ψ . Analogously, M | = max { ϕ, ψ } if and only if M | = ϕ or M | = ψ .We will write ϕ ∧ ψ and ϕ ∨ ψ freely. In addition, continuous functionsare also introduced as connectives. Notice that only finitary formulas areallowed. (iii) We can express that the equality a = a holds in a given M by saying M | = “1 − d ( a , a )”, as this means d M ( a M , a M ) = 0 where d M is a metricso a M = a M . 10 iv) One of the main differences that occurs when considering continuous logicsis that negation is usually not available, i.e. given a formula ϕ , there is notnecessarily a formula ψ such that M | = ϕ if and only if M = ψ . We willsee in Section 6 that negation seems to require an infinitary disjunctionto be expressed. (v) All formulas from discrete first-order logic can easily be translated intoformulas of continuous first-order logic. See [9] for a detailed discussion.In Section 6, we will see an example of a formula that cannot be writtenin discrete first-order logic, but allows us to approximate estimates amongnorms involved in the construction of Tsirelson’s space. (vi) In general, the relation ∈ from the language of set theory does not belongto languages for metric structures and so it does not appear in formulasin these languages. As a consequence, an object that can be defined inZFC, using the axiom of choice for instance, might not be definable ina given language for metric structures. The point of Gowers’ problem,however, is to ask whether a space like Tsirelson’s can be defined usingordinary analytic notions, which we take to mean whether it is definable ina language suitable for talking about such analytic notions, as contrastedwith a language that arguably is suitable for discussing all of mathematics,but not necessarily in a pleasing, transparent way.We restate the Compactness Theorem for continuous first-order logic and omitthe proof as it is analogous to the discrete case. Theorem 4.5 Let L be a language for metric structures and T a theory in con-tinuous first-order logic. If T is finitely satisfiable then T is consistent. The general definition of a logic is one that varies from author to author andrequires a discussion of foundational issues that might arise. We omit such ageneral definition in favour of an intuitive description which fits the purposes ofworking with compact continuous logics and continuous L ω ,ω (defined below).A logic for metric structures L assigns to each language for metric structures L the set Sent( L ) of all the sentences together with the space of structures Str( L )as defined in Section 1, and a function Val : Str( L ) × Sent( L ) → [0 , 1] given byVal( M , ϕ ) = ϕ ( a ) where a is any element of M ; Val assigns to each pair ( M , ϕ )the truth value of ϕ in M according to L .In what follows, we will only be interested in languages for metric structures,regardless of the logic. As a consequence, we can regard each case as a naturalgeneralization of discrete first-order logic: if L is a language, an L -structure M is discrete if it is based on a discrete metric space ( M, d ) and for every L ( M )-sentence ϕ we have either M | = ϕ or M | = 1 − ϕ . Thus, first-order logic is aspecial case of what we are considering here. The work of Casazza and Iovino[8] on the undefinability of Tsirelson’s space is centred around the notion of(finitary) compact continuous logics, i.e. logics for metric structures for whichStr( L ) is compact. 11 C p -Theory meets Continuous Model Theory Given a topological space X , we will denote by C p ( X ) the space of continuousreal-valued functions from X with the pointwise convergence topology, i.e. thetopology induced on C p ( X ) as a subspace of the product topology on [0 , X .For our purposes, we will only be interested in sets of functions from X to [0 , C p ( X ) and so no difficulty will arise if one considers the closedsubspace C p ( X, [0 , X to [0 , 1] instead of C p ( X ).Unless otherwise specified, we suppose that we are working with an arbitrarylogic for metric structures in this section.Some definitions and statements differ from the original ones in [8] as thoseare aimed at the compact case on which their work is centred. Surprisingly, thediscrete version of what is proved in [8] or here for continuous logic is not mucheasier to prove; the reason for this is that C p ( X, 2) is not much simpler than C p ( X, [0 , Definition 5.1(i) Let L be a language. L ′ ⊇ L is a language for pairs of structures from L if L ′ includes two disjoint copies L and L of L and there is a mapStr ( L ) × Str ( L ) → Str ( L ′ ) which assigns to every pair of L -structures ( M , N ) an L ′ -structure h M , N i , where M is an L -structure and N is an L -structure. We say that L ′ is a language for pairs of structures if it isso for some L . (ii) If L ′ is a language for pairs of structures from L and X, Y are functionsymbols from L , we say that a formula ϕ ( X, Y ) is a formula for pairsof structures from L if ( M , N ) ϕ ( X M , Y N ) = Val ( ϕ ( X, Y ) , h M , N i ) isseparately continuous on Str ( L ) × Str ( L ) . For notational simplicity, wewill write ϕ ( M , N ) instead of Val ( ϕ ( X, Y ) , h M , N i ) . The definition of formulas for pairs of structures is useful for comparing normson the same metric space. Consider for example a set C of structures which arenormed spaces ( c , k k ) based on c , i.e. the structures ( c , k k l , k k , e , e , ... )where k k is a function symbol in L for an arbitrary norm and { e n : n < ω } isthe standard vector basis of c . Let L ′ be a language including two disjointcopies of L ; then two different L -structures ( c , k k ) and ( c , k k ) can becoded as the single L ′ -structure ( c , k k , k k ). In this section, “language” willmean “language for pairs of structures”. For the purposes of [8], X and Y willalways represent function symbols for norms. Definition 5.2 Let L be a language and ϕ a formula for pairs of structures (i) The left ϕ -type of M is the function ltp ϕ, M : Str ( L ) → [0 , given by ltp ϕ, M ( N ) = ϕ ( M , N ) . The space of left ϕ -types S lϕ is the closure of ltp ϕ, M : M ∈ Str ( L ) } in C p ( Str ( L )) . If C is a subset of Str ( L ) then S lϕ [ C ] is the closure of { ltp ϕ, M : M ∈ C} in C p ( C ) , called “the space of left ϕ -typesover C ”. (ii) The right ϕ -type of N is the function rtp ϕ, N : Str ( L ) → [0 , given by rtp ϕ, N ( N ) = ϕ ( M , N ) . The spaces S rϕ and S rϕ [ C ] are defined analogously. Remark 5.3 As in the discrete case, one can recover the left ϕ -type in the classicalsense by considering ( ltp ϕ, M ) − { } . Proposition 5.4 If L is a compact logic then S lϕ and S rϕ are compact. Proof. If f is a limit point of S lϕ , then there is a cardinal κ , an ultrafilter U over κ and a sequence h M α : α < κ i in Str( L ) such that lim α →U ltp ϕ, M α = f .To see this, let κ be the cardinality of the neighbourhood filter F at f , takean ultrafilter U extending F and pick a point in each member of F . The traceof U on this κ -sequence U -converges to f . Since Str( L ) is compact and ϕ isseparately continuous, there is an M ∈ Str( L ) such that lim α →U M α = M andfor any N ∈ Str( L ), lim α →U ϕ ( M α , N ) = ϕ ( M , N ) = ltp ϕ, M ∈ S lϕ . Then S lϕ isa closed subset of [0 , Str( L ) . (cid:3) Definition 5.5 Let C be a subset of Str ( L ) and suppose { M α : α < κ } ⊆ C is suchthat t = ltp ϕ, M , then we say that M is a realization of t in C and t is a left ϕ -type of M over C . We have just showed that the classical result stating that, when the logic iscompact, for every type there is a model in which the type is realized, holds inthe continuous case. Notice that Str( L ) can be replaced by any closed subsetand the result still holds with the same proof.Now we are ready to introduce the essential concept of stability in continuouslogics: Definition 5.6 Let C ⊆ Str ( L ) . A formula for pairs of structures ϕ is stable on C if and only if whenever h M i : i < ω i and h N : j < ω i are sequences in C and U and V ultrafilters on ω , respectively, we have lim i →U lim j →V ϕ ( M i , N j ) = lim j →V lim i →U ϕ ( M i , N j ) . Now we can state a C p -theoretic result of V. Pt´ak [29] (see also S. Todorce-vic [39]) and the respective model-theoretic version which gives an importantcharacterization of stability: Theorem 5.7 (Pt´ak) If X is compact then a pointwise bounded A ⊆ C p ( X ) hasa compact closure if and only if it satisfies the following double limit condi-tion: whenever h f n : n < ω i and h x n : n < ω i are sequences in A and X respec-tively, the double limits lim n →∞ lim m →∞ f n ( x m ) and lim m →∞ lim n →∞ f n ( x m ) are equal whenever they both exist. heorem 5.8 Let L be a language for pairs of structures, ϕ an L -formula forpairs of structures and C ⊆ Str ( L ) be compact. Then, the following are equiva-lent: (i) ϕ is stable on C . (ii) There is a separately continuous function F : S lϕ [ C ] × S rϕ [ C ] → [0 , suchthat F ( ltp ϕ, M , rtp ϕ, N ) = ϕ ( M , N ) for every M , N ∈ Str ( L ) . Proof. ( ii ) → ( i ) is immediate. ( i ) → ( ii ): We have A = { ltp ϕ, M : M ∈ C} ⊆ C p (Str( L )). Also, S lϕ [ C ] = ¯ A ∩ C p (Str( L )). Clearly, stability implies the dou-ble limit condition of Pt´ak’s theorem and so S lϕ [ C ] is compact. By symmetry, F ( ltp ϕ, M , rtp ϕ, N ) = ϕ ( M , N ) is separately continuous. (cid:3) Now we can introduce the notion of definability: Definition 5.9 Let L be a language for pairs of structures, ϕ be a formula for pairsof structures and C ⊆ Str ( L ) . A function τ : S rϕ [ C ] → [0 , is a left global ϕ -typeover C if there is a sequence h M α : α < κ i in C such that for every type t ∈ S rϕ ,say t = lim β →V rtp ϕ, N β , we have τ ( t ) = lim α →U lim β →V φ ( M α , N β ) . We saythat τ is explicitly definable if it is continuous. If a left ϕ -type p ∈ S lϕ [ C ] is givenby p = lim α →U ltp ϕ, M α , we say that p is explicitly definable if τ : S rϕ → [0 , given by τ ( t ) = lim α →U lim β →V ϕ ( M α , N β ) yields an explicitly definable leftglobal ϕ -type. The previous definition is sound because the continuity of τ makes it into anallowable formula which determines, as in the discrete case, which formulasbelong to the type t , by taking the pre-image of 1 as usual.Now Pt´ak’s theorem (or its model-theoretic version) can be restated as theequivalence between stability and definability in compact continuous logics, fol-lowing [8]: Theorem 5.10 Let L be a compact logic, L a language for pairs of structures, C ⊆ Str ( L ) compact and ϕ a formula for pairs of structures. Then, the followingare equivalent: (i) ϕ is stable on C . (ii) If τ is a global left (or right) ϕ -type over C , then τ is explicitly definable. (iii) For every L -structure M , the left and right ϕ -types of M are explicitlydefinable. This model-theoretic machinery that is built upon C p -theoretic results consti-tutes the grounds on which Casazza and Iovino proved the undefinability ofTsirelson’s space from sets of Banach spaces including l p or c in compact con-tinuous logics. Most of the remainder of the proof of this fact involves analysisrather than model theory or topology so the interested reader is referred to [8].Pt´ak’s theorem relies on a well-known theorem of Grothendieck which statesthat, when X is countably compact, the closure of any A ⊆ C p ( X ) which is14ountably compact in C p ( X ) (every infinite subset of A has a limit point in C p ( X )) is compact. The spaces for which this property holds, i.e. the clo-sure of any subspace which is countably compact in C p ( X ) is compact, will becalled weakly Grothendieck spaces; when C p ( X ) has this property hereditarily,we say that X is a Grothendieck space. Grothendieck spaces have turned outto be relevant in work in progress concerning definability, allowing one to provestronger results, but they are not involved in what we present here and so wework only with weakly Grothendieck spaces. See [4] for a more detailed discus-sion on variations of Grothendieck spaces. Notice that C p ( X, [0 , C p ( X ) and so if X is a (weakly) Grothendieck space, any closedcountably compact subset of C p ( X, [0 , C p ( X, 2) is aclosed subspace of C p ( X, [0 , X so that ¯ A ∩ C p ( X ) beingcompact is equivalent to a double (ultra)limit condition as in Pt´ak’s theorem.A natural place to start is to consider weakly Grothendieck spaces in order totest statements such as the ones in [8] in the context of logics for which thespace of types is weakly Grothendieck, thus generalizing the compact case. Problem 1: What conditions are to be satisfied by X so that it is a weaklyGrothendieck space?There is a variety of disparate conditions which imply X is a weakly Grothendieckspace: A. V. Arhangel’ski˘ı showed in [4] that all Lindel¨of Σ-spaces (i.e. contin-uous images of spaces that can be perfectly mapped onto separable metrizablespaces) are weakly Grothendieck, and V. Tkachuk [37] proved that every σ -compact space is weakly Grothendieck. We collect some of these results in thefollowing theorem: Theorem 5.11 A space X is weakly Grothendieck if it satisfies any of the follow-ing conditions: (i) X is countably compact. (ii) X is σ -compact. (iii) X is Lindel¨of Σ . Theorem 5.11 (i) is Grothendieck’s theorem. The proof of Theorem 5.11 (ii) canbe reduced to (iii) as every σ -compact space is Lindel¨of Σ - see [37]. The prooffor the case in which X is Lindel¨of Σ follows from a variation by Arhangel’ski˘ıof a famous theorem of Baturov (see [2] and [3]). Remark 5.12 It is a non-trivial fact that if Y is a dense subspace of X and Y isGrothendieck, then X is Grothendieck; the proof requires the full strength of be-ing a Grothendieck space; see, for instance, [4]. Then, for Grothendieck spaces,we can restrict our attention to dense subspaces in Theorem 5.11. More specifi-cally, we have: if having a property P implies that X is Grothendieck, then hav-ing a dense subspace with the property P also implies that X is Grothendieck.15o, if X is Hausdorff k -separable , i.e. has a dense σ -compact subspace, then itis Grothendieck because Lindel¨of Σ spaces, in particular σ -compact spaces, areGrothendieck (see [4]). We thank V. V. Tkachuk for pointing this out to us. Inthe next section, we will be able to extend results in [8] for logics for which thespaces of types are weakly Grothendieck and first countable. As an example,we will work with countable fragments of a continuous infinitary logic, L ω ,ω ,for which the spaces of types are metrizable and separable, in which case theyhave a σ -compact dense subspace and thus are (weakly) Grothendieck.The separable case follows from our previous discussion: countable spaces areGrothendieck since they are σ -compact. By having a Grothendieck dense sub-space, separable spaces are Grothendieck.It can be seen in Arhangel’ski˘ı [3] that the conditions in theorem 5.11 canyield stronger results, e.g. if X is countably compact, then being pointwisebounded is enough for a subset of C p ( X ) to have compact closure. Moreover,if A is countably compact in C p ( X ) for countably compact X , then, if ¯ A is theclosure of A in R X , then the closure ¯ A ∩ C p ( X ) is an Eberlein compactum (i.e.metrizable and homeomorphic to a compact subset of C p ( Y ) for some compactmetrizable Y ). This motivates the following problem: Problem 2: Find weaker conditions under which the closure of subspaces thatare countably compact in C p ( X ) are just compact.Answers to the previous questions would light the road towards extending theresults of Casazza and Iovino to further logics. In [8], most results assume thatthe logic is countably compact as this allows them to work only with sequencesand ultrafilters over ω . This is useful because it allows one to use Theorem 3.8to test stability (and definability). In the next section, we generalize some ofthese results for logics for which the space of types is a first countable weaklyGrothendieck space. Infinitary Continuous Logics and Final Remarks In this section we present a double (ultra) limits condition which is equivalent to¯ A ∩ C p ( X ) being compact when X is first countable and weakly Grothendieck.As an application, we extend some results from [8] to non-compact continuouslogics with more expressive power for doing analysis.Recall that a subspace Y of an space X is relatively compact if ¯ Y is compactin X . Theorem 6.1 Suppose X is a first countable weakly Grothendieck space. If A is a subset of C p ( X, [0 , , then A is relatively compact in C p ( X ) if and onlyif it satisfies the following double limit condition: for every pair of sequences h f n : n < ω i and h x m : m < ω i and ultrafilters U and V in βω lim n →U lim m →V f n ( x m ) = lim m →V lim n →U f n ( x m )16 henever the limit lim m →V x m exists. Proof. Suppose ¯ A ∩ C p ( X ) is not compact in C p ( X ). Then ¯ A ∩ C p ( X ) is notcountably compact, so let h f n : n < ω i be a sequence in A with no limit pointsin ¯ A ∩ C p ( X ). Notice that ¯ A is compact in [0 , X since it is a closed subset,so each ultralimit of the sequence exists, is a limit point and is discontinuous.Take a non-principal ultrafilter U over ω and let lim n →U f n = g . Notice that g is discontinuous by assumption. Then there is an ε > y ∈ X such thatfor every neighborhood U of y , there is a y ′ ∈ U such that | g ( y ) − g ( y ′ ) | > ε .Let B be a local base at y and consider B ′ = {∩S : S ∈ [ B ] <ω } . Clearly B ′ isalso a local base at y . Take an enumeration B ′ = { B m : m < ω } and, for each m < ω , choose x m ∈ B m such that | g ( x m ) − g ( y ) | > ε . Now, for every k < ω let A k = { m < ω : x m ∈ B k } ; notice that { A k : k < ω } is centred so it can beextended to an ultrafilter V ∈ βω . Let U be a neighbourhood of y and let k < ω be such that B k ⊆ U . Then A k ⊆ { m < ω : x m ∈ U } . Thus, lim m →V x m = y and lim n →U lim m →V f n ( x m ) = g ( y )since each f n is continuous. On the other hand, lim m →V lim n →U f n ( x m ) =lim m →V g ( x m ) exists by the compactness of [0, 1]. However by our construction, | g ( x m ) − g ( y ) | > ε for every m < ω . Then the ultralimits lim n →U lim m →V f n ( x α )and lim m →V lim n →U f n ( x m ) are different, a contradiction. Conversely, sup-pose lim m →V x m exists and ¯ A ∩ C p ( X ) is compact. Then for any sequence h f n : n < ω i in A and ultrafilter U over ω , there is a continuous g = lim n →U f n .Thus we have that lim n →U lim m →V f n ( x m ) = lim n →U f n ( y ) = g ( y ) and alsothat lim m →V lim n →U f n ( x m ) = lim m →V f m ( y ) = g ( y ). (cid:3) “First countability” is not best, but is convenient to use. In work in progress,we aim for the optimal hypotheses for Theorem 6.1. Now we can restate Theo-rem 6.1 as a model-theoretic result: Theorem 6.2 Let L be a logic, L a language, and suppose C ⊆ Str ( L ) is suchthat the space of ϕ -types on C is first countable and weakly Grothendieck. Then,the following are equivalent: (i) Whenever a left type over C is given by t = lim i →U ltp ϕ, M i and h N j : j < ω i is a sequence in C and V ∈ βω we have lim i →U lim j →V ϕ ( M i , N j ) = lim j →V lim i →U ϕ ( M i , N j ) . (ii) Whenever a left type is given by t = lim i →U ltp ϕ, M i and h N j : j < ω i is asequence in C we havesup i 1] by f α (lim i →U ltp ϕ, M i ) = lim i →U ϕ ( M i , N α ). Then A = { f α : α < κ } is a subset of C p ( S lϕ [ C ] , [0 , A ∩ C p ( S lϕ [ C ] , [0 , τ = lim α →V f j is continuous.That (iii) implies (i) is immediate. (cid:3) Let C be the set of all structures which are normed spaces based on c . Weintroduce the continuous logic formula for pairs of structures used in [8]: fornorms k k and k k , let D ( k k , k k ) = sup (cid:26) k x k k x k : k x k l = 1 (cid:27) . Then let ϕ ( k k , k k ) = log D ( k k , k k )1 + log D ( k k , k k ) . It is not difficult to see that if t = lim i →U ltp φ, k k i is realized by a structure( M , k k ∗ ), then lim i →U sup k x k l =1 k x k i k x k = sup k x k l =1 k x k ∗ k x k . (1)for any structure ( c , k x k ) in C . In particular, taking k k l = k k yields, foreach x ∈ c , lim i →U k x k i = k x k ∗ . If in addition k k ≤ k k · · · ≤ k x k n . . . ,then lim i →∞ k x k i = k x k ∗ . Conversely, if k k ≤ k k · · · ≤ k x k n . . . and alsolim i →∞ k x k i = k x k ∗ for every x ∈ c , then (1) holds for any ( c , k k ) in C andany nonprincipal ultrafilter U over ω .This motivates the following definition from [8]: Definition 6.3 Let C be the set of all structures which are normed spaces basedon c , ϕ a formula for a pair of structures and k k ∗ be a norm on c . (i) If {k k i : i < ω } is a family of norms on c we say that { ltp ϕ, k k i : i < ω } determines k x k ∗ uniquely if for every U ∈ βω , t = lim i →U ltp ϕ, k k i isrealized and k k ∗ is its unique realization. (ii) We say that k k ∗ is uniquely determined by its ϕ -type over C if there is afamily of norms {k k i : i < ω } on c such that { ltp ϕ, k k i : i < ω } deter-mines k k ∗ uniquely. Notice that if k k i denotes the i -th iterate of the Tsirelson norm and k k T is theTsirelson norm, then lim i →∞ k x k i = k x k T for each x ∈ c and so we have thefollowing result from [8]: Proposition 6.4 Let L be a logic for metric structures, L a language for pairs ofstructures, C the class of structures ( c , k k l , k k ) such that the norm comple-tion of ( c , k k ) is a Banach space including l p or c and let ϕ ( X, Y ) be the for-mula defined above. Suppose { ( c , k k l , k k i ) : i < ω } is a family of structures n C such that k k ≤ k k · · · ≤ k x k n . . . and the ϕ -type t = lim i →U ltp ϕ, k k i is realized by ( c , k k l , k k ∗ ) in Str ( L ) , then { ltp ϕ, k k i : i < ω } determinesuniquely k k ∗ over C . In particular, the Tsirelson norm is uniquely determinedby its ϕ -type over C . The following result is proved in [8] and is purely analytical: Proposition 6.5 Let k k i be the i -th iterate in the construction of the Tsirelsonnorm. Then the following hold: (i) sup k x k l =1 k x k i k x k j ≤ for i < j . (ii) sup k x k l =1 k x k i k x k j ≥ j for i > j .Thus, sup i Let L be a logic for which the space of types is first countable andweakly Grothendieck, let L be a language for pairs of structures, let C be theclass of structures ( c , k k l , k k ) such that the norm completion of ( c , k k ) isa Banach space including l p or c , and let k k T be the Tsirelson norm. Thenthere is a formula for pairs of structures φ such that k k T is uniquely determinedby its ϕ -type over C and that ϕ -type is not explicitly definable over C . Remark 6.7 The previous result states that, although Tsirelson’s space T is con-structed via a limiting process involving only explicitly definable spaces basedon c , the completions of which include l p or c , T itself is not explicitly defin-able. A much more general result is proved in [8] for countably compact logics,namely: if a space is explicitly definable from a class of spaces based on c which include l p or c , then it must also include l p or c .Now we will introduce a continuous infinitary logic which extends the com-pact case, and we will use our C p -theoretic results to prove the undefinability ofTsirelson’s space in this logic.In discrete model theory, there are natural generalizations of first-order logicdenoted by L κ,ω where κ is an infinite cardinal. L κ,ω is a logic satisfying: (i) All first-order formulas are formulas in L κ,ω . (ii) If λ < κ and χ = { ϕ α : α < λ } is a subset of formulas in L κ,ω , then ∧ χ and ∨ χ are also formulas in L κ,ω . (iii) If ϕ is a formula in L κ,ω , then ( ∃ x ) ϕ ( x ) and ( ∀ x ) ϕ ( x ) are formulas in L κ,ω .One of the main differences between L κ,ω for κ > ω and first-order logic ( L ω,ω ) isthat the compactness theorem does not hold. Moreover, if a weak version of the19ompactness Theorem holds for L κ,ω , namely if whenever T is a theory in L κ,ω such that if every subsets of T of size < κ is satisfiable then T is satisfiable, then κ is a weakly compact cardinal and so cannot be proved to exist in ZFC . See,for instance, [22] and [23]. Thus, in general, one has to do without compactness.The omitting types theorem (an analogue of the Baire category theorem), is auseful substitute. For an in-depth study of this result in abstract logics, see [14].Here, we will present L ω ,ω , its continuous version, and some of their topo-logical aspects following Eagle [12]. Many concepts useful for analysis can bedefined in continuous L ω ,ω , but cannot be defined in the usual continuous logic[12]. Definition 6.8 Let L be a language for metric structures. The formulas of L ω ,ω ( L ) or L ω ,ω ( L ) -formulas are defined recursively as follows: (i) All first-order L -formulas are L ω ,ω ( L ) -formulas. (ii) If ϕ , . . . , ϕ n are L ω ,ω ( L ) -formulas and g : [0 , n → [0 , is continuousthen g ( ϕ , . . . , ϕ n ) is an L ω ,ω ( L ) -formula. (iii) If { ϕ n : n < ω } is a family of L ω ,ω ( S ) -formulas, then inf n ϕ n and sup n ϕ n are L ω ,ω ( L ) -formulas. These can be also denoted as ∧ n ϕ n and ∨ n ϕ n respectively. (iv) If ϕ is an L ω ,ω ( L ) -formula and x is a variable, then inf x ϕ and sup x ϕ are L ω ,ω ( L ) -formulas. An interesting feature of continuous L ω ,ω is that negation ( ¬ ) becomes availablein the classical sense [12]: if L is a language and ϕ is an L ω ,ω ( L )-formula, thenwe can define: ψ ( x ) = ∨ n { ϕ ( x ) + n , } . Then, M | = ψ ( a ) if and only if thereis an n < ω such that M | = max { ϕ ( a ) + n , } ; this is max { ϕ ( a ) + n , } = 1,which is the same as ϕ ( a ) ≤ − n , i.e. M = ϕ ( a ). Then, M | = ψ ( x ) if andonly if M = ϕ ( x ), and so ψ corresponds to ¬ ϕ .It is sometimes useful to restrict one’s attention to (countable) fragments of L ω ,ω ( S ), which are easier to work with than the full logic. Definition 6.9 Let L be a language. A fragment F of L ω ,ω ( L ) is a set of L ω ,ω ( L ) -formulas satisfying: (i) Every first-order formula is in F . (ii) F is closed under finitary conjunctions and disjunctions. (iii) F is closed under inf x and sup x . (iv) F is closed under subformulas. (v) F is closed under substituting terms for free variables. L ω ,ω ( L ) generates a fragment, and that every finiteset of formulas generates a countable fragment. There are two arguments tosupport the idea of working with countable fragments of L ω ,ω ( L ): firstly, noticethat a given proof involves only finitely many formulas which then generate acountable fragment of L ω ,ω ( L ). Secondly (as noted independently by C. J.Eagle (personal communication)), if an object is definable in L ω ,ω ( L ), then itis definable in a countable fragment by the same argument.As the work of Casazza and Iovino deals with continuous logics which arefinitary in nature, it is natural to ask whether their result on the undefinabilityof Tsirelson’s space can be proved for continuous L ω ,ω , which is arguably amore natural language from the point of view of Banach space theorists.In discrete model theory, countable fragments of L ω ,ω have been studiedpreviously, for instance M. Morley [27] showed that the space of types of acountable fragment of L ω ,ω is Polish. Remark 6.10 In the continuous case, we note that the space of types of a count-able fragment F of continuous L ω ,ω ( L ) can be seen as a subspace of [0 , F ,which is metrizable and second countable, and so it is separable – hence it isGrothendieck – and first countable.The following result follows from Theorem 6.7 and Remark 6.11: Theorem 6.11 Let L be a language for pairs of structures, C the class of struc-tures ( c , k k l , k k ) such that the norm completion of ( c , k k ) includes l p or c and let k k T be the Tsirelson norm. Then there is an L ω ,ω ( L ) -formula forpairs of structures φ such that k k T is uniquely determined by its ϕ -type over C and that ϕ -type is not explicitly definable over C . Remark 6.12 Note that even for discrete L ω ,ω this result is new. In ongoing research, we extend this non-definability to even wider classes oflogics. Applications of C p -theory to model theory are not confined to definabilityquestions. Stay tuned! Acknowledgements We thank Jose Iovino, Christopher Eagle and Xavier Caicedo for valuable com-ments that have helped us to understand [8], especially the spaces of types. Thesecond author thanks the Mathematics Department of the University of Texasat San Antonio for its gracious hospitality at a workshop in May 2018 wherehe had the opportunity to interact with these three and with Eduardo Due˜nez.Various subgroups of these four model theorists are doing important work. Wethank Vladimir Tkachuk for pointing us in the right direction toward findingthe results we needed in C p -theory. 21 Postscript We have not meant to give the impression that extensions of [CI] constitute theonly applications of C p -theory to model theory. In particular, at the suggestionof the referee we shall say a few words about the work of P. Simon and K.Khanaki.The great utility of stability in model theory led to the investigation of lessstringent conditions, in particular what is now known as NIP (the failure of theIndependence Property). The “bible” for NIP is Simon’s book [34]. Definition 8.1 A formula ϕ ( x, y ) has the independence property in a model U ifthere is an infinite subset A of the universe for which there is a family { b I : I ⊆ A } such that, for each a ∈ A , M | = ϕ ( a, b I ) ⇔ a ∈ I . The formula ϕ ( x, y ) is NIP (or dependent ) if it does not have the independence property. As noted in [35] and elsewhere, this is equivalent to the condition that for anymodel M of the theory, the closure in the type space of a subset of size at most κ has cardinality at most 2 κ . The reader will immediately be reminded of thesimilar condition characterizing (model-theoretic) stability. Building on work ofRosenthal [30], [31], Bourgain, Fremlin, and Talagrand [7] studied the closure in R X for X Polish, of subsets A of C p ( X ). As Simon notes, [7] proves that eitherthat closure ¯ A contains non-measurable functions or every element of ¯ A is apointwise limit of a sequence of elements of A , in which case | ¯ A | ≤ | A | . Simonproves that this dichotomy explicitly corresponds to the dichotomy betweentheories that do or do not satisfy the independence property.The Baire class 1 functions are classically defined as the pointwise limits ofsequences of continuous real-valued functions. Compact subspaces of B ( X ),the collection of Baire class 1 functions for the topological space X , are called Rosenthal compacta and have an extensive literature. We will not try to beexhaustive in our references here. In addition to the already mentioned [7] and[35], we call attention to the surveys of Negrepontis [28] and Debs [11] and thedeep work of Todorcevic [40]. Debs concentrates on separable Rosenthal com-pacta; a noteworthy aspect of [35] is that by replacing sequential convergenceby filter convergence, Simon mainly eliminates what Aleksandrov [1] called the“parasite of countability”. We say “mainly”, because the use of descriptive settheory in [11] in the separable case cannot be replicated in the general case.Moreover, it is in the case of countable theories and models that Simon [35]establishes the connection between type spaces and Rosenthal compacta: Theorem 8.2 (Proposition 2.16 of [35]) Let T and M be countable and ϕ ( x, y ) NIP. Then Inv ϕ ( M ) is a Rosenthal compactum. The relevant definitions are: Definition 8.3 Given a model U and a submodel M with universe M , the set ofautomorphisms of M fixing U is denoted by Aut ( U / M ) . A ϕ -type p ( x ) over U is M - ϕ -invariant if σp ( x ) = p ( x ) for every σ ∈ Aut ( U / M ) . The set of all invariant M - ϕ -types is denoted by Inv ϕ ( M ) . monster model is a model-theoretic technicality whichconstitutes an important part of the literature and is relevant in this context.In the previous definition, one is usually interested in U being the monstermodel (a class-size model which embeds all set-size models) and M being a set-size submodel. For a detailed introduction to the monster model, see [36] andfor more on invariant types see [34].Strictly speaking, the study of Rosenthal compacta is not a part of C p -theory,since it involves Baire class 1 functions rather than continuous functions. How-ever it is a natural extension of C p -theory, and in his address at the conferencein the proceedings of which this paper will appear, the second author suggestedthat topologists study B ( X ) with the same vigour they have applied to C p ( X ).An interesting question is where do Rosenthal compacta appear in the spec-trum of special compacta studied by C p -theorists. Are they Eberlein , Gul’ko , Corson , etc.? So far, the only positive result we have found in the literature isthat Theorem 8.4 [11] Gul’ko compacta of weight ≤ ℵ are Rosenthal. The deep analysis of [40] provides many negative results distinguishing Rosen-thal compacta from the other well-known compacta studied by C p -theorists. See[11]. Mentioning properties of interest to those who study such compacta, wehave for example: Theorem 8.5 [40] In every Rosenthal compactum, the set of G δ -points includesa dense metrizable subspace. Theorem 8.6 [40] Rosenthal compacta have σ -disjoint π -bases. Corollary 8.7 Rosenthal compacta satisfying the countable chain condition areseparable. In a series of mainly unpublished papers, Karim Khanaki has explored con-nections between model theory and Banach space theory. His latest, [25], coversground familiar to us, including stability, definability, Grothendieck, etc., as wellas topics we are just starting to research, such as NIP. 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