aa r X i v : . [ m a t h . L O ] M a y MODEL THEORY FOR REAL-VALUED STRUCTURES
H. JEROME KEISLER
Abstract.
We consider general structures where formulas have truth valuesin the real unit interval as in continuous model theory, but whose predicatesand functions need not be uniformly continuous with respect to a distancepredicate. Every general structure can be expanded to a pre-metric structureby adding a distance predicate that is a uniform limit of formulas. Moreover,that distance predicate is unique up to uniform equivalence. We use this toextend the central notions in the model theory of metric structures to generalstructures, and show that many model-theoretic results from the literatureabout metric structures have natural analogues for general structures.
Contents
1. Introduction 22. Basic Model Theory for General Structures 32.1. General Structures 42.2. Ultraproducts 52.3. Definability and Types 72.4. Saturated and Special Structures 92.5. Pre-metric Structures 102.6. Some Variants of Continuous Model Theory 123. Turning General Structures into Metric Structures 173.1. Definitional Expansions 173.2. Pre-metric Expansions 183.3. The Expansion Theorem 203.4. Absoluteness 234. Properties of General Structures 254.1. Types in Pre-metric Expansions 264.2. Definable Predicates 274.3. Topological and Uniform Properties 284.4. Infinitary Continuous Logic 304.5. Many-sorted Metric Structures 334.6. Bounded and Unbounded Metric Structures 334.7. Imaginaries 354.8. Definable Sets 374.9. Stable Theories 404.10. Building Stable Theories 434.11. Simple and Rosy Theories 455. Conclusion 47References 47
Date : May 26, 2020. Introduction
We show that much of the model theory of metric structures carries over togeneral structures that still have truth values in [0 , , , , , , The main advantage of their for-malism is that it is easy to describe a metric structure by specifying the universe,functions, constants, relations, and moduli of uniform continuity. Here we go astep further—the notion of a general structure is even less technically involvedthan the notion of a metric structure, since it does not require one to specify ametric signature giving a distance predicate and moduli of uniform continuity.Formally, the key concepts that make things work in this paper are the notion ofa pre-metric expansion of a theory (Definition 3.2.1), and the notion of an absoluteversion of a property of pre-metric structures (Definition 3.4.1).Let T be a theory (or set of sentences) with vocabulary V . A pre-metric ex-pansion of T is a metric theory T e that makes every general model M of T intoa pre-metric model M e of T e in the following way. T e has a metric signature over V ∪ { D } with distance D . Moreover, there is a sequence h d m i m ∈ N of formulas of V , called an approximate distance for T e , such that for each general model M of T , M e = ( M , D ) with D = lim m →∞ d m .By adapting arguments of Iovino [I94] and Ben Yaacov [BY05] to our setting,and using Theorem 4.23 of [BU], one can prove that every complete theory of Previous formalisms included metric open Hausdorff cats in [BY03a] and Henson’s Banachspace logic in [He], [HI].
ODEL THEORY FOR REAL-VALUED STRUCTURES 3 general structures with a countable vocabulary has a pre-metric expansion. TheExpansion Theorem 3.3.4 improves that by showing that every (not necessarilycomplete) theory T with a countable vocabulary has a pre-metric expansion inwhich each of the formulas d m ( x, y ) defines a pseudo-metric in T . We will see inSection 4 that the Expansion Theorem has far-reaching consequences. Proposition3.3.7 shows that the pre-metric expansion of a theory is unique up to uniformequivalence (but far from unique).We say that a property P of general structures with parameters is an absoluteversion of a property Q of pre-metric structures if whenever M is a general modelof a theory T and T e is a pre-metric expansion of T , M has property P if and onlyif M e has property Q . If Q has an absolute version, its absolute version is unique.We regard the absolute version of a property of pre-metric structures as the “right”extension of that property to general structures.General structures correspond to first order structures without equality in thesame way that metric structures with a distance predicate correspond to first orderstructures with equality. In first order model theory, each structure without equalitycan be expanded to a pre-structure with equality in a unique way, so there is verylittle difference between structures without equality and structures with equality.In [0 , Basic Model Theory for General Structures
We assume that the reader is familiar with the model theory of metric structuresas developed in the paper [BBHU], and we will freely use notation from that paper.Our focus in this section will be on the definitions. Only brief comments will begiven on the proofs, which will be similar to the proofs of the corresponding classicalresults that can be found, for example, in [CK12], as well as to proofs in [BBHU], In [BU], Theorem 4.23 was used to show that their formalism of continuous model theory isequivalent to the formalism of open Hausdorff cats in [BY03a].
H. JEROME KEISLER and in the much earlier monograph [CK66] that treated metric structures with thediscrete metric.2.1.
General Structures.
The space of truth values will be [0 , vocabulary V consists of a set of predicate symbols P of finite arity,a set of function symbols F of finite arity, and a set of constant symbols c . A general ([0 , M consists of a vocabulary V , a non-empty universe set M ,an element c M ∈ M for each constant symbol c , a mapping P M : M n → [0 ,
1] foreach predicate symbol P of arity n , and a mapping F M : M n → M for each func-tion symbol F of arity n . A general structure determines a vocabulary, but doesnot determine a metric signature.The formulas are as in [BBHU], with the connectives being all continuous func-tions from finite powers of [0 ,
1] into [0 , x , inf x . The truthvalue of a formula ϕ ( ~x ) at a tuple ~a ∈ M | ~x | in a general structure M is an elementof [0 ,
1] denoted by ϕ M ( ~a ). It is defined in the usual way by induction on thecomplexity of formulas. The syntax and semantics of general structures describedhere are the same as in [AH], and are the same as the restricted continuous logic in[Ca], except that there the value 1 denotes truth.We deviate slightly from [BBHU] by defining a theory to be a set of sentences(rather than a set of statements of the form ϕ = 0). Similarly, we define an n -type over a set of parameters A to be a set of formulas with free variables in ~x = h x , . . . , x n i and parameters from A . A general model of a theory T is ageneral structure M such that ϕ M = 0 for each ϕ ∈ T . An n -tuple ~b in a generalstructure M satisfies a formula ϕ ( ~x ) in M if ϕ M ( ~b ) = 0, and satisfies , or realizes ,an n -type p if ~b satisfies every formula ϕ ( ~x ) ∈ p .Hereafter, M , N will denote general structures with universe sets M, N and vo-cabulary V , and S, T, U will denote sets of sentences (that is, theories) with thevocabulary V .The notions of substructure (denoted by ⊆ ), elementary equivalence (denoted by ≡ ), elementary substructure and extension (denoted by ≺ and ≻ ), and elementarychain are as defined as in [BBHU], but applied to general structures as well asmetric structures. M | = T means that M is a general model of T , M | = ϕ ( ~b ) meansthat ϕ M ( ~b ) = 0, and T | = U means that every general model of T is a general modelof U . T and U are equivalent if they have the same general models. The completetheory Th( M ) of M is the set of all sentences true in M . Thus Th( M ) = Th( M ′ ) ifand only if M ≡ M ′ . We state two elementary results. Fact 2.1.1. (Downward L¨owenheim-Skolem) For every general structure M , thereis a general structure M ′ ≺ M such that | M ′ | ≤ ℵ + | V | . Fact 2.1.2. (Elementary Chain Theorem) The union of an elementary chain ofgeneral structures h M α : α < β i is an elementary extension of each M α . By an embedding h : M → N we mean a function h : M → N such that h ( c M ) = c N for each constant symbol c ∈ V , and for every n and ~a ∈ M n , h ( F M ( ~a )) = F N ( h ( ~a )) for every function symbol F ∈ V of arity n , and P M ( ~a ) = P N ( h ( ~a ))for every predicate symbol P ∈ V of arity n . We say that M is embeddable in we consider the elements of [0 ,
1] to be 0-ary connectives, so each r ∈ [0 ,
1] is a formula. To avoid confusion, we will always write “general model” instead of just “model”, because inthe literature on continuous model theory, “model” is used to mean what we call “metric model”.
ODEL THEORY FOR REAL-VALUED STRUCTURES 5 N if there is an embedding h : M → N . Note that the image of an embedding h : M → N is a substructure of N . An elementary embedding h : M ≺ N is anembedding that preserves the truth value of every formula. We say that M is elementarily embeddable in N if there exists an h : M ≺ N .In [BBHU], the reduction of a pre-metric structure was defined by identifyingelements that are at distance zero from each other. Some care is needed to choosethe right notion of reduction for general structures. In first order logic withoutequality, the reduction of a structure is formed by identifying two elements thatcannot be distinguished by atomic formulas. We do the analogous thing here forgeneral structures. Definition 2.1.3.
For a, b ∈ M , we write a . = M b if for every atomic formula ϕ ( x, ~z ) and tuple ~c ∈ M | ~z | , ϕ M ( a, ~c ) = ϕ M ( b, ~c ) . M is reduced if whenever a . = M b we have a = b .The relation . = M is a very old idea that goes back to Leibniz around 1840, andis called Leibniz equality . Remark 2.1.4.
For any general structure M and a, b ∈ M , a . = M b if and only ifin M , ( a, b ) satisfies the set of formulas { sup ~z | ϕ ( a, ~z ) − ϕ ( b, ~z ) | : ϕ is atomic } . The reduction map for M is the mapping that sends each element of M toits equivalence class under . = M . The reduction of the general structure M is thereduced structure N such that N is the set of equivalence classes of elements of M under . = M , and the reduction map for M is an embedding of M onto N . We saythat M , M ′ are isomorphic , in symbols M ∼ = M ′ , if there is an embedding from thereduction of M onto the reduction of M ′ . We write h : M ∼ = N if h : M → N , andfor each b ∈ N there exists a ∈ M such that h ( a ) . = N b . Remark 2.1.5. • ∼ = is an equivalence relation on general structures. • Every general structure is isomorphic to its reduction. • M , N are isomorphic if and only their reductions are isomorphic. • If there is an embedding of M onto N , then M ∼ = N . • M ∼ = N implies M ≡ N . • M ∼ = N if and only there exists h such that h : M ∼ = N . If V ⊆ V , and M is obtained from M by forgetting every symbol of V \ V ,we call M an expansion of M to V , and call M the V -part of M . Remark 2.1.6.
Suppose V ⊆ V , and M is the V -part of M . Then for everyformula ϕ ( ~x ) in the vocabulary V , and every tuple ~b ∈ M | ~x | , we have ϕ M ( ~b ) = ϕ M ( ~b ) . Remark 2.1.7. If M is reduced and M is an expansion of M , then M is reduced.Proof. For all a, b ∈ M , a . = M b implies a . = M b . (cid:3) Ultraproducts.
The ultraproduct of an indexed family of general structureswill be defined below as the reduction of the pre-ultraproduct, which is a generalstructure whose universe is the cartesian product. As we will see later, this will be a
H. JEROME KEISLER direct generalization of the ultraproduct of metric structures as defined in [BBHU].Recall that for any ultrafilter D over a set I and function g : I → [0 , r = lim D g in [0 ,
1] such that for each neighborhood Y of r , the set of i ∈ I such that g ( i ) ∈ Y belongs to D . Definition 2.2.1.
Let D be an ultrafilter over a set I and M i be a general structurefor each i ∈ I . The pre-ultraproduct Q D M i is the general structure M ′ = Q D M i such that: • M ′ = Q i ∈ I M i , the cartesian product. • For each constant symbol c ∈ V , c M ′ = h c M i i i ∈ I . • For each n -ary function symbol G ∈ V and n -tuple ~a in M ′ , G M ′ ( ~a ) = h G M i ( ~a ( i )) i i ∈ I . • For each n -ary predicate symbol P ∈ V and n -tuple ~a in M ′ , P M ′ ( ~a ) = lim D h P M i ( ~a ( i )) i i ∈ I . The ultraproduct Q D M i is the reduction of the pre-ultraproduct Q D M i . Foreach a ∈ M ′ we also let a D denote the equivalence class of a under . = M ′ .The following fact is the analogue for general structures of the fundamentaltheorem of Lo´s. Fact 2.2.2.
Let M i be a general structure for each i ∈ I , let D be an ultrafilterover I , and let M = Q D M i be the ultraproduct. Then for each formula ϕ and tuple ~b in the cartesian product Q i ∈ I M i , ϕ M ( ~b D ) = lim D h ϕ M i ( ~b i ) i i ∈ I . The special case of Fact 2.2.2 where each M i has a symbol = i for the discretemetric was already stated and proved in [CK66]. The general case can be obtainedfrom that special case by observing that if each M i has vocabulary V and N i =( M i , = i ), then Q D M i is the reduction of the V -part of Q D N i . Alternatively, Fact2.2.2 can be proved from scratch by induction on the complexity of formulas, as isdone in first order logic.If M i = M for all i ∈ I , the ultraproduct Q D M i is called the ultrapower of M modulo D , and is denoted by M I / D . Corollary 2.2.3.
For each M and ultrafilter D , M is elementarily embeddable into M I / D . Corollary 2.2.4.
Suppose M i ∼ = N i for each i ∈ I , and D is an ultrafilter over I .Then Q D M i ∼ = Q D N i .Proof. By Remark 2.1.5, for each i ∈ I there is a map h i : M i ∼ = N i . Let h : Q i ∈ I M i → Q i ∈ I N i be the mapping such that for a ∈ Q i ∈ I M i , h ( a ) = h h i ( a i ) i i ∈ I . For eachcontinuous formula ϕ ( ~v ) and tuple ~a ∈ ( Q i ∈ I M i ) | ~v | , it follows from Fact 2.2.2 that ϕ Q D M i ( ~a ) = lim D ( ϕ M i ( ~a i )) = lim D ( ϕ N i ( h i ( ~a i ))) = ϕ Q D N i ( h ( ~a )) . Therefore h : Q D M i → Q D N i .By Fact 2.2.2, if b, c ∈ Q i ∈ I N i and b i . = N i c i for each i ∈ I , then b . = Q D N i c . Itfollows that h : Q D M i ∼ = Q D N i , so by Remark 2.1.5, Q D M i ∼ = Q D N i . (cid:3) ODEL THEORY FOR REAL-VALUED STRUCTURES 7
The following is proved in the usual way, using Fact 2.2.2.
Fact 2.2.5. (Compactness) If every finite subset of T has a general model, then T has a general model. Definability and Types.
We write • r ∔ s for min( r + s, • r ≤ . s for max( r − s, • r ≤ . s ≤ . t for max( r ≤ . s, s ≤ . t ) . Thus r ≤ s if and only if r ≤ . s = 0. In the literature, r ≤ . s is sometimes written r − . s . Note that for any general structure M , the following are equivalent: • M | = sup ~x [ ϕ ( ~x ) ≤ . ψ ( ~x ) ≤ . θ ( ~x )]. • ( ∀ ~a ∈ M | ~x | )[ ϕ M ( ~a ) ≤ ψ M ( ~a ) and ψ M ( ~a ) ≤ θ M ( ~a )].In what follows, all formulas mentioned are understood to be in the vocabularyof a theory T . Let ~x be a tuple of variables, and ~y be a finite or infinite sequence ofvariables, where all the symbols x i and y j are distinct. Given a formula θ ( ~x, ~y ), welet sup ~y θ ( ~x, ~y ) denote the formula sup ~u θ ( ~x, ~y ) where ~u is the (necessarily finite)tuple of variables from ~y that occur freely in θ .We say that a sequence h ϕ m ( ~x, ~y ) i m ∈ N of formulas is Cauchy in T if for each ε > m such that for all k ≥ m , T | = sup ~x sup ~y | ϕ m ( ~x, ~y ) − ϕ k ( ~x, ~y ) | ≤ . ε. Cauchy in M means Cauchy in the complete theory Th( M ).If h ϕ m ( ~x, ~y ) i m ∈ N is Cauchy in T , then for each general model M of T there is aunique mapping from M | ~x | × M | ~y | into [0 , ϕ m ] M , such that( ∀ ~b ∈ M | ~x | )( ∀ ~c ∈ M | ~y | )[lim ϕ m ] M ( ~b, ~c ) = lim m →∞ ϕ M m ( ~b, ~c ) . We say that h ϕ m ( ~x, ~y ) i m ∈ N is exponentially Cauchy in T if whenever m ≤ k wehave T | = sup ~x sup ~y | ϕ m ( ~x, ~y ) − ϕ k ( ~x, ~y ) | ≤ . − m . Note that every exponentially Cauchy sequence of formulas in T is Cauchy, andevery Cauchy sequence in T has an exponentially Cauchy subsequence. Definition 2.3.1.
We say that a mapping P : M | ~x | × M | ~y | → [0 ,
1] is defined by h ϕ m ( ~x, ~y ) i m ∈ N in a general structure M , and is definable in M , if h ϕ m ( ~x, ~y ) i m ∈ N isCauchy in M and P = [lim ϕ m ] M .Note that for each general structure M , δ M ( ϕ, ψ ) := sup ~x sup ~y | ϕ ( ~x, ~y ) − ψ ( ~x, ~y ) | M is a pseudo-metric on the set of all formulas with free variables from ( ~x, ~y ), and theabove definition says that h ϕ m ( ~x, ~y ) i m ∈ N is Cauchy in T if and only if it is Cauchywith respect to δ M uniformly for all M | = T .We often consider the case where ~y is empty, or equivalently, where only finitelymany of the variables in ~y actually occur in some ϕ m ( ~x, ~y ). In that case, we havethe notion of a Cauchy sequence of formulas h ϕ m ( ~x ) i m ∈ N and a definable mapping P : M | ~x | → [0 ,
1] in M . H. JEROME KEISLER
We now introduce complete n -types. For each theory T and n ∈ N , a complete n -type over T is an n -type p ( ~x ) that is maximal with respect to being satisfiablein a general model of T . S n ( T ) denotes the set of all complete n -types over T . Inparticular, S ( T ) is the set of all complete extensions of T . For each p ∈ S n ( T )and formula ϕ ( ~x ) with n free variables, we let ϕ ( ~x ) p be the unique r such that | ϕ ( ~x ) − r | ∈ p . The logic topology on S n ( T ) is the topology whose closed sets arethe sets of the form { p ∈ S n ( T ) : Γ( ~x ) ⊆ p } for some n -type Γ( ~x ). It follows fromthe Compactness Theorem that: Fact 2.3.2.
For each theory T and n ∈ N , the logic topology on S n ( T ) is compact. Given a general structure M and a set A ⊆ M , let M A = ( M , a ) a ∈ A . The complete type of an n -tuple ~b over A in M is the set tp M ( ~b/A ) of all formulassatisfied by ~b in M A . The set S n (Th( M A )) of all complete n -types over A realizedin models of Th( M A ) is denoted by S M n ( A ). Thus the logic topology on S M n ( A ) iscompact.We say that a mapping P ( ~x, ~y ) is definable over A in M if P is definable in M A .The following lemma gives a relationship between P ( ~x ) being definable over acountable sequence of parameters, and P ( ~x, ~y ) being definable without parameters. Lemma 2.3.3.
Let B ⊆ M . A mapping P ( ~x ) is definable over B in M if and onlyif there is a countable sequence ~b of elements of B and an exponentially Cauchysequence h ϕ ( ~x, ~y ) i of formulas in M such that for all ~a ∈ M | ~x | we have P ( ~a ) = Q ( ~a,~b ) where Q is the mapping defined by h ϕ ( ~x, ~y ) i in M .Proof. It is clear that if there is such a sequence of formulas, then P ( ~x ) is definableover B in M . Suppose P ( ~x ) is definable over B in M , by a sequence h θ m ( ~x ) i m ∈ N with parameters in B . By taking a subsequence if necessary, we may insure thatfor some sequence ~b of elements of B , each θ m has parameters from ~b , and for each m we have(1) M | = sup ~x | θ m ( ~x,~b ) − θ m +1 ( ~x,~b ) | ≤ − ( m +1) . We now use the forced convergence trick in [BU] to find a new sequence of formulas ϕ m ( ~x, ~y ) in such a way that(2) M | = sup ~x sup ~y | ϕ m ( ~x, ~y ) − ϕ m +1 ( ~x, ~y ) | ≤ − ( m +1) , and that ϕ M m ( ~a,~b ) = θ M m ( ~a,~b ) whenever ~a ∈ M | ~x | and m ∈ N . To do that weinductively define ϕ = θ , and ϕ m +1 = max( ϕ m − − ( m +1) , min( ϕ m + 2 − ( m +1) , θ m +1 )) . Condition (2) implies that ϕ m ( ~x, ~y ) is exponentially Cauchy in M , and defines amapping Q in M such that P ( ~a ) = Q ( ~a,~b ) for all ~a ∈ M | ~x | . (cid:3) The following result is the analogue for general structures of Theorem 9.9 of[BBHU].
Proposition 2.3.4.
Let M be a general structure, A ⊆ M , and P : M n → [0 , .The following are equivalent: (i) P is a definable predicate over A in M . ODEL THEORY FOR REAL-VALUED STRUCTURES 9 (ii)
There is a unique function
Ψ : S M n ( A ) → [0 , such that P ( ~c ) = Ψ(tp M ( ~c/A )) for all ~c ∈ M n . Furthermore, Ψ is continuous in the logic topology on S M n ( A ) Proof.
The proof that (ii) ⇒ (i) is the same as the proof of Theorem 9.9 in [BBHU],which did not use a metric on M .(i) ⇒ (ii): Suppose that P is defined by h ϕ m i m ∈ N over A in M . Then h ϕ m i m ∈ N isCauchy in Th( M A ). There is a unique Ψ : S M n ( A ) → [0 ,
1] such that Ψ(tp M ( ~c/A )) =[lim ϕ m ] M A ( ~c ) = P ( ~c ) for each ~c ∈ M n . For each closed interval I ⊆ [0 , n -type Γ I ( ~x ) with parameters in A such that for all ~c ∈ M n , Ψ(tp M ( ~c/A )) ∈ I if andonly if M A | = Γ I ( ~c ). Therefore Ψ − ( I ) is closed in S M n ( A ), so Ψ is continuous. (cid:3) Remark 2.3.5.
Let A ⊆ M and ~b in M n . • If N ≻ M then tp M ( ~b/A ) = tp N ( ~b/A ) . • ϕ M ( ~b, A ) = r if and only if | ϕ ( ~x, A ) − r | belongs to tp M ( ~b/A ) .Using the compactness theorem, we have • If N ≻ M then S M n ( A ) = S N n ( A ) . • There exists N ≻ M in which every type in S n S M n ( A ) is realized. Saturated and Special Structures.
For an infinite cardinal κ , we say thata general structure M is κ -saturated if for every set A ⊆ M of cardinality | A | < κ ,every set of formulas in the vocabulary of M with one free variable and parametersfrom A that is finitely satisfiable in M A is satisfiable in M A . Remark 2.4.1. If M is κ -saturated, A ⊆ M , and | A | < κ , then every completetype p ∈ S n ( A ) is realized in M . Remark 2.4.2. M is κ -saturated if and only if the reduction of M is κ -saturated. Definition 2.4.3.
By a special cardinal we mean a cardinal κ such that 2 λ ≤ κ for all λ < κ . We say that M is special if | M | is an uncountable special cardinaland M is the union of an elementary chain of structures h M λ : λ < | M |i such thateach M λ is λ + -saturated. M is κ -special if κ is special and M is the reduction of aspecial structure of cardinality κ .Note that every strong limit cardinal is special, and if 2 λ = λ + then 2 λ is special.Thus every κ -special structure is reduced and has cardinality ≤ κ . A κ -specialstructure may have cardinality less than κ . For example, every metric structurewith a compact metric is κ -special for every special cardinal κ but has cardinalityat most 2 ℵ . Remark 2.4.4. If M is κ -special and V ⊆ V , then the reduction of the V -partof M is κ -special. Remark 2.4.5. If κ is an uncountable inaccessible cardinal, then M is κ -special ifand only if M is a reduced κ -saturated structure of cardinality κ . Fact 2.4.6. (Uniqueness Theorem for Special Models) If T is complete and M , N are κ -special models of T , then M and N are isomorphic. An easy consequence is:
Fact 2.4.7.
Suppose M is κ -special, A ⊆ M , and | A | < cofinality of κ . Then forall tuples ~b, ~c in M , tp M ( ~b/A ) = tp M ( ~c/A ) if and only if M has an automorphismthat sends ~b to ~c and is the identity on A . The following result is proved using the Compactness, Downward L¨owenheim-Skolem, and Elementary Chain Theorems.
Fact 2.4.8. (Existence Theorem for Special Models) If κ is a special cardinal and ℵ + | V | < κ , then every reduced structure M such that | M | ≤ κ has a κ -specialelementary extension. The statement of Fact 2.4.8 above differs slightly from the corresponding resultin [CK66], because we do not have a symbol for the equality relation here.2.5.
Pre-metric Structures. A metric signature L over V specifies a distin-guished binary predicate symbol d ∈ V for distance, and equips each predicateor function symbol S ∈ V with a modulus of uniform continuity △ S : (0 , → (0 , d .In the literature on metric structures, one usually fixes a metric signature L once and for all, but here we focus on general structures that are not equipped withmetric signatures, and often consider many metric signatures at the same time. Forthat reason, we will officially define a pre-metric structure to be pair consisting ofa general structure and a metric signature.A pre-metric structure M + = ( M , L ) consists of a general structure M withvocabulary V , and a metric signature L over V , such that d M is a pseudo-metricon M , and for each predicate symbol P and function symbol F of arity n , and forall ~a,~b ∈ M n and ε ∈ (0 , k ≤ n d M ( a k , b k ) < △ P ( ε ) ⇒ | P M ( ~a ) − P M ( ~b ) | ≤ ε. and max k ≤ n d M ( a k , b k ) < △ F ( ε ) ⇒ d M ( F M ( ~a ) , F M ( ~b )) ≤ ε. A metric structure is a pre-metric structure ( M , L ) such that ( M, d M ) is a completemetric space. The paper [BBHU] emphasized metric structures, but in this paperwe will focus more on pre-metric than metric structures.Given a pre-metric structure M + = ( M , L ), we will call M the downgrade of M + ,and call M + the upgrade of M to L . Note that two different pre-metric structurescan have the same downgrade, because they may have different metric signatures.We will slightly abuse notation and say that M is a pre-metric (or metric)structure for L when ( M , L ) is a pre-metric (or metric) structure. We say thata pre-metric structure M + for L is κ -saturated if its downgrade is κ -saturated.Similarly for other properties of general structures, such as being reduced, or beingan ultraproduct of a family of structures.We remind the reader that every pre-metric structure for L has a unique com-pletion up to isomorphism, that this completion is a metric structure for L , andthat every pre-metric structure is elementarily embeddable in its completion. Remark 2.5.1.
Every pre-metric structure for L that is reduced and ℵ -saturatedis a metric structure for L . In [BBHU], when M + is a metric structure, a mapping from M n into [0 ,
1] thatis uniformly continuous with respect to d M is called a predicate on M + .We will make frequent use of the following result from [BBHU]. ODEL THEORY FOR REAL-VALUED STRUCTURES 11
Fact 2.5.2. (Theorem 3.5 in [BBHU]) For every metric signature L and formula ϕ ( ~x ) in the vocabulary of L , there is a function △ ϕ : (0 , → (0 , that is a mod-ulus of uniform continuity for the mapping ϕ M : M | ~x | → [0 , in every pre-metricstructure M + for L . Fact 2.5.3. (See [Ca], page 112, and Definition 2.4 in [AH].) For each metricsignature L over V , there is a theory met( L ) whose general models are exactly thedowngrades of pre-metric structures for L . Each sentence in met( L ) consists offinitely many sup quantifiers followed by a quantifier-free formula. The sentences in met( L ) formally express that d is a pseudo-metric, and thatthe functions and predicates in V respect the moduli of uniform continuity for L . Corollary 2.5.4.
Every ultraproduct of pre-metric structures for L is a pre-metricstructure for L .Proof. By Facts 2.2.2 and 2.5.3. (cid:3) A metric theory ( T, L ) consists of a metric signature L and a set T of sentencesin the vocabulary of L such that T | = met( L ). A pre-metric (or metric) model of ametric theory ( T, L ) is a pre-metric (or metric) structure M + = ( M , L ) for L suchthat ϕ M = 0 for each sentence ϕ ∈ T .When ( T, L ) is a metric theory, we also say that T is a metric theory withsignature L . Note that for every metric theory ( T, L ), pre-metric structure ( M , L ),and continuous sentence ϕ , we have T | = M iff ( T, L ) | = ( M , L ), and T | = ϕ iff( T, L ) | = ϕ . Also, if ( M , L ) is a pre-metric model of a metric theory ( T, L ), then M is a general model of T .Part (i) of the next lemma shows that when M + is a metric structure, the defi-nition of a definable predicate here agrees with the notion of a definable predicatein [BBHU]. Lemma 2.5.5. (i)
In every pre-metric structure M + for L , every mapping that is definable inthe sense of Definition 2.3.1 is uniformly continuous. (ii) For every metric theory T with signature L , and sequence of formulas h ϕ m ( ~x ) i m ∈ N that is Cauchy in T , there is a function △ h ϕ i that is a mod-ulus of uniform continuity for the definable predicate [lim ϕ m ] M in everypre-metric model M + of T .Proof. (ii) trivially implies (i). We prove (ii). Given ε >
0, take the least m so thatfor all n ≥ m , T | = sup ~x | ϕ m ( ~x ) − ϕ n ( ~x ) | ≤ . ε/ . Let △ h ϕ i ( ε ) = △ ϕ m ( ε/
3) . Suppose M + is a pre-metric model of T , and max k ≤ n d M ( a k , b k ) < △ h ϕ i ( ε ). Then | [lim ϕ m ] M ( ~a ) − [lim ϕ m ] M ( ~b ) | ≤| [lim ϕ m ] M ( ~a ) − ϕ M m ( ~a ) | + | ϕ M m ( ~a ) − ϕ M m ( ~b ) | + | ϕ M m ( ~b ) − [lim ϕ m ] M ( ~b ) | ≤ ε/ ε/ ε/ ε. (cid:3) The next fact follows at once from Theorem 3.7 in [BBHU].
Fact 2.5.6. If M + is a pre-metric structure, a . = M b if and only if d M ( a, b ) = 0 . As mentioned above, in [BBHU], the notion of a reduction of a pre-metric struc-ture M + was defined as the structure obtained by identifying elements x, y when d ( x, y ) = 0, rather than when x . = y . So the reduction of a pre-metric structureas defined here is the same as the reduction of a pre-metric structure as definedin [BBHU]. It follows that the ultraproduct of a family of metric structures withsignature L as defined here is the same as the ultraproduct of a family of metricstructures with signature L as defined in [BBHU]. Fact 2.5.7. (By Proposition 5.3 in [BBHU].) Every ultraproduct of metric struc-tures for L is a metric structure for L . Corollary 2.5.8. (Metric Compactness Theorem) If T is a metric theory withsignature L , and every finite subset of T has a metric model, then T has a metricmodel. In [BBHU], the definitions of a pre-metric structure and of an ultraproduct weredesigned to insure that Fact 2.5.8 and the Metric Compactness Theorem hold.There were pitfalls to avoid in extending the notion of an ultraproduct to generalstructures. For example, if one adds a discontinuous predicate or function to apre-metric structure and tries to define an ultraproduct by identifying elementsat distance 0 from each other, the added predicate or function symbol would beundefined in the ultraproduct.Such pitfalls were avoided here by defining the reduction of a general structureto be the general structure formed by identifying x with y when x . = M y . Usingthat notion of reduction, we obtained a well-defined notion of ultraproduct thatcoincides with the notion in [BBHU] for metric structures, satisfies the theorem of Lo´s (Fact 2.2.2) in all cases, and leads to the Compactness Theorem for generalstructures.2.6. Some Variants of Continuous Model Theory.
In this subsection we dis-cuss some cases from the literature where variants of continuous model theory havebeen developed in order to study special classes of general structures that sharesome features with pre-metric structures. In each of these cases, one can insteadwork with the model theory of general structures as developed here.
Example 2.6.1. (Infinitary Continuous Logic) Christopher Eagle, in [Ea14] and[Ea15], developed an infinitary logic L ω ω that is analogous to continuous logic, buthas much greater power of expression and fails to satisfy the compactness theorem.The L ω ω -formulas in a vocabulary V are built from the atomic formulas usingthe connectives and quantifiers of continuous logic and, in addition, the operationssup m ϕ m and inf m ϕ m whenever h ϕ m i m ∈ M is a sequence of formulas in which onlyfinitely many variables occur freely. Given an L ω ω -formula ϕ ( ~x ) and a pre-metricstructure M , the truth value function ϕ M : M | ~x | → [0 ,
1] is defined in [Ea15] as onewould expect by induction on complexity.We adopt here the same definition for the truth value ϕ M of an L ω ω -formulain a general structure M . Then ( M , ϕ M ) is a general structure whose vocabularyhas an extra | ~x | -ary predicate symbol. However, even if M is a metric structure,( M , ϕ M ) is not necessarily a metric structure, and ϕ M may even be discontinuous.We will revisit this concept in Subsection 4.4. ODEL THEORY FOR REAL-VALUED STRUCTURES 13
Example 2.6.2. (Geodesic Logic) The paper [Cho17] introduced a variant of con-tinuous model theory, called geodesic logic , whose structures are pre-metric struc-tures with extra functions that are possibly discontinuous. He used this to obtainapproximate fixed point results and metastability results for certain discontinuousfunctions.A geodesic signature G consists of a metric signature H over a vocabulary V ,and a set Y of extra symbols for possibly discontinuous functions, including aset { L t : t ∈ [0 , } of binary function symbols (the linear structure). A geodesicstructure with signature G is a general structure in our present sense, that has avocabulary V ∪ Y whose V -part is a pre-metric structure with signature H , thatsatisfies the following sentences for each s, t ∈ [0 , x sup y [ | d ( L s ( x, y ) , L t ( x, y )) − | s − t | d ( x, y ) | ≤ . , (3) sup x sup y [ d ( L t ( x, y ) , L − t ( y, c )) ≤ . . (4)Thus a geodesic structure is just a general model of the theory T G = met H ∪ { (3) , (4) } . For example, in our setting, Theorem 6.6 (a) of [Cho17] can be stated as follows:
Let λ ∈ [0 , and let G be a geodesic signature with distance predicate d , and anextra unary function symbol F . Let M be a general model of the theory (5) T G ∪ { sup x [ d ( F x, F L λ ( x, F x )) ≤ . d ( x, F x )] } . In M , when x n +1 = L λ ( x n , F x n ) for all n , we have lim n →∞ d ( x n , x n +1 ) = 0 . The compactness theorem can then be used to get a uniform metastability boundacross all general models of (5) (Theorem 6.6 (b) of [Cho17]).
Example 2.6.3. (Sorted Vocabularies) Many-sorted metric structures are promi-nent in the literature. Here we will work exclusively with general structures, asdefined at the beginning of this section, but introduce the notions of a sorted vo-cabulary, and of a general structure that respects the sorts in that vocabulary. Ageneral structure that respects sorts cannot be a pre-metric structure. We have theflexibility of starting with a general structure that respects sorts, and adding a newunsorted distance predicate to form a pre-metric structure that does not respectsorts. As we will see later, that flexibility will be useful when we consider suchtopics as unbounded metrics and imaginary elements.A sorted vocabulary W consists of a set of sorts, sets of finitary predicate andfunction symbols, and a set of constant symbols. Each argument of a predicate orfunction symbol is equipped with a sort, and each function and constant symbol isequipped with a sort for its value. Moreover, W contains a unary predicate symbol U S associated with each sort S , and a constant symbol u (for “unsorted”) that hasno sort.We say that a general structure M with vocabulary W respects sorts if: • The universe M of M contains a family of pairwise disjoint non-emptyuniverse sets S M corresponding to the sorts S of W . • For each sort S of W and a ∈ M , U M S ( a ) = 0 when a ∈ S M , and U M S ( a ) = 1when a / ∈ S M . • For each k -ary predicate symbol P of W and ~a ∈ M k , P M ( ~a ) = 1 when atleast one argument is of the wrong sort. • For each k -ary function symbol F of W with value sort S and ~a ∈ M k , F M ( ~a ) belongs to S M when each argument is of the correct sort, and is u M when at least one argument is of the wrong sort. • The value c M of each constant symbol c of sort S is an element of S M . • The constant u M does not belong to any of the sets S M .The unsorted constant symbol u serves two purposes: It allows one to interpreteach function symbol as a total function rather than as a partial function from aproduct of sorts to a sort, and it insures that each structure that respects sorts hasan unsorted element. Lemma 2.6.4.
Let W be a sorted vocabulary and let M be a general structure withvocabulary W . (i) If M respects sorts, then the reduction of M also respects sorts. (ii) If M is reduced and respects sorts, then u M is the unique element of M thatdoes not belong to S M for any sort S of W . (iii) There is a set rs( W ) of sentences such that for every reduced structure M with vocabulary W , M respects sorts if and only if M | = rs( W ) .Proof. (i) is clear.(ii): Let a be an element of M that does not belong to S M for any sort S of W . Then for any tuple ~c in M and atomic formula ϕ ( x, ~c ) in which the variable x occurs, we have ϕ M ( a, ~c ) = 1 and ϕ M ( u M , ~c ) = 1. Thus a . = M u M , so (ii) holds.(iii): Each of the requirements for respecting sorts can be expressed in M be a setof sentences. For example, the requirement that two sorts S and S are pairwisedisjoint is expressed by the sentencesup x (1 ≤ . max( U S ( x ) , U S ( x ))) . By Remark 2.1.4, F ( x, ~y ) . = u can be expressed in M by a set of formulas. One canuse that to show that when M is reduced, the requirement that F M ( x, ~y ) = u M when x does not have sort S can be expressed by a set of sentences. The otherrequirements are similar. With a bit more work, one can take each sentence inrs( W ) to be a finite set of sup quantifiers followed by a quantifier-free formula. (cid:3) Definition 2.6.5.
Given a sorted vocabulary W , a sorted metric signature L over W consists of a distinguished distance predicate symbol d S for each sort S of W ,and a modulus of uniform continuity for each predicate and function symbol withrespect to these distance predicates. By a sorted metric (or pre-metric) structure with the sorted signature L we mean a reduced general structure M with vocabulary W that respects sorts, such that in M , each d S is a complete metric (or metric) on S , and each symbol of W satisfies the modulus of uniform continuity given by L with respect to these metrics.We will return to sorted metric structures in Subsection 4.5. Example 2.6.6. (Bounded Continuous Logic)The version of continuous logic as defined in Section 2 of [BBHU], which we willhere call bounded continuous logic , is slightly broader than the [0 , bounded vocabulary is a vocabulary V equipped ODEL THEORY FOR REAL-VALUED STRUCTURES 15 with a bounded real interval [ r P , s P ] where r P < s P , for each predicate symbol P ∈ V . The metric signatures, continuous formulas, and metric structures withthe bounded vocabulary V are as in the [0 , , ∞ ) into (0 , ∞ ), the distinguished metric d is a metric with values in [0 , s d ], the logical connectives are continuous functionsfrom R n into R , and the predicates P have truth values in the bounded interval[ r P , s P ]. The truth values of formulas in a bounded metric structure are defined byinduction on complexity of formulas in the usual way, and the truth values are in R . Many-sorted bounded metric structures are treated in a similar way.One can regard [0 , P , including the distinguishedmetric d , has the interval [0 , , , , M to a [0 , M by replacing each predicate P M by the [0 , P M where P M ( ~v ) = ( P M ( ~v ) − r P ) / ( s P − r P ) . It then turns out that a predicate P : [0 , n → [0 ,
1] is definable in M if and only if itis definable in M . In that sense, the [0 , , P with a bounded real interval. Formulas and truth values aredefined in the same way for bounded general structures as they are for boundedmetric structures in [BBHU]. Example 2.6.7. (Unbounded Continuous Logic) Many important mathematicalstructures, such as Banach spaces, valued fields, and C ∗ -algebras, are structureswith an unbounded complete metric d , functions that are uniformly continuouswith respect to d , predicates with truth values in R that are uniformly continuouswith respect to d , and constant symbols. We will call such structures unboundedmetric structures .The inductive definition of the truth value of a formula in a bounded metricstructure does not automatically carry over to unbounded metric structures, be-cause the sup of a set of values may not exist in R . In the literature, there are atleast three approaches to the study of an unbounded metric structure N by applyingthe existing model theory to some metric structure that is associated with N .One approach is to add a constant symbol 0 for a distinguished element of N ,and look at the many-sorted bounded metric structure M that has a sort S m for theclosed m -ball around 0 for each 0 < m ∈ N , and, for each 0 < m < k , a function i mk of sort S m → S k for the inclusion map. For simplicity, suppose N has no functionsymbols. For every constant symbol c and n -ary predicate symbol P of N , and every sort S m , M will have a constant symbol c m and an n -ary predicate symbol P m ofsort S m . The uniform continuity property of N guarantees that for each predicatesymbol P of N , and each m > P N maps the closed m -ball around 0 into abounded interval [ r Pm , s Pm ]. For each predicate symbol P , the bounded vocabularyof M will be equipped with both the arity of P and the bounded interval [ r Pm , s Pm ].In [Fa], C ∗ -algebras are treated as many-sorted bounded metric structures asin the preceding paragraph. In order to make things work properly, axioms areexplicitly added to guarantee that the inclusion map sends the sort S m onto theclosed m -ball in S k . In [BBHU], Sections 15 and 17, Hilbert spaces and Banachlattices are treated in a similar way (see also Remark 4.6 in [BU], and [BY09]).In those cases, in any metric model of the many-sorted theory, the inclusion mapsends the sort S m onto the closed m -ball around 0 in S k .A second, and simpler, approach is to look at the single-sorted [0 , N to the unit ball around0, and normalizing each predicate symbol so that it takes truth values in [0 , L p spaces, the many-sorted metric structure hasessentially the same model-theoretic properties as the single-sorted structure on theunit ball, and are able to successfully simplify things by working exclusively withthe unit ball.However, as pointed out in [BY08] and [BY14], there are other cases, such as innon-archimedean valued fields, where neither of the above approaches is adequate.The many-sorted structure M may have definable predicates that would not beconsidered definable in the original structure N . For example, if m < k , the distancedist( x, B ) from a point x ∈ S k to the set B = range( i mk ) might not be definable in N , but is always definable in M by the formula inf y d k ( x, i mk ( y )).To deal with that problem, Ben Yaacov [BY08] developed a variant of continuouslogic with an unbounded metric and modified notions of formula and ultraproduct.That logic has a Lipschitz gauge function ν : N → [0 , ∞ ) that is thought of as thesize of an element, and determines the closed m -balls B m = { x : ν ( x ) ≤ m } . Inmost examples, the gauge ν ( x ) is d ( x, x to a distinguishedpoint 0. The universe of the modified pre-ultraproduct is the set of elements x ofthe usual pre-ultraproduct such that h ν i ( x i ) i i ∈ I is bounded. This means that thegauge of x is finite, which in most examples means that d ( x,
0) is finite. Usingthat logic, [BY08] converted an unbounded metric structure N into an ordinary[0 , N ∞ via his emboundment construction. The intuitiveidea is to add a point at infinity to N , and to add a new metric to form a metricstructure N ∞ such that the theories of N and N ∞ have the same model-theoreticproperties. The reason for doing that is to study the unbounded metric structure N by applying ordinary continuous model theory to N ∞ .The logic developed in [BY08] is in many ways equivalent to the logic initiated in1976 by Henson [He], and developed further by Henson and Iovino [HI], and Dueezand Iovino [DI]. That approach uses a similar notion of unbounded metric struc-ture and modified ultraproduct as [BY08], but with a different notion of formula,and a semantics based on approximate truth. Another form of unbounded contin-uous logic, that uses a similar notion of unbounded metric structure and modifiedultraproduct but a different notion of formula, is developed in [Lu].The bounded continuous logic of [BBHU] is better suited for applications thaneither of the logics developed in [BY08] or in [HI] and [DI], because the notions ODEL THEORY FOR REAL-VALUED STRUCTURES 17 of formula and truth value are simpler and more natural, and the formulas areeasier to understand, in [BBHU] than in the other approaches. For that reason, itdesirable to find a way to use bounded continuous logic to study unbounded metricstructures when possible.We will return to unbounded metric structures in Subsection 4.6 below.3.
Turning General Structures into Metric Structures
In this section we define the key notion of a pre-metric expansion, and show thatfor every theory T there exists a pre-metric expansion of T .3.1. Definitional Expansions.
In this subsection we introduce definitional ex-pansions of a theory, and in the next subsection we will introduce pre-metric ex-pansions as a special case.
Definition 3.1.1.
Let T be a general theory in a vocabulary V , let D be a predicatesymbol that may or may not belong to V , and let V D = V ∪ { D } . A definitionalexpansion of T over V D is a theory T e with vocabulary V D such that for somesequence h d i = h d m ( ~x ) i m ∈ N of formulas of V that is Cauchy in T :(i) For every general model M of T , M e = ( M , [lim d m ] M ) is the a uniqueexpansion of M to a general model of T e .(ii) Every general model of T e is equal to ( M , [lim d m ] M ) where M is a generalmodel of T .We say that the sequence h d i approximates D in T e . Note that if h d i approximates D in T e , then so does every subsequence of h d i . Therefore, for every definitionalexpansion T e of T over V D , there is an exponentially Cauchy sequence of formulasof V that approximates D in T e . Lemma 3.1.2. (Axioms for T e ) Suppose T e is a definitional expansion of T , and h d m ( x, y ) i m ∈ N is exponentially Cauchy in T and approximates D in T e . Let T ′ e bethe union of T and the set of sentences (6) { sup x sup y ( | d m ( x, y ) − D ( x, y ) | ≤ . − m ) : m ∈ N } . Then T e is equivalent to T ′ e .Proof. Let N be a general structure with vocabulary V D , and let M be the V -partof N . Then N satisfies (6) if and only if D N = [lim d m ] M . By 3.1.1 (i), every generalmodel of T ′ e is a general model of T e . By 3.1.1 (ii), every general model of T e is ageneral model of T ′ e . (cid:3) Remark 3.1.3.
Suppose T e is a definitional expansion of T with vocabulary V D . (i) For each sequence h d m i m ∈ N of formulas of V that approximates D in T e ,and each general model M of T , D M e = [lim d m ] M is defined by h d i in M . (ii) If T e , T f are definitional expansions of T with vocabulary V D , and there isa sequence of formulas of V that approximates D in both T e and T f , then T e and T f are equivalent. (iii) If T ⊆ U , then U e := T e ∪ U is a definitional expansion of U , every sequencethat approximates D in T e approximates D in U e , and for every generalmodel M of U , the definitional expansion M e of M is the same for U e asfor T e . (iv) For each general model M of T , Th( M e ) is a definitional expansion of Th( M ) . The next remark says that definitional expansions are preserved under the ad-dition of constant symbols.
Remark 3.1.4.
Suppose V ′ = V ∪ C where C is a set of constant symbols. If T e is a definitional expansion of the theory T with vocabulary V D , then T e is still adefinitional expansion of T with vocabulary V ′ D . Lemma 3.1.5.
Suppose T e is a definitional expansion of a theory T , and M is ageneral model of T . (i) ( . = M ) = ( . = M e ) . (ii) M e is reduced if and only if M is reduced. (iii) If M ′ is the reduction of M , then M ′ e is the reduction of M e . (iv) If M ′ is a general model of T and h : M ∼ = M ′ , then h : M e ∼ = M ′ e .Proof. Let h d m ( ~x ) i m ∈ N approximate D in T d .(i): It is trivial that x . = M e y implies x . = M y . Suppose x . = M y . Consider anatomic formula D ( ~τ ( x, ~z )), where ~τ is a tuple of terms, and none of the variables x, y, ~z occur freely in the any of the approximating formulas d m . Then for all ~z in M e we have D ( ~τ ( x, ~z )) = lim m →∞ d m ( ~τ ( x, ~z )) = lim m →∞ d m ( ~τ ( y, ~z )) = D ( ~τ ( y, ~z )) , so x . = M e y. Therefore ( . = M e ) = ( . = M ).(ii): Expansions of reduced structures are always reduced. Suppose M e is re-duced. By the proof of (ii), if x . = M y then x . = N y , and since M e is reduced, x = y .Hence M is reduced.(iii) For each x ∈ M let x ′ ∈ M ′ be the equivalence class of x under . = M . Then D M e ( ~x ) = lim m →∞ d M m ( ~x ) = lim m →∞ d M ′ m ( ~x ′ ) = D M ′ e ( ~x ′ ) , so M ′ e is the reduction of M e .(iv): By (ii), we may assume that M , M ′ are reduced. Then any isomorphism h : M ∼ = M ′ sends [lim d m ] M to [lim d m ] M ′ . (cid:3) Proposition 3.1.6. (Definitional expansions commute with ultraproducts.) Sup-pose T e is a definitional expansion of T , D is an ultrafilter over a set I , and M i | = T for each i ∈ I . Then ( Q D M i ) e = Q D (( M i ) e ) , and ( Q D M i ) e = Q D (( M i ) e ) .Proof. By Fact 2.2.2, Q D M i , Q D M i are models of T , and Q D (( M i ) e ), Q D (( M i ) e )are models of T e . ( Q D M i ) e = Q D (( M i ) e ) because Q D M i is the V -part of Q D (( M i ) e ). Therefore, by Lemma 3.1.5 (ii), ( Q D M i ) e = Q D (( M i ) e ). (cid:3) Pre-metric Expansions.
We assume hereafter that D is a binary predicatesymbol. Definition 3.2.1.
Let T be a theory in a vocabulary V . We say that T e is apre-metric expansion of T (with signature L e ) if:(i) ( T e , L e ) is a metric theory whose signature L e has vocabulary V D and dis-tance predicate D . ODEL THEORY FOR REAL-VALUED STRUCTURES 19 (ii) There is a Cauchy sequence h d i of formulas in T such that the generalmodels of T e are exactly the structures of the form M e = ( M , [lim d m ] M ),where M is a general model of T .Since ( T e , L e ) is a metric theory, we have T e | = met( L e ), and every general modelof T e is the downgrade of a pre-metric structure with signature L e . Condition (ii)in the above definition just says that T e is a definitional expansion of T . So eachof the results 3.1.2 – 3.1.6 hold for pre-metric expansions because they hold for alldefinitional expansions.We call ( M e , L e ) the pre-metric expansion of M for T e , and call M the non-metric part of M e . We abuse notation by using M e to denote both the pre-metricexpansion ( M e , L e ) of M , and its downgrade M e . We call a sequence h d i thatapproximates D in T e an approximate distance for T e , and also for M e . Everypre-metric expansion of T has approximate distances, but they are not unique.Here are three rather trivial examples of pre-metric expansions. Example 3.2.2.
Suppose V has no predicate symbols. Then up to equivalence,the unique pre-metric expansion of T is the theory T e = T ∪ { sup x sup y D ( x, y ) ≤ . } in which the distance between any two elements is 0. Every reduced model of T orof T e is a one-element structure. Example 3.2.3.
Suppose the predicate symbol D already belongs to V , L is ametric signature over V with distance predicate D , and T | = met( L ). Let T e bethe metric theory with signature L and the same set of sentences as T . Then T e isa pre-metric expansion of T , and for each general model M of T , the upgrade of M to L is the pre-metric expansion of M for T e . Example 3.2.4.
If there is a formula D ( x, y ) in the vocabulary V such that D M is . = M in every general model M of T , then T has a pre-metric expansion with thedistinguished distance D and the trivial moduli of uniform continuity. We call sucha pre-metric expansion discrete . A theory T has a discrete pre-metric expansion ifand only if T has a Cauchy sequence of formulas h d m ( x, y ) i m ∈ N such that in everyreduced model N of T , [lim d m ] N is the discrete metric = N . (Hint: Take d m ( x, y )to be exponentially Cauchy, so that d is always within 1 / D ( x, y ) = C ( d ( x, y )) for an appropriate connective C .)Note that a classical structure without equality (where every atomic formulahas truth value 0 or 1) will not necessarily have a discrete pre-metric expansion.But by the Expansion Theorem below, it will have a pre-metric expansion if V iscountable. From this point on, except when we specify otherwise, every general structurewe consider will be understood to be a general structure whose vocabulary has atmost countably many predicate and function symbols. We let V be a vocabularythat contains at most countably many predicate and function symbols (we make norestriction on the number of constant symbols). We let D be a binary predicatesymbol and let V D = V ∪ { D } . Formulas in the vocabulary V will be called V -formulas , and general structuresfor V will be called V -structures. Similarly for terms and sentences. Unless we sayotherwise, T will be a V -theory, that is, a set of V -sentences. Definition 3.2.5.
Let T be a complete V -theory. We call h d i = h d m ( x, y ) i m ∈ N an exact distance in T if(i) h d i is Cauchy in T .(ii) [lim d m ] M is a pseudo-metric in every general model of T .(iii) For each general model M | = T and V -formula ϕ ( ~x ), the mapping ϕ M : M | ~x | → [0 ,
1] is uniformly continuous in the pseudo-metric space ( M, [lim d m ] M ). Proposition 3.2.6.
Suppose T e is a pre-metric expansion of a complete V -theory T with approximate distance h d i = h d m ( x, y ) i m ∈ N . Then h d i is an exact distancein T .Proof. By the definition of a pre-metric expansion, Definition 3.2.5 (i) and (ii) hold,and for every M | = T , M e = ( M , [lim d m ] M ) is a pre-metric structure with signature L e . Then by Fact 2.5.2, (iii) holds. (cid:3) We now state two results, Theorems 3.2.7 and 3.2.8, that follow from the proofsof results from [I94] in the context of Henson’s Banach space model theory [He],and from [BU] in the context of open Hausdorff cats. Those results together implythat every complete theory has a pre-metric expansion. In order to convert theproofs from [I94] and [BU] to proofs of Theorems 3.2.7 and 3.2.8 in our presentsetting, we would need a long detour through positive bounded formulas and openHausdorff cats. Instead, in the next subsection we will give a different and self-contained proof of a stronger result, that every (not necessarily complete) theoryhas a pre-metric expansion with an approximate distance h d m i m ∈ N such that each d m defines a pseudo-metric in every general model of T . Theorem 3.2.7. (By the proof of Proposition 53 of [I94], Theorem 5.1 of [I99],or Theorem 2.20 of [BY05].) Every complete V -theory T has an exact distance. Theorem 3.2.8. (By Theorem 4.23 of [BU].) Let T be a complete V -theory. If T has an exact distance, then T has a pre-metric expansion. Note that by Proposition 3.2.6, every complete V -theory has a pre-metric ex-pansion if and only if both Theorem 3.2.7 and Theorem 3.2.8 hold.3.3. The Expansion Theorem.
We will show that every V -theory T has a pre-metric expansion with an approximate distance h d m ( x, y ) i m ∈ N such that each d m defines a pseudo-metric in every general model of T . In Lemma 3.3.3 below weprove this in the special case that V has no function symbols. We will use thatto prove the general result in Theorem 3.3.4. By Remark 3.1.3 (iii), it is enoughto show that the empty set of sentences with vocabulary V has such a pre-metricexpansion. Definition 3.3.1.
A formula ϕ ( ~x, ~y ) is pseudo-metric in T if | ~x | = | ~y | and ϕ M is a pseudo-metric on M | ~x | for every general model M | = T . By a pseudo-metricapproximate distance for a pre-metric expansion T e of T , we mean an approximatedistance h d m ( x, y ) i m ∈ N for T e such that each d m is a pseudo-metric formula in T . Remark 3.3.2. (i) If h ϕ m ( ~x, ~y ) i m ∈ N is Cauchy in T and each formula ϕ m ispseudo-metric in T , then [lim ϕ m ] M is a pseudo-metric on M | ~x | for every M | = T .(ii) If T ⊆ U and T e is a pre-metric expansion of T , then U e = T e ∪ U is a pre-metric expansion of U with the same metric signature and approximate distance as T e . ODEL THEORY FOR REAL-VALUED STRUCTURES 21
Proof. (i) is clear. (ii) follows from Remark 3.1.3 (iii) and part (i) above. (cid:3)
Lemma 3.3.3.
If the vocabulary V has countably many predicate symbols and nofunction symbols, then every V -theory T has a pre-metric expansion with a pseudo-metric approximate distance.Proof. Let x, y, u, z , z , . . . be distinct variables. Arrange all atomic formulas withno constant symbols whose variables are among u, z , z , . . . in a countable list α , α , . . . . For each m , let α m ( x, ~z ) and α m ( y, ~z ) be the formulas formed from α m ( u, ~z ) by replacing u by x and y respectively. For each m , let β m ( x, y ) be the V -formula β m ( x, y ) = sup ~z | α m ( x, ~z ) − α m ( y, ~z ) | . Let d ( x, y ) = β ( x, y ), and for each m > d m ( x, y ) be the V -formula d m ( x, y ) = max( d m − ( x, y ) , − m β m ( x, y )) . Then for every V -structure M and every m , β M m and d M m are pseudo-metrics on M .Moreover, | = d m ( x, y ) ≤ . d m +1 ( x, y ) ≤ . d m ( x, y ) ∔ − ( m +1) , so h d i := h d m ( x, y ) i m ∈ N is exponentially Cauchy in T . For every general model M of T , let M e = ( M , D M e ) where D M e = [lim d m ] M . Since each d M m is a pseudo-metricon M , the limit D M e is a pseudo-metric on M .For every M | = T , M e is a general model of the V D -theory T h d i = { sup x sup y [ d m ( x, y ) ≤ . D ( x, y ) ≤ . d m ( x, y ) ∔ − m ] : m ∈ N } . We will find a metric signature L e such that T e = T ∪ T h d i is a pre-metric expansionof T with approximate distance h d i . It suffices to specify a modulus of uniformcontinuity △ P for each k -ary predicate symbol P ∈ V that is satisfied with respectto D in every general model of T h d i . For each 1 ≤ i ≤ k , let P ( u, ~z ) i be the atomicformula obtained from P ( z , . . . , z k ) by replacing z i by u . Then P ( u, ~z ) i is α m i forsome m i ∈ N . Let m = max( m , . . . , m k ). For each 1 ≤ i ≤ k , we have T h d i | = sup ~z | P ( x, ~z ) i − P ( y, ~z ) i | = β m i ( x, y ) ≤ . m d m ( x, y ) ≤ . m D ( x, y ) . Therefore whenever ~x, ~y differ only in the i -th argument we have T h d i | = | P ( ~x ) − P ( ~y ) | ≤ . m D ( x i , y i ) . Since one can change any k -tuple ~x to ~y in k steps by changing one variable at atime, for every pair of k -tuples ~x, ~y we have T h d i | = | P ( ~x ) − P ( ~y ) | ≤ . m k max( D ( x , y ) , . . . , D ( x k , y k )) . It follows that P has modulus of uniform continuity △ P ( ε ) = 2 − m k − ε with respectto D in each general model of T h d i . Therefore T e = T ∪ T h d i is a pre-metric expansionof T . (cid:3) In the general case that V contains function symbols, we must also specify amodulus of uniform continuity for each function symbol with respect to D . In a pre-metric structure with distance d , one can eliminate function symbols by using theformula d ( F ( ~x ) , y ) for the graph of F ( ~x ). This will not work in a general structure,because the formula D ( F ( ~x ) , y ) for the graph of F ( ~x ) is not a V -formula. We willcircumvent that difficulty by an “atomic Morleyization”, adding a new predicatesymbol for each atomic V -formula. Theorem 3.3.4. (Expansion Theorem) For every vocabulary V with countablymany predicate symbols and countably many function symbols, every V -theory T has a pre-metric expansion with a pseudo-metric approximate distance.Proof. By Remark 3.3.2 (ii), we may assume that T is the empty set of sentences.Let V ′ be the union of V and a k -ary predicate symbol P α for each atomic V -formula α ( ~x ) with k variables and no constant symbols. Let T ′ be the set of V ′ -sentencessup ~x | P α ( ~x ) − α ( ~x ) | for each α ( ~x ). Then V ′ \ V is a countable set of new predicatesymbols, and every V -structure M has a unique expansion to a general model M ′ of T ′ . It can be shown by induction on complexity that every V ′ -formula ϕ ( ~x ) is T ′ -equivalent to a V -formula ϕ ( ~x ), and is also T ′ -equivalent to a V ′ -formula ϕ ′′ ( ~x )that has no function symbols.Let V ′′ be the set of all predicate and constant symbols in V ′ , and for each V -structure M let M ′′ be the V ′′ -part of M ′ . Let T ′′ be the set of all V ′′ -sentencesthat hold in all general models of T ′ , which is the same as the set of all V ′′ -sentencesthat hold in M ′′ for every V -structure M . Note that V ′′ ⊆ V ′ , T ′′ ⊆ T ′ , and foreach V ′ -formula ϕ ( ~x ), the formula ϕ ′′ ( ~x ) defined in the previous paragraph is a V ′′ -formula. By Lemma 3.3.3, T ′′ has a pre-metric expansion T ′′ e with a pseudo-metric approximate distance h d i = h d m ( x, y ) i m ∈ N . Then for each V -structure M , M ′′ e = ( M ′′ , [lim d m ] M ′′ ).We next find a metric signature L ′ e over V ′ D such that T ′ e = T ′′ e ∪ T ′ with signature L ′ e is a pre-metric expansion of T ′ with the same approximate distance. Since each M ′ is an expansion of M ′′ , d M ′ m = d M ′′ m , so [lim d m ] M ′ = [lim d m ] M ′′ . We mustfind a modulus of uniform continuity for each k -ary function symbol F in V . Foreach m , let θ m ( ~x, y ) be a V ′′ -formula that is T ′ -equivalent to d m ( F ( ~x ) , y ). Then h θ m ( ~x, y ) i m ∈ N is Cauchy in T ′′ , and for each M | = T and all ( ~b, c ) ∈ M k +1 , D M ′′ e ( F M ( ~b ) , c ) = [lim d m ] M ′′ ( F M ( ~b ) , c ) = [lim θ m ] M ′′ ( ~b, c ) = [lim θ m ] M ′′ e ( ~b, c ) . By Lemma 2.5.5, there is a function △ F that is a modulus of uniform continuityfor [lim θ m ] M ′′ e in the pre-metric structure M ′′ e for every general model M of T .We note that Proposition 9.23 of [BBHU] on definable functions holds for pre-metric structures as well as metric structures. Therefore in M ′′ e , the function F M is definable and has the same modulus of uniform continuity △ F . This shows that M ′ e = ( M ′ , [lim d m ] M ′ ) is a pre-metric structure with the metric signature L ′ e thatagrees with L ′′ e on V ′′ and gives F the modulus of uniform continuity △ F .Finally, we show that there is a pre-metric expansion T e of the empty theory T with a pseudo-metric approximate distance. We take L e to be the restrictionof L ′ e to V D (so the symbols of V D have the same moduli of uniform continuityin L e as in L ′ e ). Then L e is a metric signature with distance predicate D over V D . For each m let b d m ( x, y ) be a V -formula that is T ′ -equivalent to d m ( x, y ),and let h b d i = h b d m ( x, y ) i m ∈ N . Since h d i is Cauchy in T ′ and each d m is pseudo-metric in every general model of T ′ , and every M | = T has a unique expansion to ageneral model M ′ | = T ′ , h b d i is Cauchy in T ′ and each b d m is pseudo-metric in eachgeneral model of T . We let T e be the union of T and a set of sentences saying that D ( x, y ) = lim m →∞ b d m ( x, y ) with signature L e . Then T e is a pre-metric expansionof T with pseudo-metric approximate distance h b d i . (cid:3) ODEL THEORY FOR REAL-VALUED STRUCTURES 23
Theorem 3.3.4 shows that a pre-metric expansion with a pseudo-metric approx-imate distance always exists, but the distance predicate built in the proof dependson an arbitrary enumeration of the atomic formulas, and may not be natural.
Corollary 3.3.5.
Every complete V -theory T has an exact distance h d m i m ∈ N inwhich each d M m is a pseudo-metric for each M | = T .Proof. By Theorem 3.3.4 and Proposition 3.2.6. (cid:3)
The above corollary is an improvement of Theorem 3.2.7, which is our trans-lation of the result from [BY05]. The construction of the approximate distancesin the proof of Theorem 3.3.4 is different, and perhaps easier to follow, than theconstruction in [BY05].
Question 3.3.6.
Does every pre-metric expansion of T have a pseudo-metric ap-proximate distance? We now show that the pre-metric expansion of a V -structure M is unique up toa uniformly continuous homeomorphism. Proposition 3.3.7.
Let T e , T f be pre-metric expansions of T . There is a function △ ( · ) : (0 , → (0 , such that for every general model M of T , the identity functionis a uniformly continuous homeomorphism from ( M, D M e ) onto ( M, D M f ) withmodulus △ ( · ) .Proof. We must find a function △ ( · ) that is a modulus of uniform continuity for D M f in the pre-metric structure M e for each M | = T . Let h d m i m ∈ N be an ap-proximate distance for f . For each M | = T , M e is a pre-metric structure withsignature L e . h d m i m ∈ N is Cauchy in the V -theory T . T is also a V D -theory, and D M f = [lim d m ] M = [lim d m ] M e . By Lemma 2.5.5 (ii), there is a function △ ( · ) withthe required property. (cid:3) Let us look at the pre-metric expansion process in the reverse direction. Thegeneral structure M can by obtained from the pre-metric expansion M e by simpli-fying in two steps: first downgrade M e to a general structure N by forgetting themetric signature, and then take the V -part of N by forgetting the distance to getthe original general structure M . M is the non-metric part of M e . We thus havetwo equivalence relations on the class of pre-metric structures with vocabulary V D ,the fine relation of having the same downgrade, and the coarse relation of havingthe same non-metric part.Formally, the class of general structures is disjoint from the class of pre-metricstructures because only the latter includes a metric signature. But intuitively, weregard M e and the downgrade of M e as the same structure presented in two differentways.3.4. Absoluteness.
In this rather philosophical subsection we develop a frame-work that allows one to apply known model-theoretic results about metric struc-tures to general structures.
For the remainder of this paper, we let T be a V -theory, T e be an arbitrary pre-metric expansion of T with signature L e , M be an arbitrary general model of T ,and M e be the pre-metric expansion of M for T e . By a general property we mean a class of general structures that is preservedunder automorphism (i.e., bijective embeddings), and by a pre-metric property we mean a class of pre-metric structures, again preserved under automorphisms. When P is a property, we interchangeably use the phrases “ N belongs to P ”, “ N hasproperty P ”, “ N satisfies P ”, and “ P holds in N ”. Definition 3.4.1. An absolute version of a pre-metric property Q is a generalproperty P such that whenever M e is a pre-metric expansion of M , M has property P if and only if M e has property Q .The following corollary was stated in the Introduction. Corollary 3.4.2.
Every pre-metric property Q has at most one absolute version.Proof. Suppose P and P ′ are absolute versions of Q , and consider a general structure M . By Theorem 3.3.4, M has a pre-metric expansion M e . Then the following areequivalent: M has property P , M e has property Q , M has property P ′ . (cid:3) Definition 3.4.3.
We say that a general property P is absolute if whenever M e isa pre-metric expansion of M , M has property P if and only if the downgrade of M e has property P . Corollary 3.4.4.
Suppose P is an absolute version of Q . Then P is absolute.Proof. By Example 3.2.3, each pre-metric structure N is a pre-metric expansion ofthe downgrade of N . Therefore for each M and M e , the following are equivalent: M has property P , M e has property Q , the downgrade of M e has property P . (cid:3) By Lemma 3.1.5 (i) we have:
Proposition 3.4.5.
The property [ M is reduced] in Definition 2.1.3 is absolute. In view of Remark 3.1.4, we often consider properties of a general structure M with additional parameters from M . Proposition 3.4.6.
Let p ( ~x, A ) be a set of V -formulas with parameters from M .The property [ p ( ~x, A ) is realized in M A ] is absolute.Proof. Every pre-metric expansion of M is an expansion of M . It follows fromRemark 2.1.6, that if M ′ is an expansion of M , A ⊆ M , and ~b ⊆ M , then ~b realizes p ( ~x, A ) in M ′ A if and only if ~b realizes p ( ~x, A ) in M A . (cid:3) For the same reason, Proposition 3.4.6 also holds when p ( ~x, A ) is an infinitary L ω ω -formula in the sense of [Ea15]In view of Corollaries 3.4.2 and 3.4.4, we consider the absolute version of a pre-metric property Q , if there is one, to be the “right” way to extend Q to all generalstructures. When we have a name for structures with a pre-metric property Q ,it will be convenient to use the same name for general structures that satisfy anabsolute version of Q . So we adopt the following convention. Definition 3.4.7.
A pre-metric property Q is said to be absolute if Q has anabsolute version. If Q is an absolute pre-metric property, a general structure thatsatisfies the absolute version of Q will be called a general structure with property Q . Proposition 3.4.8.
For any pre-metric property Q , the following are equivalent: (i) Q is absolute. (ii) The class of general structures with property Q is absolute. ODEL THEORY FOR REAL-VALUED STRUCTURES 25 (iii)
For every general structure M and all pre-metric expansions M e , M f of M , M e has property Q if and only if M f has property Q .Proof. (i) and (ii) are equivalent by Corollary 3.4.4. Assume (i), so Q has anabsolute version P . Let M be a general structure and let M e and M f be twopre-metric expansions of M . Then the following are equivalent: M e has Q , thedowngrade of M e has P , M has P , the downgrade of M f has P , M f has Q . Therefore(iii) holds.Assume (iii). Let P be the property such that for each general structure M , P holds for M if and only if Q holds for some pre-metric expansion of M . By (iii), foreach M and M e , P holds for M if and only if Q holds for M e . By Example 3.2.3,each pre-metric structure N is a pre-metric expansion of the downgrade of N , so Q holds for N if and only if P holds for the downgrade of N . In particular, for each M and M e , Q holds for M e if and only if P holds for the downgrade of M e . It followsthat P is an absolute version of Q , so (i) holds. (cid:3) Here are some examples of pre-metric properties that are not absolute.
Example 3.4.9. • Let 0 < r ≤
1. The property of being a pre-metric structure whose dis-tinguished distance has diameter r is not absolute. Similarly for diameter ≤ r , and for diameter ≥ r . • Let d ( x, y ) be a V -formula that defines a pseudo-metric in every V -structure.The property of being a pre-metric structure M e with distinguished dis-tance D such that M e satisfies sup x sup y ( d ( x, y ) ≤ . D ( x, y )), is not absolute.(Hint: If M e has an approximate distance h d m i m ∈ N , then ( M , max( D , d M ))is a pre-metric expansion of M with approximate distance h max( d m , d ) i m ∈ N and the same signature L e .) • Say that a pre-metric structure is
Lipschitz if every predicate and functionsymbol S has a modulus of uniform continuity △ S that is linear, that is, foreach S there is a positive real ℓ such that △ S ( x ) = ℓx for all x ∈ (0 , n -dimensional vectorspace over the reals.)In the next section, we will show that many pre-metric properties Q in theliterature have absolute versions, and also give necessary and sufficient conditionsfor M to be a general structure with property Q . Note that the characterizationof the absolute version of Q given by the proof of Proposition 3.4.8 mentions pre-metric expansions. In the results that follow, we will always give necessary andsufficient conditions for a general structure M to have property Q that are about M itself, rather than conditions that mention pre-metric expansions of M . This isdesirable for potential applications, because a general structure M may have a verysimple description even though all its pre-metric expansions are complicated.4. Properties of General Structures
In this section we use absoluteness to extend known results about metric struc-tures to general structures.
Types in Pre-metric Expansions.Proposition 4.1.1. If M | = T and N ≡ M , then N e ≡ M e .Proof. Let κ be a special cardinal such that | V | + ℵ < κ . By the Existence Theoremfor Special Models, there are κ -special V D -structures M ′ ≡ M e and N ′ ≡ N e . Let M ′′ , N ′′ be the V -parts of M ′ , N ′ respectively. Then M ′′ , N ′′ | = T , M ′ = M ′′ e and N ′ = N ′′ e . By Remark 2.4.4, the reductions of M ′′ and N ′′ are κ -special. By theUniqueness Theorem for Special Models, we have M ′′ ∼ = N ′′ . By Lemma 3.1.5 (iii)we have M ′′ e ∼ = N ′′ e , or in other words, M ′ ∼ = N ′ . Therefore N e ≡ N ′ ≡ M ′ ≡ M e . (cid:3) Corollary 4.1.2.
Suppose A ⊆ M and h : A → N . Then ( M , a ) a ∈ A ≡ ( N , ha ) a ∈ A if and only if ( M e , a ) a ∈ A ≡ ( N e , ha ) a ∈ A Proof.
By Remark 3.1.4 and Proposition 4.1.1. (cid:3)
Corollary 4.1.3. h : M ≺ N if and only if h : M e ≺ N e .Proof. By Corollary 4.1.2 with A = M . (cid:3) Corollary 4.1.4.
The property of two tuples realizing the same type over A isabsolute— tp M ( ~b/A ) = tp M ( ~c/A ) if and only if tp M e ( ~b/A ) = tp M e ( ~c/A ) . Also,indiscernibility over A is absolute.Proof. Apply Corollary 4.1.2 with N = M and where h is the identity on A andmaps ~b to ~c . (cid:3) The next corollary shows that for each complete theory T , T e has essentially thesame complete types as T . Corollary 4.1.5. If T = Th( M ) and T e = Th( M e ) , there is a homeomorphism h from S n ( T ) onto S n ( T e ) such that for each ~c ∈ M n , h (tp M ( ~c )) = tp M e ( ~c ) .Proof. This follows from Corollary 4.1.4. (cid:3)
Proposition 4.1.6.
Let κ be an infinite cardinal. The κ -saturation property isabsolute—for each general model M of T , M is κ -saturated if and only if M e is κ -saturated.Proof. We prove the non-trivial direction. Suppose M is κ -saturated. By Remark2.4.2 and Lemma 3.1.5 (i), we may assume without loss of generality that M isreduced. Then M e is reduced. Let A ⊆ M with | A | < κ , and let Γ( x ) be a setof V D -formulas that is finitely satisfiable in ( M e ) A . By the Existence Theoremfor Special Models, there is a reduced κ -saturated elementary extension N ′ ≻ M e .Then N ′ is equal to N e where N is the V -part of N ′ , and N ≻ M . Since N e is κ -saturated, some c ∈ N satisfies Γ( x ) in ( N e ) A . Since M is κ -saturated, there exists b ∈ M such that tp M ( b/A ) = tp N ( c/A ) , so tp N ( b/A ) = tp N ( c/A ) . By Corollary4.1.4, tp N e ( b/A ) = tp N e ( c/A ) . Therefore b satisfies Γ( x ) in N e , and hence alsosatisfies Γ( x ) in M e . (cid:3) Corollary 4.1.7. M is κ -special if and only if M e is κ -special.Proof. By Corollary 4.1.3 and Proposition 4.1.6. (cid:3)
ODEL THEORY FOR REAL-VALUED STRUCTURES 27
Definable Predicates.
The notion of a definable predicate in M was intro-duced in Definition 2.3.1. In Proposition 4.2.2 below we will show that the generalproperty [ P is a definable predicate] is absolute. The next lemma will be usedseveral times in this paper. Lemma 4.2.1.
Let T e be a pre-metric expansion of T . For every V D -formula ϕ ( ~x ) , there is a Cauchy sequence of V -formulas h ϕ m ( ~x ) i m ∈ N in T such that forevery general model M of T , ϕ M e = [lim ϕ m ] M . Hence ϕ M e is definable in M .Proof. Let h d i = h d m ( u, v ) i m ∈ N be an approximate distance for T e , such that nobound variable in d m ( u, v ) occurs in ϕ ( ~x ). Let Ψ be the set of all subformulasof ϕ ( ~x ). For every ψ ∈ Ψ, let ψ m be the V -formula obtained by replacing everysubformula of ψ of the form D ( σ, τ ) by d m ( σ, τ ), where σ, τ are V -terms. It thenfollows by induction on complexity that for every ψ ∈ Ψ we have: h ψ M m i m ∈ N is Cauchy in T and for each M | = T, ψ M e = [lim ψ m ] M . In particular, this holds when ψ = ϕ ( ~x ), as required. (cid:3) Proposition 4.2.2.
Let P : M k → [0 , . The general property [ P is a definablepredicate] is absolute.Proof. It is clear that if P is definable in M , then P is definable in M e . Supposethat P is definable in M e . Then P = [lim ϕ m ] M e for some sequence h ϕ m i m ∈ N of V D -formulas that is Cauchy in Th( M e ). By Lemma 4.2.1, each ϕ M e m is definable in M , so for each m there is a V -formula ψ m ( ~x ) such that( ∀ ~b ∈ M k ) | ψ M m ( ~b ) − ϕ M e m ( ~b ) | ≤ − m . Then P = [lim ϕ m ] M e = [lim ψ m ] M , so P is definable in M . (cid:3) The following corollary shows that when we add countably many predicatesthat are definable in M to a pre-metric expansion of M , we still have a pre-metricexpansion. Corollary 4.2.3.
Suppose V ′ = V ∪ W where W is a countable set of newpredicate symbols, M is a V -structure, M e is a pre-metric expansion of M , and M ′ = ( M , P M ′ ) P ∈ W is a V ′ -structure such that P M ′ is a definable predicate in M for each P ∈ W . Then the V ′ D -structure ( M e , P M ′ ) P ∈ W is a pre-metric expansionof M ′ .Proof. Let h d i = h d m i m ∈ N be an approximate distance for T e . By Lemma 2.5.5, foreach predicate symbol P ∈ W , P M ′ is uniformly continuous with respect to D M e with some modulus of uniform continuity △ P . Let L f agree with L e on V and giveeach new predicate symbol P ∈ W the modulus of uniform continuity △ P . Then M ′ f := ( M ′ , [lim d m ] M ′ ) = ( M ′ , D M e ) = ( M e , P M ′ ) P ∈ W is a pre-metric expansion of M ′ with signature L f . (cid:3) Given a set of parameters A ⊆ M , we say that a mapping P : M k → [0 ,
1] isa definable predicate over A in M if P is a definable predicate in M A . In view ofRemark 3.1.4, all of the results in this section hold for definable predicates over A in M . Definition 4.2.4.
We say that a general structure N whose vocabulary W contains V admits quantifier elimination over V if for every W -formula ϕ ( ~x ), ϕ N is definedby a sequence of quantifier-free V -formulas in N .Note that admitting quantifier elimination over V is a stronger property thanadmitting quantifier elimination over V D . Corollary 4.2.5. (i) M admits quantifier elimination over V if and only if M e admits quantifierelimination over V . (ii) If D M e ( x, y ) is defined in M e by a sequence of quantifier-free V -formulas,and M e admits elimination of quantifiers over V D , then M admits elimina-tion of quantifiers over V .Proof. This follows easily from Lemma 4.2.1. (cid:3)
Topological and Uniform Properties.
The following definition was givenin [BBHU] for metric structures, but makes sense for all general structures.
Definition 4.3.1.
A set C ⊆ M k is said to be type-defined by Φ( ~x ) in a generalstructure M , and that C is type-definable in M , if Φ( ~x ) is a set of formulas in thevocabulary of M with parameters in M , and C = { ~c ∈ M k : M | = Φ( ~c ) } . Note that type definability in M is preserved under finite unions and arbitraryintersections. Definition 4.3.2.
Let N be a pre-metric structure with distinguished distancepredicate d . A set C ⊆ N k is closed in N if it is closed with respect to the pseudo-metric d ( ~x, ~y ) = max i ≤ k d ( x i , y i ) on N k . Lemma 4.3.3.
Let N be a pre-metric structure. A set C ⊆ N k is closed in N ifand only if C is type-definable in N .Proof. Assume C ⊆ N k is closed in N . For each ~b ∈ N k \ C let ε ~b = inf ~c ∈ C max i ≤ k d ( b i , c i ) , so ε ~b >
0. Then C is type-defined in N by the set of formulasΦ( ~x ) = { ε ~b ≤ . max i ≤ k d ( b i , x i ) : ~b ∈ N k \ C } . Now suppose C is type-defined in N by some set of formulas Ψ( ~x ). By Fact 2.5.2,for each ψ ( ~x ) ∈ Ψ( ~x ), the set { ~c ∈ N k : N | = ψ ( ~c ) } is closed in N . Therefore C isclosed in N . (cid:3) Proposition 4.3.4.
The property of a set C ⊆ M k being closed in M is absolute. C is closed in M if and only if C is type-definable in M .Proof. By taking a subsequence if necessary, we can find an approximate distance h d m i m ∈ N for T e such that T e | = sup x sup y | d m ( x, y ) − D ( x, y ) | ≤ . − m for each m . Let C ⊆ M k . By Corollary 3.4.2 and Lemma 4.3.3, it suffices to provethat C is type-definable in M e if and only if C is type-definable in M . It is clearthat if C is type-definable in M then C is type-definable in M e ODEL THEORY FOR REAL-VALUED STRUCTURES 29
Suppose C is type-defined in M e by Φ( ~x ). Let Ψ( ~x ) be the set of all V -formulas ψ ( ~x ) with parameters in M such that C ⊆ ψ M e , and let B = { ~b : M e | = Ψ( ~b ) } .Then B = { ~b : M | = Ψ( ~b ) } , so B is type-definable in M . To prove that C is type-definable in M we show that B = C . Clearly C ⊆ B . Let ~b ∈ B \ C . By Lemma4.3.3, C is closed in M e , so there is an ε > ε ≤ max i ≤ k D M e ( b i , c i ) forall ~c ∈ C . Hence for each m ∈ N , we have ε ≤ (max i ≤ k d M m ( b i , c i ) + 2 − m ) for all ~c ∈ C . Therefore the V -formula ε ≤ . (max i ≤ k d m ( b i , x i )) ∔ − m )belongs to Ψ( ~x ). But then M e | = ( ε ≤ . max i ≤ k d m ( b i , b i )) ∔ − m for each m ∈ N , which contradicts the fact that D ( x, y ) = lim m →∞ d m ( x, y ) in M e . (cid:3) It follows that every pre-metric expansion M e has the same topology as M . Henceall topological properties of subsets C ⊆ M k or sequences in M k are absolute. Forexample, the properties that C is dense, that C is compact, and that lim n →∞ c n . = c ,are absolute. We let cl M ( C ) denote the closure of C in M . Then cl M ( C ) = cl M e ( C ).We now obtain absolute versions of properties related to uniform convergence. Proposition 4.3.5.
The property [ h c n i n ∈ N is Cauchy in M ] is absolute. A sequence h c n i n ∈ N is Cauchy in a general structure M if and only if h c n i n ∈ N converges to somepoint in some elementary extension M ′ ≻ M .Proof. It suffices to show that the following are equivalent:(a) h c n i n ∈ N is Cauchy in M e .(b) h c n i n ∈ N converges in some elementary extension of M e .(c) h c n i n ∈ N converges in some elementary extension of M .It is clear that (a) ⇔ (b).(b) ⇒ (c): Assume (b). Then there is a general structure N and a point c ∈ N such that N e ≻ M e and lim n →∞ c n . = c in N e . By Corollary 4.1.3, N ≻ M . Byabsoluteness of convergence, lim n →∞ c n . = c in N , so (c) holds..(c) ⇒ (b): Suppose N ≻ M , c ∈ N , and lim n →∞ c n . = c in N . Since convergenceof a sequence is absolute, lim n →∞ c n . = c in N e . By Corollary 4.1.3, N e ≻ M e , so(b) holds. (cid:3) We say that a general structure M is complete if every pre-metric expansion M e of M is a metric structure. Corollary 4.3.6.
The property of being complete is absolute. A general structure M is complete if and only if M is reduced and every Cauchy sequence in M convergesto some point in M . Corollary 4.3.7.
Every ℵ -saturated reduced structure is complete.Proof. Suppose M is reduced and ℵ -saturated. By Lemma 3.1.5 (ii) and Propo-sition 4.1.6, M e is a reduced ℵ -saturated pre-metric structure, and hence is com-plete. Let h c n i n ∈ N be Cauchy in M . By Proposition 4.3.5, h c n i n ∈ N is Cauchy in M e . Therefore h c n i n ∈ N converges to some point c in M e . By the absoluteness ofconvergence, h c n i n ∈ N converges to some point c in M . (cid:3) Recall that if M is a pre-metric structure, then a completion of M is a metricstructure N such that the reduction of M is a dense elementary substructure of N . Definition 4.3.8.
We say that a general structure N is a completion of a gen-eral structure M if N is complete, and the reduction of M is a dense elementarysubstructure of N .The elementary substructure requirement in the above definition ensures thatthe approximate distances are preserved when passing from M to N . Corollary 4.3.9.
Let M , N be general models of T and let T e be a pre-metricexpansion of T . Then N is a completion of M if and only if N e is a completion of M e .Proof. Let M ′ be the reduction of M . By Lemma 3.1.5 (ii), M ′ e is the reduction of M e . By Corollary 4.3.6, N e is reduced and complete if and only if N is reduced andcomplete. By Proposition 4.3.4, M ′ e is dense in N e if and only if M ′ is dense in N .By Corollary 4.1.3, M ′ e ≺ N e if and only if M ′ ≺ N . (cid:3) Proposition 4.3.10.
Every general structure M has a completion, which is uniqueup to an isomorphism that is the identity on the reduction of M .Proof. By Theorem 3.3.4, there is a pre-metric expansion T e of Th( M ). Let M ′ bethe reduction of M . Then M e is a pre-metric structure, M ′ e is the reduction of M e ,and there exists a completion N of M e that is unique up to an isomorphism thatis the identity on M ′ . Moreover, N ≻ M ′ e . Then N = N e where N is the V -partof N . By Corollary 4.3.9, N is a completion of M .If N ′ is another completion of M , then by Corollary 4.3.9, N ′ e is a completionof M e , so there is an isomorphism h : N e ∼ = N ′ e that is the identity on M ′ . Then h : N ∼ = N ′ , as required. (cid:3) Let κ be an infinite cardinal. A complete metric theory T is κ -categorical ifevery two complete models of T of density character κ are isomorphic. Corollary 4.3.11.
The property of having a κ -categorical complete theory is abso-lute. A complete general theory T is κ -categorical if and only if every two completegeneral models of T of density character κ are isomorphic.Proof. By Proposition 4.3.4 (closed is absolute), Corollary 4.3.6 (being complete isabsolute), and Lemma 3.1.5 (iii) (being isomorphic is absolute). (cid:3)
Infinitary Continuous Logic.
We return to the infinitary continuous for-mulas that were introduced in [Ea15] and discussed in Example 2.6.1. We assumein this subsection that | V | ≤ ℵ . L ω ω ( V ) denotes the set of all continuous L ω ω -formulas over the vocabulary V . Lemma 4.4.1.
Let T e be a pre-metric expansion of a V -theory T . For everyformula ψ ( ~x ) ∈ L ω ω ( V D ) , there is a formula ϕ ( ~x ) ∈ L ω ω ( V ) such that ψ M e = ϕ M for every general V -model M of T .Proof. The pre-metric expansion T e has an exponentially Cauchy approximate dis-tance h d m ( u, v ) i m ∈ N such that no bound variable in d m ( u, v ) occurs in ψ ( ~x ) or in ~z . Then for each m and pair of V -terms σ ( ~z ) , τ ( ~z ), d m ( σ ( ~z ) , τ ( ~z )) is a V -formula,and for every general model M of T and tuple ~z in M we have D M e ( σ ( ~z ) , τ ( ~z )) − − m ≤ d M m ( σ ( ~z ) , τ ( ~z )) ≤ D M e ( σ ( ~z ) , τ ( ~z )) + 2 − m . ODEL THEORY FOR REAL-VALUED STRUCTURES 31
It follows that D M e ( σ ( ~z ) , τ ( ~z )) = inf m sup k d M m + k ( σ ( ~z ) , τ ( ~z )) = sup m inf k d M m + k ( σ ( ~z ) , τ ( ~z )) . Let ψ ∈ L ω ω ( V D ), and let ϕ be the L ω ω -formula in the vocabulary V ob-tained by replacing each atomic subformula of ψ of the form D ( σ ( ~z ) , τ ( ~z )) byinf m sup k d m + k ( σ ( ~z ) , τ ( ~z )). It follows by induction on the complexity of ψ that ψ M e = ϕ M for every general V -model M of T . (cid:3) We say that a mapping P : M k → [0 ,
1] is L ω ω -definable in a general structure M if P = ϕ M for some L ω ω -formula ϕ ( ~x ) with | ~x | = k in the vocabulary of M . Itis easily seen that if P is definable in M then P is L ω ω -definable in M . Proposition 4.4.2.
The property [ P is L ω ω -definable in M ] is absolute.Proof. Let M e be a pre-metric expansion of M . It is easily seen by induction oncomplexity that for each L ω ω -formula ϕ ( ~x ) in the vocabulary V of M we have ϕ M = ϕ M e , so L ω ω -definability in M implies L ω ω -definability in M e . Conversely,by Lemma 4.4.1, L ω ω -definability in M e implies L ω ω -definability in M . (cid:3) We now generalize several results in [Ea15] from metric theories to general the-ories.
Proposition 4.4.3.
Let M be a separable complete V -structure. There is an L ω ω -sentence ϕ in the vocabulary V , called a Scott sentence of M , such that for everyseparable complete V -structure N , ϕ N = 0 if M ∼ = N and ϕ N = 1 otherwise.Proof. By Theorem 3.2.1 of [Ea15] (which follows from [BDNT]), the result holdsfor metric structures. By the Expansion Theorem 3.3.4, the empty V -theory T has a pre-metric expansion T e with signature L e . By the preceding section, M e is a separable complete pre-metric structure, and hence is a metric structure withsignature L e . Therefore there is an L ω ω -sentence ψ in the vocabulary V D such thatfor every separable metric structure N ′ with signature L e , ψ N ′ = 0 if M e ∼ = N ′ , and ψ N ′ = 1 otherwise. By Lemma 4.4.1, there is an L ω ω -sentence ϕ in the vocabulary V such that ψ N e = ϕ N for every V -structure N . Moreover, every metric structure N ′ with signature L e is a general model of T e , and hence is equal to N e where N isthe V -part of N ′ , and N is a separable complete V -structure. By Lemma 3.1.5 (iii), M ∼ = N if and only if M e ∼ = N e . Therefore ϕ has the required property that for everyseparable complete V -structure N , ϕ N = 0 if M ∼ = N and ϕ N = 1 otherwise. (cid:3) Proposition 4.4.4.
Let M be a separable complete general structure and let P bea mapping from M n into [0 , . The following are equivalent: (i) P is L ω ω -definable in M . (ii) P is fixed by all automorphisms of M .Proof. By Theorem 3.2.3 of [Ea15], the result holds for metric structures. By thepreceding section, Lemma 3.1.5 (iii), and Proposition 4.4.2, both (i) and (ii) areabsolute for complete separable general structures. (cid:3)
Proposition 4.4.7 below extends the Omitting Types Theorem 3.3.4 of [Ea15] togeneral theories T . The result in [Ea15] is about reduced pre-metric structures,which are pre-metric structures whose distinguished distance is a metric, but is notnecessarily complete. We first need the notion of a strong countable fragment of L ω ω ( V ). By [BBHU],Section 6, we may fix a countable set F of connectives such that F is closed undercomposition and projections, and for each n ∈ N the set of n -ary connectives C ∈ F is uniformly dense in the set of all n -ary connectives. We also assume that F contains the connective ≤ . and each rational constant in [0 , countable fragment of L ω ω ( V ) we mean a countable set L ⊆ L ω ω ( V ) that containsthe set of atomic formulas over V , and is closed under the connectives in F , sup x ,inf x , subformulas, and substituting terms for free variables. The smallest countablefragment of L ω ω ( V ) is the set of all (finitary) V -formulas with connectives in F . Definition 4.4.5.
We say that L is a strong countable fragment of L ω ω ( V ) if thereis a countable fragment L D of L ω ω ( V D ) such that L = L D ∩ L ω ω ( V ), and for eachfinitary V -formula θ ( x, y ) with connectives in F and pair of V -terms σ ( ~u ) , τ ( ~v ), L D is closed under the operation of replacing each subformula of the form D ( σ ( ~u ) , τ ( ~v ))by θ ( σ ( ~u ) , τ ( ~v )).Clearly, every strong countable fragment of L ω ω ( V ) is a countable fragment of L ω ω ( V ). Also, the smallest countable fragment of L ω ω ( V ) is a strong countablefragment of L ω ω ( V ). It is easily seen that every countable subset of L ω ω ( V ) iscontained in a strong countable fragment of L ω ω ( V ). Definition 4.4.6.
Let L be a countable fragment of L ω ω ( V ), and let T be a setof sentences in L . We say that a set Σ( ~x ) ⊆ L is principal over ( T, L ) if there isa formula ϕ ( ~x ) ∈ L , an | ~x | -tuple of V -terms ~τ ( ~y ), and a rational r ∈ (0 , • T ∪ { ϕ ( ~y ) } is satisfiable, • T ∪ { ϕ ( ~y ) ≤ . r } | = Σ( ~τ ( ~y )). Proposition 4.4.7.
Let L be a strong countable fragment of L ω ω ( V ) . Let T be aset of sentences in L , and for each n ∈ N , suppose Σ n ( ~x n ) ⊆ L but Σ n ( ~x n ) is notprincipal over ( T, L ) . Then there is a reduced separable model of T in which noneof the sets Σ n ( ~x n ) is realized.Proof. By hypothesis, L = L D ∩ L ω ω ( V ) for some countable fragment L D of L ω ω ( V D ) as in Definition 4.4.5. Let U e be a pre-metric expansion of the empty V -theory U . Then every pre-metric model of T ∪ U e is of the form M e for somegeneral model M of T . Claim.
For each n ∈ N , Σ n ( ~x n ) is not principal over ( T ∪ U e , L D ).To prove this Claim, suppose that Σ n ( ~x n ) is principal over ( T ∪ U e , L D ), wit-nessed by ϕ ( ~x n ) ∈ L D , a tuple of V -terms ~τ ( ~y ), and a rational r ∈ (0 , F and taking a subsequence, wesee that U e has an exponentially Cauchy approximate distance h d m i m ∈ N where each d m is a finitary continuous formula that belongs to L . Hence by Fact 2.5.3 andLemma 3.1.2, we may take U e to be a set of sentences of L D . For each m ∈ N , let ϕ m ( ~x ) be the formula obtained from ϕ ( ~x ) by replacing every subformula of ϕ ofthe form D ( σ ( ~x ) , σ ( ~x )) by d m ( σ ( ~x ) , σ ( ~x )). Then ϕ m ∈ L D ∩ L ω ω ( V ) = L . Itfollows by induction on the complexity of formulas that for each ε > m ∈ N such that for all k ≥ m , N | = T e , and ~b ∈ N | ~x | , ϕ N m ( ~b ) is within ε of ϕ N ( ~b ).Taking ε = r/
2, we obtain a formula ϕ m ( ~x ) ∈ L such that ϕ m ( ~x ) , ~τ ( ~y ), and r/ n ( ~x ) is principal over ( T, L ). This contradicts our hypothesis thatΣ n ( ~x n ) is not principal over ( T, L ), and proves the Claim. ODEL THEORY FOR REAL-VALUED STRUCTURES 33
Now, by Theorem 3.3.4 of [Ea15], there is a reduced separable pre-metric model N of T ∪ U e in which none of the sets Σ n is realized. Then the V -part of N is areduced separable model of T in which none of the sets Σ n is realized. (cid:3) Since the set L ( V ) of all V -formulas built from connectives in F is a strongcountable fragment of L ω ω ( V ), we have: Corollary 4.4.8.
Let T ⊆ L ( V ) , and for each ∈ N , let Σ n ( ~x n ) be a non-principalsubset of L ( V ) over ( T, L ( V )) . Then there is a reduced separable model of T inwhich none of the sets Σ n ( ~x n ) is realized. Many-sorted Metric Structures.
In Definition 2.6.5 we defined a sortedmetric structure M with sorted vocabulary W to be a reduced structure that re-spects the sorts of W and has a complete metric d S on each sort S . As a sortedvocabulary, W must have an unsorted constant symbol u and a unary predicatesymbol U S for each sort S . Definition 4.5.1.
We say that Th( M ), Th( M e ) have essentially the same types ifthere is a homeomorphism h from S n (Th( M )) onto S n (Th( M e )) such that for each ~c ∈ M n , h (tp M ( ~c )) = tp M e ( ~c ). Proposition 4.5.2.
Let W be a sorted vocabulary with sorts S , S , . . . , and M bea sorted metric structure over W . Then any pre-metric expansion M e = ( M , D ) of M is a metric structure, and Th( M ) , Th( M e ) have essentially the same types.Proof. Note that in M e , the new distance symbol D does not have sorts assignedto its arguments. Since M is reduced, M e is a reduced pre-metric structure, so D is a metric. Moreover, u M is the only sortless element by Lemma 2.6.4 (ii). Weshow that D is complete. Let h d m ( x, y ) i m ∈ N be an approximate distance for M e .Suppose that h a k i k ∈ N is Cauchy convergent in M e .Case 1: For some sort S n , a k ∈ S M n for infinitely many k . By taking a subse-quence, we may assume that a k ∈ S M n for all k . Since M e is a pre-metric structure, d M m is uniformly continuous with respect to D , so h a k i k ∈ N is Cauchy with respectto d M m . d M m is a complete metric on S M m , so h a k i k ∈ N converges to some point b ∈ S M m with respect to d M m . The set S M n is defined by the formula U S n ( x ) in M , and is there-fore closed in M e . By Proposition 4.3.4, the topologies of M e and M restricted to S M n are the same. Since M is a sorted metric structure, that topology is the metrictopology of d M m restricted to S M . Therefore h a k i k ∈ N converges to b with respect to D . Case 2. For each sort S n , a k / ∈ S M n for all but finitely many k . Then h a k i k ∈ N converges to some point b in the completion M ′ e ≻ M e . For each n , S n is open in M ′ e , so b / ∈ S M ′ e n . Therefore b = u M ′ e = u M e . It follows that h a k i k ∈ N converges to u M e with respect to D .In both cases, h a k i k ∈ N converges to a point with respect to D , so D is complete.The existence of the homeomorphism h follows from Corollary 4.1.5. (cid:3) Bounded and Unbounded Metric Structures.
Bounded vocabularies,metric structures, and general structures, were discussed in Example 2.6.6. Likethe metric case, the model theory of [0 , , structure M has a pre-metric expansion M e = ( M , D ), which is a bounded pre-metric structure. By normalizing D , one can even get a pre-metric expansion where D has values in [0 , N has been treated in theliterature using the model theory of metric structures. One way, as in [BBHU],[Fa], and [BY09], was to look at the corresponding bounded many-sorted metricstructure M with a sort for the closed m -ball around a distinguished constant 0 foreach positive m ∈ N . M can be viewed either as a many-sorted structure with ametric in each sort, or as a single sorted general structure with a unary predicatefor each sort. By the Expansion Theorem, as a single-sorted general structure, M has a pre-metric expansion M e , and by Proposition 4.5.2, Th( M e ) has essentiallythe same types as Th( M ).The Expansion Theorem opens up additional possibilities, which have not yetbeen explored in the literature, for using the model theory of metric structuresto treat unbounded metric structures. It allows the flexibility to look at generalstructures that are built in some way from an unbounded metric structure, andthen use the Expansion Theorem to get a related metric structure.For instance, consider an unbounded metric structure N with a real-valued gaugepredicate ν , as in the paper [BY08] that was discussed in Example 2.6.7 above. Tostudy N , one might look at the bounded general structure N ′ such that N ′ has thesame universe and functions as N plus a point at ∞ , and for each predicate P of N (including d and ν ), and positive m ∈ N , N ′ has the predicate P m formed bytruncating P at m , that is, P N ′ m ( ~v ) = − m if P N ( ~v ) ≤ − mm if P N ( ~v ) ≥ mP N ( ~v ) otherwise.This construction merely truncates the predicates of N rather than adding sortsfor the closed m -balls. Again, N ′ will have a pre-metric expansion N ′ e = ( N ′ , D ),and the theories of N ′ and N ′ e will have essentially the same types. As usual,the distinguished metric D of N ′ e may be wildly behaved and unnatural, but itsexistence does make the model theory of metric structures available for the studyof the unbounded metric structure N .Note that for any x in N , ν N ′ m ( x ) = m whenever m ≤ ν N ( x ), and ν N ′ m ( x ) = ν N ( x )whenever m > ν N ( x ). So in any model of Th( N ′ ), lim m →∞ ν m ( x ) is finite if andonly if ν m ( x ) is eventually constant as m → ∞ .Instead of looking at an arbitrary model of Th( N ′ ), it may be useful to look atthe substructure consisting of those elements y such that lim m →∞ ν m ( y ) is finite,plus the point at ∞ . This corresponds to looking at the modified pre-ultraproductof unbounded metric structures that was discussed in Example 2.6.7. The universeof the modified pre-ultraproduct M = Q D N i is the set of elements y of the ordinarypre-ultraproduct M ′ = Q D N ′ i such that the set { ν N i ( y i ) : i ∈ I } has a finite bound,plus the point at ∞ .To see the connection, one can check that for each element x of M ′ , the followingare equivalent, where x = D y means { i ∈ I : x i = y i } ∈ D : • For some y = D x , y belongs to M . • For some y = D x , the set { ν N i ( y i ) : i ∈ I } has a finite bound. ODEL THEORY FOR REAL-VALUED STRUCTURES 35 • For some y = D x , the set { lim m →∞ ν N ′ i m ( y i ) : i ∈ I } has a finite bound. • For some y = D x , lim m →∞ ν M ′ m ( y ) is finite.4.7. Imaginaries.
For each metric structure M , [BU] (Section 5) introduces asorted metric structure M eq that has M as its home sort and infinitely many sortsof imaginary elements. In a similar way, we will now introduce imaginary elementsfor any reduced general structure M with the vocabulary V .In the following, we let T be a complete V -theory, ~x be a tuple of variables and ~y be a countable sequence of variables, and h ϕ i = h ϕ m ( ~x, ~y ) i m ∈ N be an exponentiallyCauchy sequence of formulas in T . Let V h ϕ i be the two-sorted vocabulary obtainedfrom V by adding a home sort S , an imaginary sort S i , unary predicate symbols U S for the home sort and U S i for the imaginary sort, a predicate symbol d h ϕ i ofsort S i × S i , a predicate symbol P h ϕ i of sort S | ~x | × S i , and the unsorted constantsymbol u .Intuitively, the imaginary elements will be equivalence classes of infinite se-quences of parameters in the home sort, P h ϕ i ( ~x, ~y ) will be lim m →∞ ϕ m ( ~x, ~y ), and d h ϕ i ( ~y, ~z ) will be lim m →∞ (sup ~x | ϕ m ( ~x, ~y ) − ϕ m ( ~x, ~z ) | ) . Definition 4.7.1.
Let T h ϕ i be the set of V h ϕ i -sentencessup wz [ | sup ~x | P h ϕ i ( ~x, w ) − P h ϕ i ( ~x, z ) | − d h ϕ i ( w, z ) | ≤ . , { sup z inf ~y sup ~x [ | ϕ m ( ~x, ~y ) − P h ϕ i ( ~x, z ) | ≤ . · − m : m ∈ N } , { sup ~y inf z sup ~x [ | ϕ m ( ~x, ~y ) − P h ϕ i ( ~x, z ) | ≤ . · − m : m ∈ N } .T h ϕ i is the natural analogue, for general structures, of the theory T ψ that isdefined in [BU]. Proposition 4.7.2.
For every reduced model M of T , there is, up to isomorphism,a unique reduced model M h ϕ i of T ∪ T h ϕ i that respects sorts in V h ϕ i and agrees with M in the home sort.Proof. The argument is the same as in [BU], pages 29-30. (cid:3)
The elements of sort S i in M h ϕ i are called imaginary elements , or canonicalparameters. By Lemma 2.6.4, u M h ϕ i is the only element without a sort. Proposition 4.7.3.
For each reduced model M of T , d h ϕ i is a metric on the imag-inary sort in M h ϕ i .Proof. We work in M h ϕ i . It is clear from the definition that d h ϕ i is the limit ofa uniformly convergent sequence of pseudo-metrics on S i , and hence is itself apseudo-metric on S i . Suppose that b, c have sort S i and d h ϕ i ( b, c ) = 0. Since M h ϕ i is reduced, it suffices to show that b . = c . Let ~z be a tuple in M h ϕ i , and let α ( b, ~z )be an atomic formula. If α begins with a predicate symbol other than d h ϕ i or P h ϕ i or U S i , then b and c have the wrong sort, so α ( b, ~z ) = α ( c, ~z ) = 1.Suppose α begins with the predicate symbol d h ϕ i , so α ( b, ~z ) has the form d h ϕ i ( σ ( b, ~z ) , τ ( b, ~z ))where σ, τ are terms. No function or constant symbol in V h ϕ i has value sort S i . Soif either σ or τ starts with a function or constant symbol, then α ( b, ~z ) = α ( c, ~z ) = 1. The only other possibilities are that for some variable z , α ( b, ~z ) is either d h ϕ i ( b, b ), d h ϕ i ( z , z ), d h ϕ i ( b, z ), or d h ϕ i ( z , b ). In each of those cases, since d h ϕ i ( b, c ) = 0,we have α ( b, ~z ) = α ( c, ~z ) . If α begins with the predicate symbol P h ϕ i or U S i , then an argument that issimilar to the preceding paragraph again shows that α ( b, ~z ) = α ( c, ~z ). Thus in allcases we have α ( b, ~z ) = α ( b, ~z ) and hence b . = c . (cid:3) Proposition 4.7.4. If M is κ -special, then M h ϕ i is κ -special. If in addition κ hasuncountable cofinality, then d h ϕ i is a complete metric on the imaginary sort.Proof. By the Existence Theorem for Special Models, there is a κ -special model N of T ∪ T h ϕ i . It is easily checked that the V -part M ′ of N is κ -special, and N = M ′h ϕ i .By the Uniqueness Theorem for Special Models, M ′ ∼ = M , so M ′h ϕ i ∼ = M h ϕ i . If κ has uncountable cofinality, then M h ϕ i is ℵ -saturated, and as in Remark 2.5.1, itfollows that d h ϕ i is a complete metric on the imaginary sort. (cid:3) With more work, one can show that if M is κ + -saturated, then M h ϕ i is κ + -saturated, but we will not need that fact here. Definition 4.7.5.
We say that two exponentially Cauchy sequences h ϕ i and h ψ i are equivalent in T if [lim ϕ m ] M = [lim ψ m ] M for all general models M of T . Remark 4.7.6. If M is a reduced model of T and h ϕ i and h ψ i are equivalentexponentially Cauchy sequences in T , then M h ϕ i = M h ψ i , in the sense that theyhave the same universes and the identity function is an isomorphism between them. We now define the sorted vocabulary V eq and structure M eq . Let Φ be a setof exponentially Cauchy sequences h ϕ i for M that contains exactly one member ofeach equivalence class. For each h ϕ i ∈ Φ, take a copy of M h ϕ i in such a way thatthe home sorts are all the same, and the imaginary sorts of M h ϕ i and M h ψ i aredisjoint when h ϕ i 6 = h ψ i . Denote the imaginary sort of M h ϕ i by S h ϕ i i .Let V eq be the union of the sorted vocabularies { V h ϕ i : h ϕ i ∈ Φ } . Finally, let M eq be a V eq -structure that respects sorts and is a common expansion of the structures { M h ϕ i : h ϕ i ∈ Φ } . Remark 4.7.7.
It follows from Remark 4.7.6 that for each reduced V -structure M , M eq is unique up to isomorphism. Note that V eq may have uncountably many sorts and hence uncountably manypredicate symbols. To avoid that difficulty, one can restrict things to countable setsof sorts. For any set Θ ⊆ Φ, let V Θ and M Θ be the parts of V eq and M eq withonly the main sort and the imaginary sorts S h ϕ i i where h ϕ i ∈ Θ . Whenever Θ iscountable, the vocabulary V Θ has countably many predicate and function symbols.The next result concerns pre-metric expansions of M . Proposition 4.7.8.
Suppose M is a reduced V -structure, and M e = ( M , D ) is apre-metric expansion of M with vocabulary V D . Then ( V D ) eq has the same sorts as V eq , and ( M e ) eq is a sorted pre-metric structure whose signature has: • the distinguished distance D in the home sort, • the distinguished distance d h ϕ i in each imaginary sort S h ϕ i i , • the same modulus of uniform continuity as M e has for each symbol of V , • the modulus of uniform continuity for P h ϕ i that M e has for [lim ϕ m ] , ODEL THEORY FOR REAL-VALUED STRUCTURES 37 • the trivial modulus of uniform continuity for U S and each U S h ϕ i i .The same holds with a set Θ ⊆ Φ in place of eq .Proof. Since every exponentially Cauchy sequence of formulas in T is also expo-nentially Cauchy in T e , every sort of M eq is also a sort of ( M e ) eq . Also, for each h ϕ i ∈ Φ, ( M e ) h ϕ i = ( M h ϕ i , D ) is the structure obtained by adding the predicate D in the home sort. If h ϕ i and h ψ i are equivalent in T then they are equivalentin T e . The proof of Lemma 4.2.1 shows that every exponentially Cauchy sequence h ϕ i in T e is equivalent in T e to an exponentially Cauchy sequence h ψ i in T , so byRemark 4.7.6 we have ( M e ) h ϕ i = ( M e ) h ψ i . It follows that ( V e ) eq has the same sortsas V eq , and ( M e ) eq with the signature described above is a pre-metric expansionof M eq . (cid:3) Assume the hypotheses of Proposition 4.7.8, and let Θ ⊆ Φ. One can combine allthe distinguished distances in M Θ e into a single metric D Θ in the following way. If x, y are in the home sort, D Θ ( x, y ) = D ( x, y ). If x, y have the same imaginary sort S h ϕ i i , then D Θ ( x, y ) = d h ϕ i ( x, y ). If neither x nor y has a sort, then D Θ ( x, y ) = 0.Otherwise, D Θ ( x, y ) = 1. According to our definition, the structure ( M Θ , D Θ ) doesnot respect sorts. However, M Θ is a general structure whose vocabulary is V Θ , and( M Θ , D Θ ) is a V Θ D -structure. Corollary 4.7.9.
Assume the hypotheses of Proposition 4.7.8, and Θ ⊆ Φ is finite.Then ( M Θ , D Θ ) is a pre-metric expansion of M Θ .Proof. The vocabulary V Θ has countably many predicate and function symbols. Itis easily seen that there is a V Θ -formula θ ( x, y ) that defines D Θ ( x, y ) in M Θ . (cid:3) Proposition 4.7.10. If M is κ -special, then M eq is κ + -special, and M Θ is κ -specialfor each Θ ⊆ Φ .Proof. The proof is similar to the proof of Proposition 4.7.4, but with many imag-inary sorts. (cid:3)
Corollary 4.7.11.
Suppose T e is a pre-metric expansion of T , M is κ -special, and κ has uncountable cofinality. Then ( M e ) eq is a sorted metric structure, and ( M e ) Θ is a sorted metric structure for each Θ ⊆ Φ .Proof. By Propositions 4.7.4, 4.7.10, and Remark 2.5.1. (cid:3)
Definable Sets.
Recall that in [BBHU], in a metric structure M e , the distance between a k -tuple ~x ∈ M k and a closed set S ⊆ M k is the mappingdist M e ( ~x, S ) = inf { max i ≤ k D M e ( x i , y i ) : ~y ∈ S } , and S is a definable set over A in M e if S is a closed subset of M k and dist M e ( ~x, S )is a definable predicate over A in M e . Note that the empty set ∅ is always definableover ∅ , and dist M e ( ~x, ∅ ) = 1.The following result uses the fact that the Expansion Theorem 3.3.4 gives apseudo-metric approximate distance. Proposition 4.8.1.
The general property [ M is complete and S is a definable setin M over A ] is absolute. For each complete general structure M , set A ⊆ M , andset S ⊆ M k , the following are equivalent: (a) S is definable over A in M . (b) S is closed in M , and for each V -formula ϕ ( ~x, ~y ) , the mapping dist M ϕ ( ~x, S ) := inf { ϕ M ( ~x, ~y ) : ~y ∈ S } is a definable predicate over A in M . (c) S is closed in M , and for each V -formula ϕ ( ~x, ~y ) that is pseudo-metric in Th( M ) , dist M ϕ ( ~x, S ) is a definable predicate over A in M .Proof. We first consider an arbitrary pre-metric expansion M e of M . By Propo-sition 4.3.6, M is complete if and only if M e is complete. By Proposition 4.3.4, asubset of M k is closed in M e if and only if it is closed in M . Let (a’)–(c’) be thestatements (a)–(c) with M e in place of M . By Theorem 9.17 of [BBHU], (a’), (b’),and (c’) are equivalent. We now show that (b) is equivalent to (b’).(b’) ⇒ (b): Assume (b’). Let ϕ ( ~x, ~y ) be a V -formula. By (b’), dist M e ϕ is definablein M e over A . dist M e ϕ is definable in M over A by Proposition 4.2.2. Since ψ M = ψ M e for every V -formula ψ , dist M e ϕ = dist M ϕ . This proves (b).(b) ⇒ (b’): Assume (b). Let ϕ m ( ~x, ~y ) be the V -formula max i ≤ k d m ( x i , y i ). By(b), dist M ϕ m ( ~x, S ) = inf { max i ≤ k d M m ( x i , y i ) : ~y ∈ S } is a definable predicate over A in M . Moreover, h ϕ m i n ∈ N is Cauchy in T , andconverges uniformly to dist M e ( ~x, S ). Therefore dist M e ( ~x, S ) is a definable predicateover A in M e , so (a’) holds and hence (b’) holds.Since (b) is equivalent to (b’) for all M e , the property [ M is complete and (b)] isthe absolute version of the property [ M e is complete and (b’)]. The absolute versionis unique and (a’) is equivalent to (b’), so (a) is equivalent to (b). It is trivial that(b) implies (c).(c) ⇒ (b): Assume (c). By Theorem 3.3.4, there exists a pre-metric expansion M f of M with a pseudo-metric approximate distance h d m ( x, y ) i m ∈ N . By hypothesis,each formula d m ( x, y ) is pseudo-metric in Th( M ). Let ϕ m ( ~x, ~y ) be the V -formulamax i ≤ k d m ( x i , y i ), which is also pseudo-metric in Th( M ). By (c),dist M ϕ m ( ~x, S ) = inf { max i ≤ k d M m ( x i , y i ) : ~y ∈ S } is a definable predicate over A in M . Moreover, h ϕ m i m ∈ N is Cauchy in Th( M ), andconverges uniformly to dist M f ( ~x, S ). Therefore dist M f ( ~x, S ) is a definable predicateover A in M f . It follows that (a’) holds for M f , and therefore (b) holds. (cid:3) We now turn to the notions of definable and algebraic closure. The followingdefinitions agree with the corresponding definitions in [BBHU] in the case that M is a metric structure. Definition 4.8.2.
Let M be a complete general structure. An element b belongsto the definable closure of A in M , in symbols b ∈ dcl M ( A ), if the singleton { b } isdefinable over A in M . b belongs to the algebraic closure of A in M , b ∈ acl M ( A ),if there is a compact set C in M such that b ∈ C and C is definable over A in M .A tuple ~b belongs to dcl M ( A ) or acl M ( A ) if each term b i does.One can use Proposition 4.8.1 to obtain necessary and sufficient conditions for b ∈ dcl M ( A ) and b ∈ acl M ( A ). ODEL THEORY FOR REAL-VALUED STRUCTURES 39
Corollary 4.8.3. If M is a complete general structure, then dcl M ( A ) = dcl M e ( A ) and acl M ( A ) = acl M e ( A ) .Proof. By Propositions 4.3.4 and 4.8.1. (cid:3)
Corollary 4.8.4.
Suppose M , N are complete general structures and A ⊆ M ≺ N .Then dcl M ( A ) = dcl N ( A ) and acl M ( A ) = acl N ( A ) .Proof. By Proposition 4.3.6, M e and N e are metric structures. By Corollary 4.1.3we have M e ≺ N e . By Corollary 4.8.3 above and Corollary 10.5 of [BBHU],dcl M ( A ) = dcl M e ( A ) = dcl N e ( A ) = dcl N ( A ), and similarly for acl. (cid:3) Corollary 4.8.5.
Suppose M is a reduced ℵ -saturated general structure, A ⊆ M ,and b ∈ M . (i) b ∈ dcl M ( A ) if and only if b is the only realization of tp( b/A ) in M . (ii) b ∈ acl M ( A ) if and only if the set of realizations of tp( b/A ) is compact in M .Proof. By Exercises 10.7 (4) and 10.8 (4) of [BBHU], (i) and (ii) hold with an ℵ -saturated metric structure N in place of M . By Corollary 4.3.7, M is complete.By Proposition 4.1.6, M e is an ℵ -saturated metric structure. By Corollary 4.8.3,dcl M ( A ) = dcl M e ( A ) and acl M ( A ) = acl M e ( A ). By Corollary 4.1.4, an element c ∈ M realizes tp M ( b/A ) if and only if it realizes tp M e ( b/A ). By Proposition 4.3.4,a subset of M is compact in M if and only if it is compact in M e . Therefore (i) and(ii) hold for M . (cid:3) The paper [EG] introduced the notion of a metric structure admitting weakelimination of finitary imaginaries. We now introduce the analogous notion forgeneral complete structures.An exponentially Cauchy sequence h ϕ i in M is called finitary if there is an ℓ ∈ N such that for each m ∈ N , ϕ m has at most the free variables ( ~x, y , . . . , y ℓ ). If h ϕ i is finitary, the elements of sort S h ϕ i i in M h ϕ i are called finitary imaginaries . Definition 4.8.6.
Let M be a reduced ℵ -saturated general structure(i) M admits elimination of finitary imaginaries if for every finitary imaginary b ∈ M h ϕ i , there is a finite tuple ~c from M such that in M h ϕ i , b ∈ dcl( ~c ) and ~c ∈ dcl( b ).(ii) M admits weak elimination of finitary imaginaries if for every finitary imag-inary b ∈ M h ϕ i , there is a finite tuple ~c from M such that in M h ϕ i , b ∈ dcl( ~c )and ~c ∈ acl( b ).The next result shows that the property of being reduced and admitting (orweakly admitting) elimination of finitary imaginaries is absolute. Corollary 4.8.7.
For every reduced ℵ -saturated general structure M and pre-metric expansion M e of M , M e admits (or weakly admits) elimination of finitaryimaginaries if and only if M admits (or weakly admits) elimination of finitaryimaginaries.Proof. By Corollaries 4.7.9 and 4.8.3. (cid:3)
Stable Theories.
As is often done the literature on stable theories, we willwork in a monster structure of inaccessible cardinality. The axioms of ZFC do notimply the existence of an inaccessible cardinal. However, one can avoid inaccessiblecardinals by working in a universal domain, at the cost of some minor complications(see [BBHU]).
We assume hereafter that T is a complete V -theory, and that υ is an inaccessiblecardinal greater than | V | + ℵ . By a monster structure we mean a reduced υ -saturated structure of cardinality υ . We let M be a monster model of T , and let M e be a pre-metric expansion of M . For the rest of this paper we will work exclusively within M and M e . By a smallset we mean a set of cardinality < υ . A, B will always denote small subsets of M . Remark 4.9.1.
By Remark 2.4.5, M is υ -special. By the Uniqueness Theorem forSpecial Models, T has a unique monster model up to isomorphism. By Corollary4.1.7, M e is also monster structure. By Corollary 4.3.7, M and M e are complete,so M e is a metric structure. Let us recall the definition of a λ -stable metric theory in [BBHU], where λ is aninfinite cardinal. For each small set A , the D -metric on the type space S M e ( A ) isdefined by D M e ( p, q ) = inf { D M e ( b, c ) : tp M e ( b/A ) = p, tp M e ( c/A ) = q } . The complete metric theory Th( M e ) is λ -stable if for each A of cardinality ≤ λ there is a dense subset of S M e ( A ) of cardinality ≤ λ with respect to the D -metric.And Th( M e ) is stable if it is λ -stable for some small cardinal λ . Here, it will beconvenient to say that M e is stable instead of saying that Th( M e ) is stable. So,by a λ -stable metric structure we mean a monster metric structure whose completemetric theory is λ -stable. Proposition 4.9.2.
The property of being λ -stable is absolute. M is λ -stable ifand only if for each A of cardinality ≤ λ there is a set B ⊆ S M ( A ) of cardinality ≤ λ such that the set { b : tp M ( b/A ) ∈ B } is dense in M .Proof. By Corollary 4.1.4, for all elements b, c of M , tp M ( b/A ) = tp M ( c/A ) if andonly if tp M e ( b/A ) = tp M e ( c/A ). By Proposition 4.3.4, a subset of M is dense in M ifand only if it is dense in M e . So it suffices to prove that a set B ⊆ S M e ( A ) is dense in S M e ( A ) with respect to the D -metric if and only if the set B ′ := { b : tp M e ( b/A ) ∈ B } is dense in M e . We prove the non-trivial direction here. Let c ∈ M e and p = tp M e ( c/A ). Since B is dense in S M e ( A ), there is a sequence h p n i n ∈ N in B thatconverges to p in the D -metric. Since M e is κ + -saturated, for each n there exists b n such that tp M e ( b n /A ) = p n and D M e ( b n , c ) = D M e ( p n , p ). Then b n ∈ B ′ for each n and lim n →∞ b n = c in M e , so B ′ is dense in M e . (cid:3) Corollary 4.9.3. M is ℵ -stable if and only if M is λ -stable for all infinite λ .Proof. This follows from the corresponding result for metric structures (Remark14.8 in [BBHU]) and the fact that being λ -stable is absolute (Proposition 4.9.2). (cid:3) Exercise 4.9.4.
Suppose every predicate symbol and every function symbol in V is unary. Then T is ℵ -stable. Hint: Show that for all b.c ∈ M and all A ⊆ M , tp M ( b ) = tp M ( c ) if and only if tp M ( b/A ) = tp M ( c/A ) . Corollary 4.9.5.
Suppose λ = λ ℵ . The following are equivalent: ODEL THEORY FOR REAL-VALUED STRUCTURES 41 (i) T is stable. (ii) T is λ -stable. (iii) For every set A of cardinality ≤ λ , | S M ( A ) | ≤ λ .Proof. The equivalence of (i) and (ii) follows from the corresponding result formetric structures (Theorem 8.5 in [BU]) and the fact that λ -stability is absolute.It is trivial that (iii) implies (ii). Assume (ii). Suppose A ⊆ M , and | A | ≤ λ . By(ii) and Proposition 4.9.2, there is a set B ⊆ S M ( A ) of cardinality ≤ λ such thatthe set B ′ := { b : tp M ( b/A ) ∈ B } is dense in M . By Remark 2.4.1, every completetype in S M ( A ) is realized in M . Therefore, since B ′ is dense in M , B is dense in S M ( A ). So | S M ( A ) | ≤ λ ℵ = λ , and (iii) follows. (cid:3) In [BBHU], a stable independence relation on a monster metric structure M = M e is a ternary relation A | ⌣ C B on small subsets of M that has the followingproperties . • Invariance under automorphisms of M . • Symmetry: A | ⌣ C B if and only if B | ⌣ C A . • Transitivity: A | ⌣ C BD if d only if A | ⌣ C B and A | ⌣ BC D . • Finite character: A | ⌣ C if and only if ~a | ⌣ C B for all finite ~a ⊆ A . • Full existence:
For all
A, B, C there exists A ′ such that ( M A ′ ) C ≡ ( M A ) C and A ′ | ⌣ C B . • Strong local character:
For each finite ~a , there exists B ⊆ B of cardinality ≤ | V | + ℵ with ~a | ⌣ B B . • Stationarity:
For all small complete M ≺ M and all small A, A ′ , B , if( M A ) M ≡ ( M A ′ ) M , A | ⌣ M B, A ′ | ⌣ M B, then ( M A ) B ∪ M ≡ ( M A ′ ) B ∪ M We define a stable independence relation on a monster general structure M to bea ternary relation that satisfies the same seven properties with respect to M . Theorem 4.9.6. (i)
A relation | ⌣ is a stable independence relation on M if and only if it is astable independence relation on M e . (ii) M is stable if and only if it has a stable independence relation, and also ifand only if it has a unique stable independence relation.Proof. (i): Let | ⌣ be a ternary relation on small subsets of the universe of M . Weshow that each of the properties listed above for a stable independence relationis absolute, that is, holds for M if and only if it holds for M e . It is trivial thatSymmetry, Transitivity, Finite Character, and Local Character are absolute. ByLemma 3.1.5 (iii), Invariance under automorphisms is absolute. To show that FullExistence is absolute, use Corollary 4.1.4, which says that two tuples realize thesame complete type over C in M if and only if they realize the same complete typeover C in M e . To show that Stationarity is absolute, use Corollary 4.1.4 again andCorollary 4.1.3, which says that M ≺ M if and only if M e ≺ M e .(ii) follows immediately from (i) and Theorem 14.1 of [BBHU]. (cid:3) In naming these properties, we follow Adler [Ad], rather than [BBHU]
We now consider the approach to stability theory via definable types.
Definition 4.9.7.
We say that a complete k -type tp M ( ~a/B ) is definable over C in M if for each V -formula ϕ ( ~x, ~y ) with parameters in C there is a mapping Q : M | ~y | → [0 ,
1] that is definable over C in M such that for all ~b ∈ B | ~y | we have ϕ M ( ~a,~b ) = Q ( ~b ) . In the case that M = M e (so M is a metric structure), the above definition isthe same as the corresponding definition in [BBHU]. We now show that the generalproperty [tp( ~a/B ) is definable over C ] is absolute. Proposition 4.9.8.
Let ~a ∈ M and B, C be small subsets of M . tp M ( ~a/B ) isdefinable over C in M if and only if tp M e ( ~a/B ) is definable over C in M e .Proof. Suppose first that tp M e ( ~a/B ) is definable over C in M e . Let ϕ ( ~x, ~y ) be a V -formula with parameters in C . By hypothesis there is a mapping Q : M | ~y | → [0 , C in M e such that for all ~b ∈ B | ~y | we have ϕ M e ( ~a,~b ) = Q ( ~b ) . By Proposition 4.2.2, Q is definable over C in M , and by Remark 2.1.6, ϕ M ( ~a,~b ) = ϕ M e ( ~a,~b ) = Q ( ~b ), so tp M ( ~a/B ) is definable over C in M .Now suppose that tp M ( ~a/B ) is definable over C in M . Let ϕ ( ~x, ~y ) be a V D -formula with parameters in C . By Lemma 4.2.1, ϕ M e is definable in M over C .Then there is a sequence of V -formulas h ϕ m ( ~x, ~y ) i m ∈ N with parameters in C suchthat for each m ∈ N ,( ∀ ~a ∈ M | ~x | )( ∀ ~b ∈ M | ~y | ) | ϕ M m ( ~a,~b ) − ϕ M e ( ~a,~b ) | ≤ − m . By hypothesis, for each m ∈ N there is a mapping Q m : M | ~y | → [0 ,
1] definable over C in M such that for all ~b ∈ B | ~y | we have ϕ M m ( ~a,~b ) = Q m ( ~b ) . By Proposition 4.2.2,each Q m is definable in M over C , so there is a sequence of V -formulas h ψ m ( ~y ) i m ∈ N with parameters in C such that( ∀ ~b ∈ M | ~y | )( | ψ M m ( ~b ) − Q m ( ~b ) | ≤ − m )and ( ∀ ~b ∈ B | ~y | )( | ψ M m ( ~b ) − Q m ( ~b ) | = | ψ M m ( ~b ) − ϕ M m ( ~a,~b ) | ≤ − m ) . Then ( ∀ ~b ∈ B | ~y | )( | ψ M m ( ~b ) − ϕ M e ( ~a,~b ) | ≤ · − m . We now modify h ψ m ( ~y ) i m ∈ N to a sequence of V -formulas that is Cauchy in Th( M )using the forced convergence trick of [BU]. Note that for all ~b ∈ B | ~y | , | ψ M m ( ~b ) − ψ M m +1 ( ~b ) | ≤ · − m . We inductively define θ = ψ , and θ m +1 = max( θ m − · − m , min( θ m + 3 · − m , ψ m +1 )) . Then θ M m ( ~b ) = ψ M m ( ~b ) for all m and all ~b ∈ B | ~y | . Moreover, h θ m ( ~y ) i m ∈ N is a sequenceof V -formulas with parameters in C such that( ∀ ~b ) | θ M m ( ~b ) − θ M m +1 ( ~b ) | ≤ · − m , so h θ m ( ~y ) i m ∈ N in Cauchy in Th( M ). Therefore by Proposition 4.2.2, Q := [lim θ m ] M is a definable predicate over C in M e , and ϕ M e ( ~a,~b ) = Q ( ~b ) for all ~b ∈ B | ~y | .Therefore tp M e ( ~a/B ) is definable over C in M e . (cid:3) Definition 4.9.9.
We say that a complete type p = tp M ( ~a/B ) does not fork over C in M if p is definable over acl M ( C ). ODEL THEORY FOR REAL-VALUED STRUCTURES 43
In the case that M = M e , the above definition is the same as the definition in[BBHU]. We now show that the property [tp( ~a/B ) does not fork over C ] is absolute. Corollary 4.9.10.
For every a, B , and C , tp M ( a/B ) does not fork over C in M ifand only if tp M e ( a/B ) does not fork over C in M e .Proof. By Proposition 4.9.8, tp M ( ~a/B ) is definable over acl M ( C ) in M if and onlytp M e ( ~a/B ) is definable over acl M ( C ) in M e . We have acl M ( C ) = acl M e ( C ) byCorollary 4.8.3. (cid:3) Corollary 4.9.11. M is stable if and only if for every small N ≺ M and everytuple ~a of elements of M , tp M ( ~a/N ) is definable over N in M .Proof. By Theorem 14.16 of [BBHU], the result holds with the monster metricstructure M e in place of M . Corollary 4.1.3, N ≺ M if and only if N e ≺ M e . ByProposition 4.9.8, tp M ( ~a/N ) is definable over N in M if and only if tp M e ( ~a/N ) isdefinable over N in M e . By Proposition 4.9.2, M is stable if and only if M e isstable, and the result follows. (cid:3) Building Stable Theories.
In both first order and continuous logic, sta-ble structures are of particular interest because they are well-behaved and can beanalyzed. The literature contains a wide variety of examples of stable first orderstructures, and several examples of stable metric structures in areas such as Banachspaces and probability algebras. We now present a way to build many examples ofstable general structures from first order structures that are stable, or even stablefor positive formulas. As in [HI] and [BY03a], we exploit a connection between[0 , positive if it is built from atomic formulas using onlyquantifiers and the connectives ∧ , ∨ . A first-order structure is positively κ -saturated if every set of positive formulas with the free variable x and fewer than κ parametersthat is finitely satisfiable is satisfiable in the structure. Note that every κ -saturatedfirst-order structure is positively κ -saturated. A positive monster structure is apositively υ -saturated first order structure of cardinality υ .In a positive monster structure K , the complete positive type of a k -tuple ~b overa set A is the set of all positive formulas ϕ ( ~x ) with parameters in A satisfied by ~b .We say that a first-order structure K is positively λ -stable if K is a positive monsterstructure and, for every set A of cardinality λ , the set of complete positive 1-typesover A in K has cardinality ≤ λ . In particular, every λ -stable first order monsterstructure is a positively λ -stable.By a V - formula we mean a finite or countable conjunction of positive formulas,possibly with parameters in K . We also allow the empty conjunction, whose truthvalue is always true. Let D be the set of all dyadic rationals in [0 , q, r, s vary over D . Let J be the set of all intervals of the form [0 , r ] or [ r, r ∈ D .Let K be a first-order positive monster structure with universe M whose vocab-ulary W contains at least all the function and constant symbols of V . By a positiveinterpretation of V in K we mean a function I that associates, with each k -arypredicate symbol P ∈ V and interval J ∈ J , a V -formula I ( P, J ) in the vocabulary W with k free variables, such that whenever r < s we have:(a) I ( P, [0 , r ]) K ⊆ I ( P, [0 , s ]) K and I ( P, [ r, K ⊇ I ( P [ s, K . (b) I ( P, [0 , r ]) K ∩ I ( P, [ s, K = ∅ .(c) I ( P, [ r, K ∪ I ( P, [0 , s ]) K = M k . Theorem 4.10.1.
Suppose I is a positive interpretation of V in a positive monsterstructure K . There is a unique V -structure M = I ( K ) with universe M such that M agrees with K on all function and constant symbols in V , and for each k -arypredicate symbol P ∈ V , r ∈ D , and ~b ∈ M k , P M ( ~b ) ∈ [0 , r ] if and only if K | = ^ s>r I ( P, [0 , s ])( ~b ) and P M ( ~b ) ∈ [ r, if and only if K | = ^ s It is clear that there is at most one such M . We firstprove that such an M exists. Let I ′ ( P, [0 , r ]) = ^ s>r I ( P, [0 , s ]) , I ′ ( P, [ r, ^ s We next prove that M is υ -saturated, and thus is a monster structure. Note thatany small set of V -formulas with parameters in M that is finitely satisfiable in K is satisfiable in K . Finite disjunctions and countable conjunctions of V -formulasare logically equivalent to V -formulas. Moreover, since K is ℵ -saturated, for every V -formula Θ( ~x, y ), ( ∃ y )Θ and ( ∀ y )Θ are equivalent to V -formulas in K .By a D -interval we mean either the empty set or an interval [ r, s ] where 0 ≤ r ≤ s ≤ r, s ∈ D . A D -rectangle is a finite cartesian product of D -intervals.Note that for any continuous connective C : [0 , k → [0 , 1] and D -interval [ r, s ], C − ([ r, s ]) is a countable intersection of finite unions of D -rectangles.Using the above two paragraphs, one can show by induction on the complexityof formulas that for every V -formula ϕ ( ~x ) and D -interval [ r, s ], there is a V -formulaΘ( ~x ) such that for all ~b ∈ M k , ϕ M ( ~b ) ∈ [ r, s ] if and only if K | = Θ( ~b ). For everysmall set A , every finitely satisfiable set of V -formulas Θ( x ) with parameters in A is satisfiable in K . It follows that every finitely satisfiable set of V -formulas ϕ ( x )with parameters in A is satisfiable in M , so M is a monster structure.It also follows that if ~b, ~c have the same positive type over A in K , then tp M ( ~b/A ) =tp M ( ~c/A ). Therefore if K is positively λ -stable, then M is λ -stable. (cid:3) Simple and Rosy Theories. The notion of a simple theory was introducedin the context of cats in [BY03b]. In the literature (see [EG], for example), thedefinition of a simple complete metric theory is obtained by translating the defini-tion of a simple cat in [BY03b] into the context of continuous model theory. Fact4.11.2 below is a necessary and sufficient condition for a complete metric theoryto be simple that is proved in [BY03b]. Using that fact, we will show here thatthe property of being simple is absolute. We need the relation ≡ LsC (for Lascarequivalence) from [BY03b]. Definition 4.11.1. In a monster general structure M , we write A ≡ LsC B if A, B are small sequences of the same length and there exist finitely many sequences A , . . . , A n such that A = A , B = A n , and for each k < n , A k and A k +1 bothoccur on some C -indiscernible sequence. Fact 4.11.2. (By Theorem 1.51 of [BY03b]) A complete metric theory is simple ifand only if its monster metric model M has a ternary relation A | ⌣ C B on smallsets that has the following properties: • Invariance under automorphisms of M . • Symmetry. • Transitivity. • Finite character. • For every A and C , A | ⌣ C C . • Local character: For every A there exists a small cardinal λ such that forevery B there exists B ⊆ B with B ≤ λ and A | ⌣ B B . • Extension: If A | ⌣ C B and b B ⊇ B , there exists A ′ ≡ BC A such that A ′ | ⌣ C b B . • Independence theorem: Whenever A | ⌣ C A , B | ⌣ C A , B | ⌣ C A , and B ≡ LsC B , there exists B such that B | ⌣ C A A , B ≡ LsCA B , and B ≡ LsCA B . As in first order model theory, every stable metric theory is simple. By a simple metric structure we mean a monster metric structure whose completetheory is simple. The following corollary shows that the property of being a simplemetric structure has an absolute version. By Definition 3.4.7, this tells us that theright definition of a simple general structure is a monster general structure thatsatisfies that absolute version. Corollary 4.11.3. The property of being simple is absolute. A general monsterstructure M is simple if and only if there exists a ternary relation A | ⌣ C B on smallsubsets of M that satisfies the properties listed in Fact 4.11.2.Proof. It suffices to show that for each ternary relation | ⌣ on the universe of M ,each of the condit6ions listed in Fact 4.11.2 is absolute, that is, it holds for M if andonly if it holds for M e . We have already seen that Invariance under automorphismsis absolute. Absoluteness for the other conditions follow easily from Corollary4.1.4, which says that the property tp( ~b/A ) = tp( c/A ) is absolute, and hence thatindiscernibility over C is absolute. (cid:3) The notion of a rosy metric theory is introduced by Ealy and Goldbring [EG]. Definition 4.11.4. A metric structure M , or its complete metric theory Th( M ), is real rosy if M is a monster metric structure and has a ternary relation | ⌣ on smallsets with the following properties: • Invariance under automorphisms of M . • Monotonicity: If A | ⌣ C B , A ′ ⊆ A , and B ′ ⊆ B , then A ′ | ⌣ C B ′ . • Base monotonicity: Suppose C ∈ [ D, B ]. If A | ⌣ D B , then A | ⌣ C B . • Transitivity. • Normality: A | ⌣ C B implies AC | ⌣ C B . • Extension. • Countable character: A | ⌣ C B if and only if ~a | ⌣ C B for all countable ~a ⊆ A . • Local character. • Anti-reflexivity: a | ⌣ B a implies a ∈ acl M ( B ).A ternary relation with the above properties is called a strict independence re-lation . Every strict independence relation also satisfies Symmetry, Full Existence,and A | ⌣ C C . Corollary 4.11.5. The property of being real rosy is absolute. A general monsterstructure M is real rosy if and only if it has a strict independence relation.Proof. It is enough to show that for each ternary relation | ⌣ on the universe M , eachof the conditions in Definition 4.11.4 is absolute, that is, it holds for M if and only ifit holds for M e . We have already observed that Invariance under automorphisms isabsolute. The absoluteness of Anti-reflexivity follows from Corollaries 4.1.4 abouttypes, and Corollary 4.8.3 about algebraic closure. It is trivial that the otherproperties for a strict independence relation are absolute. (cid:3) By Proposition 4.7.10, if M is a monster general structure, then M eq is a monsterstructure, and M Θ is a monster structure for each Θ ⊆ Φ. Definition 4.11.6. A metric structure M , or its theory Th( M ), is called rosy if M is a monster metric structure and M eq is real rosy. ODEL THEORY FOR REAL-VALUED STRUCTURES 47 Lemma 4.11.7. Let M be a monster general structure, and let M e be a pre-metricexpansion of M . (i) ( M e ) eq has the same sorts, and the same universe in each sort, as M eq , (ii) If ~a,~b are tuples and C is a small set in M eq , then tp M eq ( ~a/C ) = tp M eq ( ~b/C ) ⇔ tp ( M e ) eq ( ~a/C ) = tp ( M e ) eq ( ~b/C ) . Proof. By Proposition 4.7.8, ( M e ) eq is a pre-metric expansion of M eq . Therefore(i) holds. The proof of Corollary 4.1.4 also works in this case, even though thevocabulary V eq has uncountably many predicate symbols, and shows that (ii) holds. (cid:3) Corollary 4.11.8. The property of being rosy is absolute. A general monsterstructure M is rosy if and only if M eq has a strict independence relation.Proof. By Lemma 4.11.7 (i), M eq and ( M e ) eq have the same sorts and universe sets.Therefore they have the same ternary relations. It follows from Lemma 4.11.7 (i)that for each ternary relation | ⌣ , each of the conditions in Definition 4.11.4 holdsfor M eq if and only if it holds for ( M e ) eq . (cid:3) Conclusion We have shown that every general structure with truth values in [0 , 1] can bemade into a metric structure by adding a distance predicate that is a uniformlimit of pseudo-metric formulas, and completing the metric. We used that resultto extend many notions and results from the class of metric structures to the classof general structures. Thus the model theory of metric structures is considerablymore broadly applicable than it initially appears. References [Ad] Hans Adler. A Geometric Introduction to Forking and Thorn-forking. J. Math. Logic9 (2009), 1-21.[AH] Jean-Martin Albert and Bradd Hart. 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