Continuous first order logic and local stability
aa r X i v : . [ m a t h . L O ] J a n CONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY
ITA¨I
BEN YAACOV
AND ALEXANDER USVYATSOV
Abstract.
We develop continuous first order logic, a variant of the logic described in[CK66]. We show that this logic has the same power of expression as the framework ofopen Hausdorff cats, and as such extends Henson’s logic for Banach space structures.We conclude with the development of local stability, for which this logic is particularlywell-suited.
Introduction
A common trend in modern model theory is to generalise model-theoretic notions andtools to frameworks that go beyond that of first order logic and elementary classes andproperties. In doing this, there is usually a trade-off: the more general the framework,the weaker the available tools, and one finds oneself many times trying to play this trade-off, looking for the most general framework in which a specific argument can be carriedthrough. The authors admit having committed this sin not once.The present paper is somewhat different, though: we do present what seems to be anew framework, or more precisely, a new logic, but in fact we prove that it is completelyequivalent to one that has been previously defined elsewhere, namely that of (metric)open Hausdorff cats (see [Ben05]).Another logic dealing with metric structures is Henson’s logic of positive bounded for-mulae and approximate satisfaction (see for example [HI02]). Even though Henson’s logicwas formulated for unbounded Banach space structures while ours deal with boundedmetric structures, it is fair to say that the two logics are equivalent. First of all, Henson’sapproach makes perfect sense in the bounded setting in which case the two logics areindeed equivalent. Banach space structures can (in most cases) be reduced for model-theoretic purposes to their closed unit ball (see Example 4.5). Moreover, there existsan unbounded variant of continuous logic which is equivalent with to (a somewhat ex-tended) Henson’s logic for arbitrary (bounded or unbounded) metric structures. It canbe reduced back to continuous logic as studied here (i.e., bounded) via the addition of a
Date : 28 May 2018.2000
Mathematics Subject Classification.
Key words and phrases. continuous logic, metric structures, stability, local stability.Research of the first author supported by NSF grant DMS-0500172.The authors would like to thank C. Ward Henson for stimulating discussions, and Sylvia Carlisle andEric Owiesny for a careful reading of the manuscript.
BEN YAACOV
AND ALEXANDER USVYATSOV single point at infinity. This beyond the scope of the present paper and is discussed indetail in [Benc].Finally, this logic is almost a special case of the continuous first order logic that Changand Keisler studied in [CK66]. We do differ with their definitions on several crucialpoints, where we find they were too general, or not general enough. Our logic is aspecial case in that instead of allowing any compact Hausdorff space X as a set of truthvalues, we find that letting X be the unit interval [0 ,
1] alone is enough. Indeed, asevery compact Hausdorff space embeds into a power of the interval, there is no loss ofgenerality. Similarly, as unit interval admits a natural complete linear ordering, we mayeliminate the plethora of quantifiers present in [CK66], and the arbitrary choices involved,in favour of two canonical quantifiers, sup and inf, which are simply the manifestations inthis setting of the classical quantifiers ∀ and ∃ . On the other hand, extending Chang andKeisler, we allow the “equality symbol” to take any truth value in [0 , equality symbol it becomes a distance symbol, allowing us to interpret metric structuresin the modified logic.However, continuous first order logic has significant advantages over previous for-malisms for metric structures. To begin with, it is an immediate generalisation of classi-cal first order logic, more natural and less technically involved than previous formalisms.More importantly, it allows us to beat the above-mentioned trade-off. Of course, if twologics have the same power of expression, and only differ in presentation, then an argu-ment can be carried in one if and only if it can be carried in the other; but it may stillhappen that notions which arise naturally from one of the presentations are more useful,and render clear and obvious what was obscure with the other one. This indeed seemsto be the case with continuous first order logic, which further supports our contentionthat it is indeed the “true and correct” generalisation of classical first order logic to thecontext of metric structures arising in analysis.An example for this, which was part of the original motivation towards these ideas, isa question by C. Ward Henson, which can be roughly stated as “how does one generaliselocal (i.e., formula-by-formula) stability theory to the logic of positive bounded formu-lae?”. The short answer, as far as we can see, is “one doesn’t.” The long answer is thatpositive bounded formulae may not be the correct analogues of first order formulae forthese purposes, whereas continuous first order formulae are.In Section 1 we define the syntax of continuous first order logic: signatures, connectives,quantifiers, formulae and conditions.In Section 2 we define the semantics: pre-structures, structures, the special role of themetric and truth values.In Section 3 we discuss types and definable predicates. The family of definable predi-cates is the completion, in some natural sense, of the family of continuous formulae.In Section 4 we discuss continuous first order theories and some basic properties suchas quantifier elimination. We also compare continuous first order theories with previousnotions such as open Hausdorff cats. ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 3
In Section 5 we discuss imaginaries as canonical parameters to formulae and definablepredicates.In Section 6 we define ϕ -types, i.e., types which only depend on values of instances ofa formula ϕ .In Section 7 we develop local stability, answering Henson’s question.In Section 8 we show how to deduce the standard global theory of independence fromthe local one in a stable theory.We also have two appendices:Appendix A contains a remark concerning an alternative (and useful) presentation ofcontinuity moduli.Appendix B deals with the case of a formula which is stable in a single model of atheory. 1. Continuous first order formulae
In classical (first order) logic there are two possible truth values: “true”, sometimesdenoted by ⊤ or T , and “false”, denoted by ⊥ or F . Often enough one associates theclassical truth values with numerical values, and the most common choice is probably toassign T the value 1 and F the value 0. This assignment is not sacred, however, and forour purposes the opposite assignment, i.e., T = 0 and F = 1, fits more elegantly.The basic idea of this paper is to repeat the development of first order logic withone tiny difference: we replace the finite set of truth values { , } with the compact set[0 , discrete logic , whereas the one we develop here will bereferred to as continuous logic .As in classical logic, a continuous signature L is a set of function symbols and predicatesymbols, each of which having an associated arity n < ω . In an actual continuousstructure, the function symbols will be interpreted as functions from the structure intoitself, and the predicate symbols as functions to the set of truth values, i.e., the interval[0 , Definition 1.1. A non-metric continuous signature consists of a set of function symbolsand predicate symbols, and for each function symbol f or predicate symbol P , its arity n f < ω or n P < ω .We may also consider multi-sorted signatures, in which case the arity of each symbolspecify how many arguments are in each sort, such that the total is finite, and eachfunction symbol has a target sort.Given a continuous signature L , we define L -terms and atomic L -formulae as usual.However, since the truth values of predicates are going to be in [0 , { , } , we need to adapt our connectives and quantifiers accordingly. ITA¨I
BEN YAACOV
AND ALEXANDER USVYATSOV
Let us start with connectives. In the discrete setting we use a somewhat fixed set ofunary and binary Boolean connectives, from which we can construct any n -ary Booleanexpression. In other words, any mapping from { , } n → { , } can be written usingthese connectives (otherwise, we would have introduced additional ones). By analogy, an n -ary continuous connective should be a continuous mapping from [0 , n → [0 , , n → [0 , n . However, this may be problematic, as continuummany connectives would give rise to uncountably many formulae even in a countablesignature. To avoid this anomaly we will content ourselves with a set of connectiveswhich merely allows to construct arbitrarily good approximations of every continuousmapping [0 , n → [0 , • Constants in [0 , • ¬ x = 1 − x , and x . • x ∧ y = min { x, y } , x ∨ y = max { x, y } , x − . y = ( x − y ) ∨ x ∔ y = ( x + y ) ∧ | x − y | .We can express the non-constant connectives above in terms of the connectives ¬ and − . : x ∧ y = x − . ( x − . y ) x ∨ y = ¬ ( ¬ x ∧ ¬ y ) x ∔ y = ¬ ( ¬ x − . y ) | x − y | = ( x − . y ) ∨ ( y − . x ) = ( x − . y ) ∔ ( y − . x )The expression x − . ny is a shorthand for (( x − . y ) − . y ) . . . − . y , n times. We would alsolike to point out to the reader that the expression x − . y is the analogue of the Booleanexpression y → x . For example, the continuous Modus Ponens says that if both y and x − . y are true, i.e., equal to zero, then so is x . Definition 1.2. (i) A system of continuous connectives is a sequence F = { F n : n <ω } where each F n is a collection of continuous functions from [0 , n to [0 , F is closed if it satisfies:(a) For all m < n < ω , the projection on the m th coordinate π n,m : [0 , n → [0 ,
1] belongs to F n .(b) Let f ∈ F n , and g , . . . , g n − ∈ F m . Then the composition f ◦ ( g , . . . , g n − ) ∈ F m .If F is any system of continuous connectives, then ¯ F is the closed system itgenerates.(iii) We say that a closed system of continuous connectives F is full if for every0 < n < ω , the set F n is dense in the set of all continuous functions { f : [0 , n → [0 , } in the compact-open (i.e., uniform convergence) topology. An arbitrarysystem of continuous connectives F is full if ¯ F is. ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 5 (We exclude n = 0 in order to allow full systems of connectives without truthconstants, i.e., in which F is empty.) Fact 1.3 (Stone-Weierstrass Theorem, lattice version) . Let X be a compact Hausdorffspace containing at least two points, I ⊆ R an interval, and equip A = C ( X, I ) with theuniform convergence topology. Let B ⊆ A be a sub-lattice, such that for every distinct x, y ∈ X , a, b ∈ I , and ε > , there is f ∈ B such that | f ( x ) − a | , | f ( y ) − b | < ε . Then B is dense in A .Proof. The proof of this or very similar results should appear in almost any analysistextbook. We will nonetheless include the proof for completeness.Let f ∈ A and ε > x, y ∈ X we can by hypothesisfind g x,y ∈ B for which | g x,y ( x ) − f ( x ) | , | g x,y ( y ) − f ( y ) | < ε . (In case x = y we take g x,x = g x,z for any z = x .) The set V x,y = { z ∈ X : f ( z ) − ε < g x,y ( z ) } is an openneighbourhood of y .Let us fix x . The family { V x,y : y ∈ X } is an open covering of X , and admits a finitesub-covering { V x,y i : i < n } . Let g x = W i Corollary 1.4. Let X be a compact Hausdorff space and let A = C ( X, [0 , . Assumethat B ⊆ A is closed under ¬ and − . , separates points in X (i.e., for every two distinct x, y ∈ X there is f ∈ B such that f ( x ) = f ( y ) ), and satisfies either of the following twoadditional properties: (i) The set C = { c ∈ [0 , 1] : the constant c is in B } is dense in [0 , . (ii) B is closed under x x .Then B is dense in A .Proof. Since B is closed under ¬ and − . it is also closed under ∨ and ∧ , so it is asub-lattice of A .Since B separates points it is in particular non-empty, so we have 0 = f − . f ∈ B for any f ∈ B , whereby 1 = ¬ ∈ B . In case B is closed under x we conclude that1 / n ∈ B for all n , and since B is also closed under ∔ , B contains all the dyadic constantsin [0 , 1] which are dense in [0 , B contains a dense setof constants.Let x, y ∈ X be distinct, a, b ∈ [0 , 1] and ε > 0. Let us first treat the case where X = [0 , 1] and id [0 , ∈ B . Then x, y ∈ [0 , ITA¨I BEN YAACOV AND ALEXANDER USVYATSOV assume that x < y . Assume first that a ≥ b . Let m ∈ N be such that ay − x < m , and let f ( t ) = ( a − . m ( t − . x )) ∨ b . Then f ( x ) = a ∨ b = a , and f ( y ) = 0 ∨ b = b . Replacing theconstants x, a, b ∈ [0 , 1] with close enough approximations from C we obtain f ∈ B suchthat | f ( x ) − a | , | f ( y ) − b | < ε . If a < b , find f ∈ B such that | f ( x ) − ¬ a | , | f ( y ) − ¬ b | < ε ,and then ¬ f ∈ B is as required.We now return to the general case where X can be any compact Hausdorff space.Since x = y there is g ∈ B such that g ( x ) = g ( y ), and without loss of generality we mayassume that g ( x ) < g ( y ). As in the previous paragraph construct f : [0 , → [0 , 1] suchthat | f ( g ( x )) − a | , | f ( g ( y )) − b | < ε , and observe that f ◦ g ∈ B .We have shown that B satisfies the hypotheses of Fact 1.3 and is therefore dense in A . (cid:4) Corollary 1.5. Let C ⊆ [0 , be dense, ∈ C . Then the following system is full: (i) F = C (i.e., a truth constant for each c ∈ C ). (ii) F = {− . } . (iii) F n = ∅ otherwise. Corollary 1.6. The following system is full: (i) F = {¬ , x } . (ii) F = {− . } . (iii) F n = ∅ otherwise. The system appearing in Corollary 1.5 can be viewed as the continuous analogue of thefull system of Boolean connectives { T, F, →} ( T and F being truth constants), while thatof Corollary 1.6 is reminiscent of {¬ , →} . We will usually use the latter (i.e., {¬ , x , − . } ),which has the advantage of being finite. Note however that for this we need to introducean additional unary connective x which has no counterpart in classical discrete logic. Remark . Unlike the discrete case, the family {¬ , ∨ , ∧} is not full, and this cannotbe remedied by the addition of truth constants. Indeed, it can be verified by inductionthat every function f : [0 , n → [0 , 1] constructed from these connectives is 1-Lipschitzin every argument.This takes care of connectives: any full system would do. We will usually prefer towork with countable systems of connectives, so that countable signatures give countablelanguages. When making general statements (e.g., the axioms for pseudo-metrics anduniform continuity we give below) it is advisable to use a minimal system of connectives,and we will usually use the one from Corollary 1.6 consisting of {¬ , x , − . } . On the otherhand, when spelling out actual theories, it may be convenient (and legitimate) to admitadditional continuous functions from [0 , n to [0 , 1] as connectives.As for quantifiers, the situation is much simpler: we contend that the transition from { T, F } to [0 , 1] imposes a single pair of quantifiers, or rather, imposes a re-interpretationof the classical quantifiers ∀ and ∃ (on this point we differ quite significantly from [CK66]).In order to see this, let us look for a construction of the discrete quantifiers ∀ and ∃ . ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 7 Let M be a set, and R M ( n ) be the set of all n -ary relations on M ; we may view each R ∈ R M ( n ) as a property of n free variables x , . . . , x n − . For every R ∈ R M ( n ), let j ( R ) ∈ R M ( n + 1) be defined as the same relation, with an additional dummy variable x n . Similarly, for every R ∈ R M ( n + 1) we have relations ∃ x n R and ∀ x n R in R M ( n ).Then for every R ∈ R M ( n ) and Q ∈ R M ( n + 1) the following two properties hold (here“ (cid:15) ” means “implies”): Q (cid:15) j ( R ) ⇐⇒ ∃ x n Q (cid:15) Rj ( R ) (cid:15) Q ⇐⇒ R (cid:15) ∀ x n Q. These properties actually determine the relations ∃ x n Q and ∀ x n Q , and can therefore beused as the definition of the semantics of the quantifiers.Replacing { T, F } with [0 , C M ( n ) be the set of all functions from M n to [0 , j : C M ( n ) → C M ( n + 1) as above, and inf x n , sup x n : C M ( n + 1) → C M ( n ) inthe obvious manner. Since we identify T with 0 and F with 1, the relation (cid:15) should bereplaced with ≥ , and we observe that for every f ∈ C M ( n ) and g ∈ C M ( n + 1): g ≥ j ( f ) ⇐⇒ inf x n g ≥ fj ( f ) ≥ g ⇐⇒ f ≥ sup x n g. Therefore, as in discrete logic, we will have two quantifiers, whose semantics are definedby the properties above. We will use the symbols inf and sup, respectively, to denote thequantifiers, as these best describe their semantics. Make no mistake, though: these arenot “new” quantifiers that we have “chosen” for continuous logic, but rather the onlypossible re-interpretation of the discrete quantifiers ∃ and ∀ in continuous logic. (Re-mark 2.11 below will relate our quantifiers to Henson’s sense of approximate satisfactionof quantifiers, further justifying our choice of quantifiers.)Once we have connectives and quantifiers, we define the set of continuous first orderformulae in the usual manner. Definition 1.8. A condition is an expression of the form ϕ = 0 where ϕ is a formula.A condition is sentential if ϕ is a sentence.A condition is universal if it is of the form sup ¯ x ϕ = 0 where ϕ is quantifier-freeIf r is a dyadic number then ϕ ≤ r is an abbreviation for the condition ϕ − . r = 0,and similarly ϕ ≥ r for r − . ϕ = 0. (Thus sup ¯ x ϕ ≤ r and inf ¯ x ϕ ≥ r abbreviate universalconditions.) With some abuse of notation we may use ϕ ≤ r for an arbitrary r ∈ [0 , 1] asan abbreviation for the set of conditions { ϕ ≤ r ′ : r ′ ≥ r dyadic } . We define ϕ ≥ r and ϕ = r as abbreviations for sets of conditions similarly. Notation 1.9. Given a formula ϕ we will use ∀ ¯ x ϕ = 0 as an alternative notation forsup ¯ x ϕ = 0. While this may be viewed as a mere notational convention, the semanticcontents of ∀ ¯ x ( ϕ = 0) is indeed equivalent to that of (sup ¯ x ϕ ) = 0 (notice how the ITA¨I BEN YAACOV AND ALEXANDER USVYATSOV parentheses move, though). Similarly, we may write ∀ ¯ x ϕ ≤ r for sup ¯ x ϕ ≤ r and ∀ ¯ x ϕ ≥ r for inf ¯ x ϕ ≥ r . 2. Continuous structures In classical logic one usually has a distinguished binary predicate symbol =, and thelogic requires that this symbol always be interpreted as actual equality. The definition wegave for a non-metric continuous signature is the analogue of a discrete signature withoutequality . The analogue of a discrete signature with equality is somewhat trickier, sincethe symbol taking equality’s place need no longer be discrete. Discrete equality alwayssatisfies the equivalence relation axioms: ∀ x x = x ∀ xy x = y → y = x ∀ xyz x = y → ( y = z → x = z )(ER)Still within the discrete framework, let us replace the symbol = with the symbol d .Recalling that T = 0, F = 1 we obtain the discrete metric: d ( a, b ) = ( a = b a = b Let us now translate ER to continuous logic, recalling that − . is the analogue of implica-tion: sup x d ( x, x ) = 0sup xy d ( x, y ) − . d ( y, x ) = 0sup xyz ( d ( x, z ) − . d ( y, z )) − . d ( x, y ) = 0(PM)Following Notation 1.9, we can rewrite PM equivalently as the axioms of a pseudo-metric,justifying the use of the symbol d : ∀ x d ( x, x ) = 0 ∀ xy d ( x, y ) = d ( y, x ) ∀ xyz d ( x, y ) ≤ d ( x, z ) + d ( z, y )(PM ′ )By the very definition of equality it is also a congruence relation for all the othersymbols, which can be axiomatised as: ∀ ¯ x ¯ yzw (cid:0) z = w → f (¯ x, z, ¯ y ) = f (¯ x, w, ¯ y ) (cid:1) ∀ ¯ x ¯ yzw (cid:0) z = w → ( P (¯ x, z, ¯ y ) → P (¯ x, w, ¯ y )) (cid:1) (CR) ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 9 Translating CR to continuous logic as we did with ER above would yield axioms sayingthat every symbol is 1-Lipschitz with respect to d in each of the variables. While wecould leave it like this, there is no harm in allowing other moduli of uniform continuity. Definition 2.1. (i) A continuity modulus is a function δ : (0 , ∞ ) → (0 , ∞ ) (for ourpurposes a domain of (0 , 1] would suffice).(ii) Let ( X , d ), ( X , d ) be metric spaces. We say that a mapping f : X → X is uniformly continuous with respect to a continuity modulus δ (or just that f re-spects δ ) if for all ε > x, y ∈ X : d ( x, y ) < δ ( ε ) = ⇒ d ( f ( x ) , f ( y )) ≤ ε .(For a different approach to the definition of continuity moduli see Appendix A.)Thus, for each n -ary symbol s and each i < n we will fix a continuity modulus δ s,i ,and the congruence relation property will be replaced with the requirement that as afunction of its i th argument, s should respect δ s,i . As above this can be written in purecontinuous logic or be translated to a more readable form:sup x
1] (say rational or dyadic numbers): simply, for every ε > 0, andevery r, q ∈ C such that r > ε and q < δ s,i ( ε ) (where s is either f or P ):sup ¯ x, ¯ y,z,w (cid:0) q − . d ( z, w ) (cid:1) ∧ (cid:0) d ( f (¯ x, z, ¯ y ) , f (¯ x, w, ¯ y )) − . r (cid:1) = 0sup ¯ x, ¯ y,z,w (cid:0) q − . d ( z, w ) (cid:1) ∧ (cid:0) ( P (¯ x, z, ¯ y ) − . P (¯ x, w, ¯ y )) − . r (cid:1) = 0(UC ′′L )This leads to the following definition: Definition 2.3. A (metric) continuous signature is a non-metric continuous signaturealong with the following additional data:(i) One binary predicate symbol, denoted d , is specified as the distinguished distancesymbol .(ii) For each n -ary symbol s , and for each i < n a continuity modulus δ s,i , called the uniform continuity modulus of s with respect to the i th argument.If we work with a multi-sorted signature then each sort S has its own distinguisheddistance symbol d S . BEN YAACOV AND ALEXANDER USVYATSOV Definition 2.4. Let L be a continuous signature. A (continuous) L -pre-structure is aset M equipped, for every n -ary function symbol f ∈ L , with a mapping f M : M n → M ,and for every n -ary relation symbol P ∈ L , with a mapping P M : M n → [0 , L (or equivalently PM ′ , UC ′L )hold.An L -structure is a pre-structure M in which d M is a complete metric (i.e., d M ( a, b ) =0 = ⇒ a = b and every Cauchy sequence converges).The requirement that d M be a metric corresponds to the requirement that = M beequality. Completeness, on the other hand, has no analogue in discrete structures, sinceevery discrete metric is trivially complete; still, it turns out to be the right thing torequire.As in classical logic, by writing a term τ as τ (¯ x ) we mean that all variables occurringin τ appear in ¯ x . Similarly, for a formula ϕ the notation ϕ (¯ x ) means that the tuple ¯ x contains all free variables of ϕ . Definition 2.5. Let τ ( x Let ϕ ( x 1] defined inductively as follows: • If ϕ = P ( τ , . . . , τ m − ) is atomic then ϕ M (¯ a ) = P M (cid:0) τ M (¯ a ) , . . . , τ Mm − (¯ a ) (cid:1) . • If ϕ = λ ( ψ , . . . , ψ m − ) where λ is a continuous connective then ϕ M (¯ a ) = λ (cid:0) ψ M (¯ a ) , . . . , ψ Mm − (¯ a ) (cid:1) . • If ϕ = inf y ψ ( y, ¯ x ) ϕ M (¯ a ) = inf b ∈ M ψ M ( b, ¯ a ), and similarly for sup. Proposition 2.7. Let M be an L -pre-structure, τ ( x 0, the set of conditions { ϕ i ≤ ε : i < n } is satisfiable. Definition 2.9. A morphism of L -pre-structures is a mapping of the underlying setswhich preserves the interpretations of the symbols. It is elementary if it preserves thetruth values of formulae as well. Proposition 2.10. Let M be an L -pre-structure. Let M = M/ { d M ( x, y ) = 0 } , and let d denote the metric induced by d M on M . Let ( ˆ M , ˆ d ) be the completion of the metricspace ( M , d ) (which is for all intents and purposes unique).Then there exists a unique way to define an L -structure ˆ M on the set ˆ M such that d ˆ M = ˆ d and the natural mapping M → ˆ M is a morphism. We call ˆ M the L -structure associated to M .Moreover: (i) If N is any other L -structure, then any morphism M → N factors uniquelythrough ˆ M . (ii) The mapping M → ˆ M is elementary.Another way of saying this is that the functor M ˆ M is the left adjoint of the forgetfulfunctor from the category of L -structures to that of L -pre-structures, and that it sendselementary morphisms to elementary morphisms.Proof. Straightforward using standard facts about metrics, pseudo-metrics and comple-tions. (cid:4) We say that two formulae are equivalent , denoted ϕ ≡ ψ if they define the samefunctions on every L -structure (equivalently: on every L -pre-structure). For example,let ϕ [ t/x ] denote the free substitution of t for x in ϕ . Then if y does not appear in ϕ , thensup x ϕ ≡ sup y ϕ [ y/x ] (this is bound substitution of y for x in sup x ϕ ). Similarly, providedthat x is not free in ϕ we have ϕ ∧ sup x ψ ≡ sup x ϕ ∧ ψ , ϕ − . sup x ψ ≡ sup x ( ϕ − . ψ ), etc. BEN YAACOV AND ALEXANDER USVYATSOV Using these and similar observations, it is easy to verify that all formulae written usingthe full system of connectives {¬ , x , − . } have equivalent prenex forms. In other words, forevery such formula ϕ there is an equivalent formula of the form ψ = sup x inf y sup z . . . ϕ ,where ϕ is quantifier-free. The same would hold with any other system of connectiveswhich are monotone in each of their arguments. Remark . We can extend Notation 1.9 to all conditions in prenex form, and therebyto all conditions. Consider a condition ψ ≤ r (a condition of the basic form ψ = 0 isequivalent to ψ ≤ ψ in prenex form, so the condition becomes:sup x (cid:18) inf y (cid:18) sup z . . . ϕ ( x, y, z, . . . ) (cid:19)(cid:19) ≤ r A reader familiar with Henson’s logic [HI02] will not find it difficult to verify, by inductionon the number of quantifiers, that this is equivalent to the approximate satisfaction of: ∀ x (cid:16) ∃ y (cid:16) ∀ z . . . ϕ ( x, y, z, . . . ) ≤ r (cid:17)(cid:17) (Notice how the parentheses move, though.)We view this as additional evidence to the analogy between the continuous quantifiersinf and sup and the Boolean quantifiers ∃ and ∀ . Remark . Unlike the situation in Henson’s logic, there are no bounds on the quanti-fiers as everything in our logic is already assumed to be bounded. For a fuller statementof equivalence between satisfaction in continuous logic and approximate satisfaction inpositive bounded logic, see the section on unbounded structure in [Benc]. In particular weshow there that under appropriate modifications necessitated by the fact that Henson’slogic considers unbounded structures, it has the same power of expression as continuouslogic. Definition 2.13. Let M be an L -structure. A formula with parameters in M is some-thing of the form ϕ (¯ x, ¯ b ), where ϕ (¯ x, ¯ y ) is a formula in the tuples of variables ¯ x and ¯ y ,and ¯ b ∈ M . Such a formula can also be viewed as an L ( M )-formula, where L ( M ) isobtained from L by adding constant symbols for the elements of M , in which case it maybe denoted by ϕ (¯ x ) (i.e., the parameters may be “hidden”). Definition 2.14 (Ultraproducts) . Let { M i : i ∈ I } be L -structures (or even pre-structures), and U an ultrafilter on I . Let N = Q i M i , and interpret the functionand predicate symbols on it as follows: f N (( a i ) , ( b i ) , . . . ) = ( f M i ( a i , b i , . . . )) P N (( a i ) , ( b i ) , . . . ) = lim U P M i ( a i , b i , . . . )Recall that a sequence in a compact set has a unique limit modulo an ultrafilter: for anyopen set U ⊆ [0 , P N (( a i ) , ( b i ) , . . . ) ∈ U ⇐⇒ { i ∈ I : P M i ( a i , b i , . . . ) ∈ U } ∈ U ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 13 It is immediate to verify that N satisfies PM and UC L , so N is an L -pre-structure.Finally, define Q i M i / U = N = ˆ N , and call it the ultraproduct of { M i : i ∈ I } modulo U . Theorem 2.15 ( Lo´s’s Theorem for continuous logic) . Let N = Q i M i / U as above. Forevery tuple ( a i ) ∈ Q M i let [ a i ] be its image in N . Then for every formula ϕ (¯ x ) we have: ϕ N ([ a i ] , [ b i ] , . . . ) = lim U ϕ M i ( a i , b i , . . . ) Proof. By induction on the complexity of ϕ . See also [CK66, Chapter V]. (cid:4) Corollary 2.16 (Compactness Theorem for continuous first order logic) . Let Σ be afamily of conditions (possibly with free variables). Then Σ is satisfiable in an L -structureif and only if it is finitely so, and furthermore if and only if it is approximately finitelyso (see Definition 2.8).Proof. The proof is essentially the same as in discrete logic. Replacing free variableswith new constant symbols we may assume all conditions are sentential. EnumerateΣ = { ϕ i = 0 : i < λ } , and let I = { ( w, ε ) : w ⊆ λ is finite and ε > } . For every( w, ε ) ∈ I choose M ω,ε in which the conditions ϕ i ≤ ε hold for i ∈ w . For ( w, ε ) ∈ I ,let J w,ε = { ( w ′ , ε ′ ) ∈ I : w ′ ⊇ w, ε ′ ≤ ε } . Then the collection U = { J w,ε : ( w, ε ) ∈ I } generates a proper filter on I which may be extended to an ultrafilter U . Let M = Q M w,ε / U . By Lo´s’s Theorem we have M (cid:15) ϕ i ≤ ε for every i < λ and ε > 0, so in fact M (cid:15) ϕ i = 0. Thus M (cid:15) Σ. (cid:4) Fact 2.17 (Tarski-Vaught Test) . Let M be a structure, A ⊆ M a closed subset. Thenthe following are equivalent: (i) The set A is (the domain of ) an elementary substructure of M : A (cid:22) M . (ii) For every formula ϕ ( y, ¯ x ) and every ¯ a ∈ A : inf { ϕ ( b, ¯ a ) M : b ∈ M } = inf { ϕ ( b, ¯ a ) M : b ∈ A } . Proof. One direction is by definition. For the other, we first verify that A is a substructureof M , i.e., closed under the function symbols. Indeed, in order to show that ¯ a ∈ A = ⇒ f (¯ a ) ∈ A we use the assumption for the formula d ( y, f (¯ a )) and the fact that A is complete.We then proceed to show that ϕ (¯ a ) A = ϕ (¯ a ) M for all ¯ a ∈ A and formula ϕ by inductionon ϕ , as in the first order case. (cid:4) When measuring the size of a structure we will use its density character (as a metricspace), denoted k M k , rather than its cardinality.We leave the following results as an exercise to the reader: Fact 2.18 (Upward L¨owenheim-Skolem) . Let M be a non-compact structure (as a metricspace). Then for every cardinal κ there is an elementary extension N (cid:23) M such that k N k ≥ κ . BEN YAACOV AND ALEXANDER USVYATSOV Fact 2.19 (Downward L¨owenheim-Skolem) . Let M be a structure, A ⊆ M a subset. Thenthere exists an elementary substructure N (cid:22) M such that A ⊆ N and k N k ≤ | A | + |L| . Fact 2.20 (Elementary chain) . Let α be an ordinal and ( M i : i < α ) an increasing chainof structures such that i < j < α = ⇒ M i (cid:22) M j . Let M = S i M i . Then M i (cid:22) M for all i . Types and definable predicates We fix a continuous signature L , as well as a full system of connectives (which mightas well be {¬ , x , − . } ).3.1. Spaces of complete types. Recall that a condition (or abbreviation thereof) issomething of the form ϕ = 0, ϕ ≤ r or ϕ ≥ r where ϕ is a formula and r ∈ [0 , 1] isdyadic. Definition 3.1. (i) Let M be a structure and ¯ a ∈ M n . We define the type of ¯ a in M , denoted tp M (¯ a ) (or just tp(¯ a ) when there is no ambiguity about the ambientstructure), as the set of all conditions in ¯ x = x With the topology given above, S n is a compact and Hausdorff space. ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 15 Proof. Let L ( n ) be the family of all formulae in the variables x A function f : S n → [0 , is continuous if and only if it can be uni-formly approximated by formulae.Proof. For right to left, we know that every formula defines a continuous mapping on thetype space, and a uniform limit of continuous mappings is continuous.Left to right is a consequence of Corollary 1.4 and the fact that formulae separatetypes. (cid:4) Given a formula ϕ ( x The family of sets of the form [ ϕ < r ] forms a basis of open sets for thetopology on S n . Equivalently, the family of sets of the form [ ϕ ≤ r ] forms a basis of closedsets.Moreover, if U is a neighbourhood of p , we can always find a formula ϕ ( x The discussion above, which is semantic in nature, shouldconvince the reader that uniform limits of formulae are interesting objects, which wewould like to call definable predicates . But, as formulae are first defined syntactically andonly later interpreted as truth value mappings from structures or from type spaces, itwill be more convenient later on to first define definable predicates syntactically. Sinceuniform convergence of the truth values is a semantic notion it cannot be brought intoconsideration on the syntactic level, so we use instead a trick we call forced convergence . BEN YAACOV AND ALEXANDER USVYATSOV This will be particularly beneficial later on when we need to consider sequences of for-mulae which sometimes (i.e., in some structures, or with some parameters) converge, andsometimes do not.Forced conversion is first of all an operation on sequences in [0 , , its limit. Formally: Definition 3.6. Let ( a n : n < ω ) be a sequence in [0 , a F lim ,n : n < ω ) by induction: a F lim , = a a F lim ,n +1 = a F lim ,n + 2 − n − a F lim ,n + 2 − n − ≤ a n +1 a n +1 a F lim ,n − − n − ≤ a n +1 ≤ a F lim ,n + 2 − n − a F lim ,n − − n − a F lim ,n − − n − ≥ a n +1 The sequence ( a F lim ,n : n < ω ) is always a Cauchy sequence, satisfying n ≤ m < ω = ⇒| a F lim ,n − a F lim ,m | ≤ − n .We define the forced limit of the original sequence ( a n : n < ω ) as: F lim n →∞ a i def = lim n →∞ a F lim ,n . Lemma 3.7. The function F lim : [0 , ω → [0 , is continuous, and if ( a n : n < ω ) is asequence such that | a n − a n +1 | ≤ − n for all n then F lim a n = lim a n .In addition, if a n → b ∈ [0 , fast enough so that | a n − b | ≤ − n for all n , then F lim a n = b .Proof. Continuity follows from the fact that if ( a n ) and ( b n ) are two sequences, and forsome m we have | a n − b n | < − m for all n ≤ m , then we can show by induction that | a F lim ,n − b F lim ,n | < − m for all n ≤ m , whereby | F lim a n − F lim b n | < · − m .For the second condition, one again shows by induction that | a F lim ,n − b | < − n whencethe conclusion. (cid:4) We may therefore think of F lim as an infinitary continuous connective. Definition 3.8. A definable predicate is a forced limit of a sequence of formulae, i.e., an(infinite) expression of the form F lim n →∞ ϕ n .We say that a variable x is free in F lim ϕ n if it is free in any of the ϕ n .Note that a definable predicate may have infinitely (yet countably) many free variables.In practise, we will mostly consider forced limits of formulae with a fixed (finite) tuple offree variables, but possibly with parameters, so the limit might involve infinitely manyparameters. We may write such a definable predicate as ψ (¯ x, B ) = F lim ϕ n (¯ x, ¯ b n ), andsay that ψ (¯ x, B ) is obtained from ψ (¯ x, Y ) (= F lim ϕ n (¯ x, ¯ y n )) by substituting the infinite ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 17 tuple of parameters B = S ¯ b n in place of the parameter variables Y = S ¯ y n . (Anotherway to think about this is to view parameter variables as constant symbols for whichwe do not have yet any interpretation in mind: then indeed all the free variables are in¯ x .) Later on in Section 5 we will construct canonical parameters for such instances of ψ (¯ x, Y ). We shall see then that dealing with infinitely many parameters (or parametervariables) is not too difficult as long as the tuple ¯ x of actually free variable is finite. Sofrom now on, we will only consider definable predicates in finitely many variables.The semantic interpretation of definable predicates is as expected: Definition 3.9. Let ψ (¯ x ) = F lim ϕ n (¯ x ) be a definable predicate in ¯ x = x The continuous functions S m → [0 , are precisely those given bydefinable predicates in m free variables.Proof. Let ψ ( x 1] weget ψ : S m → [0 , f : S m → [0 , 1] be continuous. Then by Proposition 3.4 f can beuniformly approximated by formulae: for every n we can find ϕ n ( x AND ALEXANDER USVYATSOV so are their uniform limits. This means, for example, that every definable predicate canbe added to the language as a new actual predicate symbol.We conclude with a nice consequence of the forced limit construction, which mightnot have been obvious if we had merely defined definable predicates as uniform limits offormulae. Lemma 3.11. Let M be a structure, and let ( ϕ n (¯ x ) : n < ω ) be a sequence of formulae,or even definable predicates, such that the sequence of functions ( ϕ Mn : n < ω ) convergesuniformly to some ξ : M m → [0 , (but need not converge at all for any other structureinstead of M ).Then there is a definable predicate ψ (¯ x ) such that ψ M = ξ = lim ϕ Mn (i.e., ξ is definablein M ).Proof. Up to passing to a sub-sequence, we may assume that | ϕ Mn − ξ | ≤ − n , so ξ = F lim ϕ Mn = ( F lim ϕ n ) M . would do. (cid:4) Partial types.Definition 3.12. A partial type is a set of conditions, usually in a finite tuple of variables.(Thus every complete type is in particular a partial type.)For a partial type p ( x Let p (¯ x, ¯ y ) be a partial type. We define ∃ ¯ y p (¯ x, ¯ y ) to be any partialtype q (¯ x ) (say the maximal one) satisfying [ q ] S n = π ([ p ] S n + m ).By the previous argument we have: Fact 3.14. For every partial type p (¯ x, ¯ y ) , a partial type ∃ ¯ y p (¯ x, ¯ y ) exists. Moreover, if M is structure and ¯ a ∈ M then M (cid:15) ∃ ¯ y p (¯ a, ¯ y ) if and only if there is N (cid:23) M and ¯ b ∈ N such that N (cid:15) p (¯ a, ¯ b ) . In case M is ω -saturated (i.e., if every -type over a finite tuplein M is realised in M ) then such ¯ b exists in M . ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 19 Theories Definition 4.1. A theory is a set of sentential conditions, i.e., things of the form ϕ = 0,where ϕ is a sentence.Thus in some sense a theory is an “ideal”. Since every condition of the form ϕ ≤ r , ϕ ≥ r or ϕ = r is logically equivalent to one of the form ϕ ′ = 0 (if r is dyadic) or toa set of such conditions (for any r ∈ [0 , ϕ = 0. Definition 4.2. Let T be a theory. A (pre-)model of T is an L -(pre-)structure M inwhich T is satisfied.The notions of satisfaction and satisfiability of sets of conditions from Definition 2.8apply in the special case of a theory (a set of conditions without free variables). Inparticular, a theory is satisfiable if and only if it has a model, and by Proposition 2.10,this is the same as having a pre-model.A theory is complete if it is satisfiable and maximal as such (i.e., if it is a complete0-type), or at least if its set of logical consequences is. The complete theories are preciselythose obtained as theories of structures:Th( M ) = { ϕ = 0 : ϕ an L -sentence and ϕ M = 0 }≡ { ϕ = ϕ M : ϕ an L -sentence } . (In the second line we interpret ϕ = r as an abbreviation for a set of conditions asdescribed earlier.)4.1. Some examples of theories. Using the metric, any equational theory (in theordinary sense) can be expressed as a theory, just replacing x = y with d ( x, y ) = 0. Example . Consider probability algebras (i.e., measure algebras, as discussed for ex-ample in [Fre04], with total measure 1). The language is L = { , , c , ∧ , ∨ , µ } , with allcontinuity moduli being the identity. The theory of probability algebras, denoted P rA ,consists of the following axioms: h equational axioms of Boolean algebras i µ (1) = 1 µ (0) = 0 ∀ xy (cid:0) µ ( x ) + µ ( y ) = µ ( x ∨ y ) + µ ( x ∧ y ) (cid:1) ∀ xy (cid:0) d ( x, y ) = µ (( x ∧ y c ) ∨ ( y ∧ x c )) (cid:1) The last two axioms are to be understood in the sense of Notation 1.9. Thus ∀ xy µ ( x ) + µ ( y ) = µ ( x ∨ y ) + µ ( x ∧ y ) should be understood as sup xy | µ ( x )+ µ ( y )2 − µ ( x ∨ y )+ µ ( x ∧ y )2 | = 0,etc. In the last expression, division by two is necessary to keep the range in [0 , BEN YAACOV AND ALEXANDER USVYATSOV get used to this we will tend to omit it and simply write sup xy | µ ( x ) + µ ( y ) − µ ( x ∨ y ) − µ ( x ∧ y ) | = 0.Note that we cannot express µ ( x ) = 0 → x = 0, but we do not have to either: if M is a model, a ∈ M , and µ M ( a ) = 0, then the axioms imply that d M ( a, M ) = 0, whereby a = 0 M .The model companion of P rA is AP A , the theory of atomless probability algebras,which contains in addition the following sentence: ∀ x ∃ y (cid:16) µ ( y ∧ x ) = µ ( x )2 (cid:17) , Following Remark 2.11 we can express this by:sup x inf y (cid:12)(cid:12)(cid:12) µ ( y ∧ x ) − µ ( x )2 (cid:12)(cid:12)(cid:12) = 0(We leave it to the reader to verify that this sentential condition does indeed if and onlyif the probability algebra is atomless.) Example . Let us now consider a signature L cvx consisting of binaryfunction symbols c λ ( x, y ) for all dyadic numbers λ ∈ [0 , T cvx consist of:( ∀ xyz ) d ( z, c λ ( x, y )) ≤ λd ( z, x ) + (1 − λ ) d ( z, y )( ∀ xyz ) c λ + λ ( c λ λ λ ( x, y ) , z ) = c λ + λ ( c λ λ λ ( y, z ) , x ) λ + λ + λ = 1( ∀ xyz ) d ( c λ ( x, z ) , c λ ( y, z )) = λd ( x, y ) . By [Mac73], the models of T cvx are precisely the closed convex subspaces of Banachspaces of diameter ≤ 1, equipped with the convex combination operations c λ ( x, y ) = λx + (1 − λ ) y . We may in fact restrict to a single function symbol c / ( x, y ), sinceevery other convex combination operator with dyadic coefficients can be expressed usingthis single operator. Since our structures are by definition complete, dyadic convexcombinations suffice. Example . Let us continue with the previous example. We may slightly modify ourlogic allowing the distance symbol to have values in the compact interval [0 , T cvx are convex sets of diameter ≤ 2. Let us add a constant symbol 0, introduce k x k as shorthand for d ( x, 0) and λx as shorthand for c λ ( x, ∀ x ) k x k ≤ ∀ x ∃ y ) d ( x, y/ ∧ (1 / − . k x k ) = 0The first axiom tells us that our model is a convex subset of the unit ball of the ambientBanach space. The second tells us that our model is precisely the unit ball: if k x k ≤ / x exists. One can add more structure on top of this, for example:(i) Multiplication by i , rendering the ambient space a complex Banach space. ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 21 (ii) Function symbols ∨ and ∧ , rendering the ambient space a Banach lattice. Inorder to make sure we remain inside the unit ball, we actually need to add( x, y ) ( x ∧ y ) / x, y ) ( x ∨ y ) / ∨ and ∧ . In particularthe model-theoretic study of independence in L p Banach lattices carried out in[BBH] fits in this setting.Alternatively, in order to study a Banach space E one could introduce a multi-sortedstructure where there is a sort E n for each closed ball around 0 in E of radius n < ω .On each sort, all the predicate symbols have values in a compact interval, and operationssuch as + go from E n × E m to E n + m . However, since every sort E n is isomorphic to E up to rescaling, this boils down to the single-sorted approach described above (and inparticular, re-scaled addition is indeed the convex combination operation).It can be shown that either approach (single unit ball sort or a sort for each radius) hasthe same power of expression as Henson’s logic, i.e., the translation from a Banach spacestructure `a la Henson to a unit ball structure (in an appropriate signature) preserves suchnotions as elementary classes and extensions, type-definable subsets of the unit ball, etc.This should be intuitively clear from Remark 2.11. Thus, continuous first order logic isindeed as good a setting for the study of such properties as stability and independence inBanach space structures as Henson’s logic, but as we show later it is much better adaptedfor such study.The reader may find in [Benc] a treatment of unbounded continuous signatures andstructures, an approach much closer in spirit to Henson’s treatment of Banach spacestructures. It is proved there that approximate satisfaction of positive bounded formu-lae (which makes sense in any unbounded continuous signature) has the same power ofexpression as satisfaction of conditions of continuous first order logic. The equivalencementioned in the previous paragraph follows. In addition, the single point compactifi-cation method defined there turns such unbounded structures into bounded structuresas studied here without chopping them into pieces as above (and again, preserving suchnotions as elementarity and definability).4.2. Type spaces of a theory. If T is a theory, we define its type spaces as in classicalfirst order logic:S n ( T ) = { p ∈ S n : T ⊆ p } = { tp M (¯ a ) : M (cid:15) T and ¯ a ∈ M n } . This is a closed subspace of S n , and therefore compact and Hausdorff in the inducedtopology. We define [ ϕ ≤ r ] S n ( T ) = [ ϕ ≤ r ] S n ∩ S n ( T ), and similarly for ϕ = r , etc. Asbefore, we may omit S n ( T ) if the ambient type space in question is clear.If M is a structure and A ⊆ M , we define L ( A ) by adding constant symbols for theelements of A and identify M with its natural expansion to L ( A ). We define T ( A ) =Th L ( A ) ( M ) and S n ( A ) = S n ( T ( A )), the latter being the space of n -types over A .This definition allows a convenient re-statement of the Tarski-Vaught Test: BEN YAACOV AND ALEXANDER USVYATSOV Fact 4.6 (Topological Tarski-Vaught Test) . Let M be a structure, A ⊆ M a closedsubset. Then the following are equivalent: (i) The set A is (the domain of ) an elementary substructure of M : A (cid:22) M . (ii) The set of realised types { tp M ( a/A ) : a ∈ A } is dense in S ( A ) .Proof. We use Fact 2.17. Assume first that the set of realised types is dense. Let ϕ ( y, ¯ a ) ∈L ( A ), r = inf y ϕ ( y, ¯ a ) M ∈ [0 , ε > ϕ ( y, ¯ a ) < r + ε ] ⊆ S ( A )is open and non-empty, so there is b ∈ A such that ϕ ( b, ¯ a ) M < r + ε , whence condition(ii) of Fact 2.17. Conversely, since the sets of the form [ ϕ ( y, ¯ a ) < r ] (with ¯ a ∈ A ) forma basis of open sets for S ( A ), condition (ii) of Fact 2.17 implies the realised types aredense. (cid:4) By previous results, a uniform limit (or forced limit) of formulae with parameters in A is the same (as functions on M , or on any elementary extension of M ) as a continuousmapping ϕ : S n ( A ) → [0 , A is called adefinable predicate over A , or an A -definable predicate.We define κ -saturated and (strongly) κ -homogeneous structures as usual, and showthat every complete theory admits a monster model, i.e., a κ -saturated and strongly κ -homogeneous model for some κ which is far larger than the cardinality of any otherset under consideration. It will be convenient to assume that there is always an ambientmonster model: every set of parameters we consider is a subset of a monster model, andevery model we consider is an elementary substructure thereof. (Even when consideringan incomplete theory, each model of the theory embeds in a monster model of its completetheory.)If M is a monster model and A ⊆ M a (small) set, we defineAut( M /A ) = { f ∈ Aut( M ) : f ↾ A = id A } . A definable predicate with parameters (in M ) is A -invariant if it is fixed by all f ∈ Aut( M /A ).All type spaces we will consider in this paper are quotient spaces of S n ( M ), where M is a fixed monster model of T , or of S n ( T ), which can be obtained using the followinggeneral fact: Fact 4.7. Let X be a compact Hausdorff space and A ⊆ C ( X, [0 , any sub-family offunctions. Define an equivalence relation on X by x ∼ y if f ( x ) = f ( y ) for all f ∈ A ,and let Y = X/ ∼ . Then: (i) Every f ∈ A factors uniquely through the quotient mapping π : X → Y as f = f Y ◦ π . (ii) The quotient topology on Y is precisely the minimal topology under which everysuch f Y is continuous. (iii) This topology is compact and Hausdorff. ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 23 Conversely, let Y be a compact Hausdorff space and π : X → Y a continuous projection.Then Y is a quotient space of X and can be obtained as above using A = { f ◦ π : f ∈ C ( Y, [0 , } .Proof. The first item is by construction. Let T be the quotient topology on Y and T theminimal topology in which every f Y is continuous. Then T is compact as a quotient of acompact topology. If y , y ∈ Y are distinct then there is a function f Y separating them,whereby T is Hausdorff. Finally, let V ⊆ [0 , 1] be open and f ∈ A , so U = f − Y ( V ) ⊆ Y is a pre-basic open set of T . Then π − ( U ) = f − ( V ) ⊆ X is open, whereby U ∈ T .Thus T refines T . Since T is compact and T is Hausdorff they must coincide.For the converse, the space Y ′ = X/ ∼ constructed in this manner can be identifiedwith Y . The original topology on Y refines the quotient topology by the second item,and as above they must coincide. (cid:4) For example, let M be a monster model and A ⊆ M a set. Then there is a naturalprojection π : S n ( M ) → S n ( A ) restricting from L ( M ) to L ( A ), and let: A = { ϕ ◦ π : ϕ ∈ C (S n ( A ) , [0 , } , A ′ = { ϕ ∈ C (S n ( M ) , [0 , ϕ is A -invariant } . Then clearly A ⊆ A ′ . On the other hand, A and A ′ separate the same types, so byFact 4.7 every f ∈ A ′ factors through S n ( A ) and A ′ = A . In other words we’ve shown: Lemma 4.8. Let A be a set (in the monster model) and let ϕ be an A -invariant definablepredicate with parameters possibly outside A . Then ϕ is (equivalent to) an A -definablepredicate A . Let us adapt the notions of algebraicity and algebraic closure to continuous logic: Lemma 4.9. Let A be a set of parameters and p ( x ) ∈ S ( A ) . Then the following areequivalent: (i) For every ε > there is a condition ( ϕ ε ( x ) = 0) ∈ p (with parameters in A ) and n ε < ω such that for every sequence ( a i : i ≤ n ε ) , if ϕ ε ( x i ) < / for all i ≤ n ε then d ( x i , x j ) ≤ ε for all i < j ≤ n ε . (ii) Every model containing A contains all realisations of p . (iii) Every indiscernible sequence in p is constant. (iv) There does not exist an infinite sequence ( a i : i < ω ) of realisations of p such that inf { d ( a i , a j ) : i < j < ω } > . (v) The set of realisations of p is compact.Proof. (i) = ⇒ (ii). We may assume that for the choice of ϕ ε , the number n ε is minimal:we can therefore find in the universal domain elements a AND ALEXANDER USVYATSOV Assume that A ⊆ M . Then the same holds in M , and we may therefore find a Let T be complete. Since any two n -types are realised inside the monster model we can define for every p, q ∈ S n ( T ): d ( p, q ) = inf { d (¯ a, ¯ b ) : ¯ a (cid:15) p and ¯ b (cid:15) q } Here d ( a 0, then so is the closed ε -neighbourhood of F : F ε = { p ∈ S n ( T ) : d ( p, F ) ≤ ε } . Indeed, since the set F is closed in can be written as [ p (¯ x )] where p is some partial type,and then F ε = [ ∃ ¯ y (cid:0) d (¯ x, ¯ y ) ≤ ε ∧ p (¯ y ) (cid:1) ] . This leads us to the following definition which will turn out to be useful later on: ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 25 Definition 4.11. A compact topometric space is a triplet h X, T , d i , where T is acompact Hausdorff topology and d a metric on X , satisfying:(i) The metric refines the topology.(ii) For every closed F ⊆ X and ε > 0, the closed ε -neighbourhood of F is closed in X as well.When dealing with topometric spaces some care must be taken about the language.We will follow the convention that whenever we use terms which come from the realmof general topology (such as compactness, closed and open sets, etc.) we refer to thetopology. When wish to refer to the metric, we will use terminology that clearly comesfrom the realm of metric spaces. When there may be ambiguity, we will say explicitly towhich part we are referring.We may therefore sum up the previous observations as: Fact 4.12. The type space S n ( T ) is a compact topometric space. We will come back to topometric spaces later. Let us now conclude with a small factabout them: Lemma 4.13. Let X be a compact topometric space (so by the terminological conventionabove, we mean to say that the topology is compact). Then it is complete (as a metricspace).Proof. Let ( x n : n < ω ) be a Cauchy sequence in X . We may assume that d ( x n , x n +1 ) ≤ − n − for all n . For each n the set { x n } − n , (the closed 2 − n -ball around x n ) is closed inthe topology, and ( { x n } − n : n < ω ) is a decreasing sequence of non-empty closed sets. Bycompactness there is some x in the intersection, and clearly x n → x in the metric. (cid:4) Quantifier elimination.Definition 4.14. A quantifier-free definable predicate is a definable predicate defined bya forced limit of quantifier-free formulae.A theory has quantifier elimination if every formula can be uniformly approximatedover all models of T by quantifier-free formulae, i.e., if every formula is equal in modelsof T to a quantifier-free definable predicate.(In order to avoid pathologies when there are no constant symbols in L , we must allowthat if ϕ is a formula without free variables, the quantifier-free definable predicate mayhave a free variable.)We introduce the following criterion for quantifier elimination, analogous to the clas-sical back-and-forth criterion: Definition 4.15. We say that a theory T has the back-and-forth property if for everytwo ω -saturated models M, N (cid:15) T , non-empty tuples ¯ a ∈ M n and ¯ b ∈ N n , and singleton c ∈ M , if ¯ a and ¯ b have the same quantifier-free type (i.e., ϕ M (¯ a ) = ϕ N (¯ b ) for all quantifier-free ϕ ) then there is d ∈ N such that ¯ a, c and ¯ b, d have the same quantifier-free type. BEN YAACOV AND ALEXANDER USVYATSOV Theorem 4.16. The following are equivalent for any continuous theory T (not neces-sarily complete): (i) The theory T admits quantifier elimination. (ii) The theory T has the back-and-forth property.Proof. Assume first that T admits quantifier elimination. Then under the assumptionswe have ¯ a ≡ ¯ b . Let p ( x, ¯ y ) = tp( c, ¯ a ). Then p ( x, ¯ b ) is consistent and is realised by some d ∈ N by ω -saturation.For the converse we introduce an auxiliary definition: The inf y -type of a tuple ¯ a ∈ M is given by the function ϕ (¯ x ) ϕ M (¯ a ), where ϕ varies over all the formulae of the forminf y ψ ( y, ¯ x ), ψ quantifier-free. We define S inf y n ( T ) as the set of all inf y -types of n -tuplesin models of T . This is a quotient of S n ( T ), and we equip it with the quotient topology,which is clearly compact and Hausdorff.We claim first that if M, N (cid:15) T and ¯ a ∈ M n , ¯ b ∈ N n have the same quantifier-freetype then they have the same inf y -type. Since we may embed M and N elementarily inmore saturated models, we may assume both are ω -saturated. Assume inf y ψ ( y, ¯ a ) = r .Then there are c m ∈ M such that ψ ( c m , ¯ a ) ≤ r + 2 − m , and by ω -saturation there is c ∈ M such that ψ ( c, ¯ a ) = r . Therefore there is d ∈ N such that ψ ( d, ¯ b ) = r , wherebyinf y ψ M ( y, ¯ a ) ≥ inf y ψ N ( y, ¯ b ). By a symmetric argument we have equality.We conclude that the quantifier-free formulae separate points in S inf y n ( T ). Sincequantifier-free formulae form a family of continuous functions on S inf y n ( T ) which is closedunder continuous connectives, the quantifier-free formulae are dense in C (S inf y n ( T ) , [0 , y -formula can be uniformly approximated byquantifier-free formulae, and by induction on the structure of the formula, every formulacan be thus approximated (on models of T , or equivalently on S n ( T )). (cid:4) Corollary 4.17. The theory of atomless probability algebras (described above) is completeand has quantifier elimination.Proof. The back-and-forth property between complete atomless probability algebras isimmediate, and does not require ω -saturation (or rather, as it turns out, all completeatomless probability algebras are ω -saturated). Indeed, two n -tuples a A theory T is model complete if for every M, N (cid:15) T , M ⊆ N = ⇒ M ≺ N .We leave the following as an exercise to the reader (see [Benb] for a complete proof): ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 27 Proposition 4.19. A theory T is model complete if and only if every formula can be uni-formly approximated on S n ( T ) by formulae of the form inf ¯ y ϕ (¯ x, ¯ y ) , where ϕ is quantifier-free. Continuous first order logic and open Hausdorff cats. We now show theequivalence between the framework of continuous first order logic and that of (metric)open Hausdorff cats. For this we assume familiarity with the latter framework, as exposedin [Ben05]. The reader who is not familiar with open Hausdorff cats may safely skip thispart.To every theory T we associate its type-space functor S( T ) in the usual manner. Forevery n < ω we defined S n ( T ) above. If m, n < ω and f : n → m is any mapping, we define f ∗ : S m ( T ) → S n ( T ) by f ∗ ( p ( x Let T be a continuous first order theory. Then its type-space functor S( T ) isan open, compact and Hausdorff type-space functor in the sense of [Ben03] .Since a type-space functor is one way to present a cat, this can be restated as: everycontinuous first order theory is an open Hausdorff cat.Proof. Clearly S( T ) is a Hausdorff compact type-space functor. To see it is open, let π n : S n +1 ( T ) → S n ( T ) consist of restriction to the n first variables (so π n = ( n ֒ → n + 1) ∗ ). Let U ⊆ S n +1 ( T ) be a basic open set, i.e., of the form [ ϕ (¯ x, y ) < r ]. Then π n ( U ) = [inf y ϕ < r ] is open as well, so π n is an open mapping. (cid:4) Recall that a definable n -ary function from a cat T to a Hausdorff space X is acontinuous mapping f : S n ( T ) → X . Equivalently, this is a mapping from the models of T to X such that for every closed subset F ⊆ X , the property f (¯ x ) ∈ F is type-definablewithout parameters (whence definable function). A definable metric is a definable binaryfunction which defines a metric on the models.Note that d is indeed a definable metric, so T is a metric cat, and the models of T (in the sense of continuous first-order logic) are precisely its complete models as a metricHausdorff cat, as defined in [Ben05].For the converse, we will use the following property of definable functions in open cats: Lemma 4.21. Let T be an open cat. Let f (¯ x, y ) be an definable n + 1 -ary function from T to [0 , (or the reals, for that matter), and let g (¯ x ) = sup y f (¯ x, y ) . Then g is also adefinable function.Proof. For every real number, we can define g (¯ x ) ≥ r by the partial type V s 1] be continuous, and let ψ : Y → [0 , 1] be defined by ψ ( y ) =sup { ϕ ( x ) : f ( x ) = y } . Then ψ is continuous. BEN YAACOV AND ALEXANDER USVYATSOV Observe that if T is a metric open Hausdorff cat and d a definable metric on somesort, then by compactness the metric is bounded. Thus, up to rescaling we may alwaysassume its range is contained in [0 , Theorem 4.23. Let T be a metric open Hausdorff cat, and let d be a definable metricon the home sort with range in [0 , .Then there exists a metric signature ˆ L whose distinguished metric symbol is ˆ d , and an ˆ L -theory ˆ T , such that S( ˆ T ) ≃ S( T ) , and such that the metric ˆ d coincide with d .Moreover, if κ is such that S n ( T ) has a basis of cardinality ≤ κ for all n < ω , then wecan arrange that |L| ≤ κ .Proof. For each n , we choose a family F n ⊆ C (S n ( T ) , [0 , {¬ , − . , x } , and such that d ∈ F . By Lemma 4.21 we may further assumethat sup y P (¯ x, y ) ∈ F n for each P (¯ x, y ) ∈ F n +1 . We can always choose the F n such that | F n | ≤ κ for all n , where κ is as in the moreover part. By Corollary 1.4, F n is dense in C (S n ( T ) , [0 , P ∈ F n is uniformly continuous with respect to d . Indeed, | P (¯ x ) − P (¯ y ) | is a continuous function from S n ( T ), and for every ε > 0, the followingpartial type is necessarily inconsistent: {| P (¯ x ) − P (¯ y ) | ≥ ε } ∪ { d ( x i , y i ) ≤ − m : m < ω, i < n } . Therefore there is m < ω such that d (¯ x, ¯ y ) ≤ − m = ⇒ | P (¯ x ) − P (¯ y ) | ≤ ε .Let ˆ L n = { ˆ P : n < ω, P ∈ F n } , where we associate to each n -ary predicate symbolˆ P the uniform continuity moduli obtained in the previous paragraph. Every universaldomain of T , or closed subset thereof, is naturally a ˆ L -structure, by interpreting each ˆ P as P . In particular, all the predicates satisfy the appropriate continuity moduli.Clearly, the family of n -ary definable functions is closed under continuous connectives.Also, if ϕ ( x 1] is a definable n -ary function and f : n → m is anymapping, then ϕ ( x i , . . . , x i n − ) = ϕ ◦ f ∗ is a definable m -ary function: in other words,the definable functions are closed under changes of variables. Finally, by Lemma 4.21,the definable functions are closed under quantification. Put together, every ˆ L -formula ϕ ( x 4, which in turn imply that ψ ≤ / ϕ ≥ / 4, so q isinconsistent with ˆ T .Thus θ n : S n ( T ) → S n ( ˆ T ) is a homeomorphism for every n , and by construction it iscompatible with the functor structure so θ : S( T ) → S( ˆ T ) is the required homeomorphismof type-space functors.It is clear from the way we axiomatised it that ˆ T has quantifier elimination. (cid:4) Theorem 4.23, combined Fact 4.20, says that framework of continuous first order theo-ries coincides with that of metric open Hausdorff cats. In fact, we know from [Ben05] thatif T is a non-metric Hausdorff cat then its home sort can be “split” into (uncountablymany) hyperimaginary metric sorts, so in a sense every Hausdorff cat is metric. Thus,with some care, this observation can be generalised to all open Hausdorff cats.Compact type-space functors are a structure-free and language-free way of present-ing cats. By [Ben03], one can associate to each compact type-space functor a positiveRobinson theory T ′ in some language L ′ and talk about universal domains for that catas L ′ -structures. Then “complete models” mean the same thing in both settings: Fact 4.24. Let T be a complete continuous theory and T ′ the corresponding cat (i.e.,positive Robinson theory in a language L ′ ).Then there is a one-to-one correspondence between ( κ -)monster models of T and( κ -)universal domains of T ′ over the same set of elements (for a fixed big cardinal κ ), insuch a way that the type of a tuple in one is the same as its type in the other.Moreover, a closed subset of such a model is an elementary submodel in the sense of T if and only if it is a complete submodel in the sense of T ′ , as defined in [Ben05] .Proof. For the first condition, the identification of the type-space functors of T and of T ′ imposes for each monster model (or universal domain) of one an interpretation of theother language, and it is straightforward (though tedious) to verify then that the newstructure is indeed a universal domain (or a monster model) for the other theory.The moreover part is a special case of Fact 4.6. (cid:4) Since T is a metric cat, its monster model is a monster metric space (momspace) asdefined in [SU], and its models (in the sense of continuous logic) are precisely the class K c studied there. So results proved for momspaces apply in our context. BEN YAACOV AND ALEXANDER USVYATSOV Imaginaries In classical first order model theory there are two common ways to view (and define)imaginaries: as canonical parameters for formulae, or, which is more common, as classesmodulo definable equivalence relations. Of course, any canonical parameters for a formulacan be viewed as an equivalence class, and an equivalence class is a canonical parameterfor the formula defining it, so both approaches are quite equivalent in the discrete setting.We have already observed that in the passage from discrete to continuous logic equiv-alence relations are replaced with pseudo-metrics. On the other hand, the notion of acanonical parameter remains essentially the same: the canonical parameter for ϕ (¯ x, ¯ a )is something (a tuple, an imaginary. . . ) c which an automorphism fixes if and only if itdoes not alter the meaning of the formula (i.e., c = f ( c ) ⇐⇒ ϕ (¯ x, ¯ a ) ≡ ϕ ( x, f (¯ a )) forevery f ∈ Aut( M )).As in the classical setting, both approaches are essentially equivalent, but in prac-tise the canonical parameter approach has considerable advantages. In particular, whendoing stability, we would need to consider canonical parameters for definable predicates ψ (¯ x, A ), which only has finitely many free variables but may have infinitely many param-eters. Canonical parameters for such definable predicates are dealt with as with canonicalparameters for formulae, and the existence of infinitely many parameters introduces veryfew additional; complications. On the other hand, if we wished to define the canonicalparameter as an equivalence class modulo a pseudo-metric we would be forced to considerpseudo-metrics on infinite tuples, the logic for whose equivalence classes could becomemessy.Other minor advantages include the fact that we need not ask ourselves whether aparticular formula defines a pseudo-metric on every structure or only on models of agiven theory, and finally the conceptually convenient fact that unlike equivalence relationswhich need to be replaced with pseudo-metrics, canonical parameters are a familiar notionwhich we leave unchanged.Let L be a continuous signature. For convenience, assume that L has a single sort S . If we wanted to work with a many-sorted language we would have to keep track onwhich variables (in the original language) belong to which sort, but other than that thetreatment is identical.Let us start with the case of a formula ϕ ( x Proposition 5.1. An L ϕ -structure is a model of T ϕ if and only if it is of the form M ϕ for some L -structure M . Therefore, if T is a complete L -theory then T ∪ T ϕ is a complete L ϕ -theory. Wediscussed the case of a single formula ϕ , but we can do the same with several (all)formulae simultaneously. BEN YAACOV AND ALEXANDER USVYATSOV Remark . As we said earlier, the continuous analogue of an equivalence relation is apseudo-metric. We can recover classes modulo pseudo-metrics from canonical parametersin very straightforward manner:(i) Assume that ϕ (¯ x, ¯ y ) defines a pseudo-metric on M n . Then the pseudo-metric d M ϕ, ϕ (defined on S M ϕ, ϕ = M n ) coincides with ϕ M , and ( S ϕ , d ϕ ) M ϕ is the com-pletion of the set of equivalence classes of n -tuples modulo the relation ϕ (¯ a, ¯ b ),equipped with the induced metric.(ii) In particular, let ξ n ( x Let M be a structure. We define S ϕ ( M ) as the quotient of S x ( M ) givenby the family of functions A M,ϕ = { ϕ ( x, b ) : b ∈ M } . An element of S ϕ ( M ) is called a (complete) ϕ -type over M .Accordingly, the ϕ -type of a over M , denoted tp ϕ ( a/M ), is given by the mappings ϕ ( x, b ) ϕ ( a, b ) where b varies over all elements of the appropriate sort of M . BEN YAACOV AND ALEXANDER USVYATSOV We equip S ϕ ( M ) with a metric structure: For p, q ∈ S ϕ ( M ) we define d ( p, q ) = sup b ∈ M | ϕ ( x, b ) p − ϕ ( x, b ) q | . Fact 6.2. Equipped with this metric and with its natural topology (inherited as a quotientspace from S x ( M ) ), S ϕ ( M ) is a compact topometric space as in Definition 4.11. Definition 6.3. Let M be a structure. A ϕ -predicate over M , or an M -definable ϕ -predicate , is a continuous mapping ψ : S ϕ ( M ) → [0 , Fact 6.4. Let ψ : S x ( M ) → [0 , be an M -definable predicate. Then the following areequivalent: (i) ψ is a ϕ -predicate (i.e., factors through the projection S x ( M ) → S ϕ ( M ) ). (ii) There are formulae ψ n ( x, ¯ b n ) , each obtained using connectives from several in-stances ϕ ( x, b n,j ) , where each b n,j ∈ M , and in M we have ψ ( x ) = F lim ψ n ( x ) . (iii) ψ can be written as f ◦ ( ϕ ( x, b i ) M : i < ω ) where f : [0 , ω → [0 , is continuousand b i ∈ M for all i < ω .Proof. (i) = ⇒ (ii). Standard application of Corollary 1.4.(ii) = ⇒ (iii) = ⇒ (i). Immediate. (cid:4) Lemma 6.5. Let M be a monster model, and M (cid:22) M a model. Let ψ ( x ) be an M -invariant ϕ -predicate over M . Then ψ is (equal to) a ϕ -predicate over M .Proof. We know that ψ ( x ) is equal to a definable predicate over M , so ψ ( x ) =lim ψ n ( x, c n ) where each ψ n ( x, z n ) is a formula and c n ∈ M . We also know that ψ ( x ) = lim χ n ( x, d n ), where each χ n ( x, ¯ d n ) is a combination of instances ϕ ( x, d n,j ) withparameters d n,j ∈ M . For all ε > n < ω such that: M (cid:15) sup x | ψ n ( x, c n ) − χ n ( x, ¯ d n ) | < ε In particular: M (cid:15) inf ¯ y n sup x | ψ n ( x, c n ) − χ n ( x, ¯ y n ) | < ε Since M (cid:22) M : M (cid:15) inf ¯ y n sup x | ψ n ( x, c n ) − χ n ( x, ¯ y n ) | < ε So there is ¯ d ′ n ∈ M such that: M (cid:15) sup x | ψ n ( x, c n ) − χ n ( x, ¯ d ′ n ) | < ε. We can therefore express ψ as lim χ n ( x, ¯ d ′ n ), which is a ϕ -predicate over M . (cid:4) This leads to the following: Definition 6.6. Let A be a (small) set in a monster model M . ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 35 (i) A ϕ -predicate over A , or an A -definable ϕ -predicate is a ϕ -predicate over M which is A -invariant.(ii) We define S ϕ ( A ) as the quotient of S ϕ ( M ) determined by the A -definable ϕ -predicates. The points of this space are called (complete) ϕ -types over A .(iii) Accordingly, the ϕ -type tp ϕ ( a/A ) is given by the mappings ψ ( x ) ψ ( a ) where ψ varies over all A -definable ϕ -predicates.By Lemma 6.5, these definitions coincide with previous ones in case that A = M (cid:22) M .We conclude with a result about compatibility of two kinds of extensions of local types:to the algebraic closure of the set of parameters, and to more (all) formulae. Lemma 6.7. Let p ∈ S ϕ ( A ) , and let q ∈ S ϕ (acl( A )) and r ∈ S x ( A ) extend p . Then q ∪ r is consistent.Proof. Let R ⊆ S x (acl( A )) be the pullback of r (i.e., the set of all its extensions to acomplete type over acl( A )), and R ⊆ S ϕ (acl( A )) the image of R under the restrictionprojection S x (acl( A )) → S ϕ (acl( A )). Then R is the set of all extensions of p to acl( A )compatible with r , and we need to show that q ∈ R (i.e., that R is the set of all theextensions of p ).Indeed, assume not. The sets R and therefore R are closed. Therefore we can separate R from q by a ϕ -predicate ψ ( x, a ), with parameter a ∈ acl( A ), such that ψ ( x, a ) q = 0and R ⊆ [ ψ ( x, a ) = 1]. Since a ∈ acl( A ), by Lemma 4.9 there is a sequence ( a i : i < ω )such that:(i) Every a i is an A -conjugate of a .(ii) For every ε > n such that every A -conjugate of a is in the ε -neighbourhood of some a i for i < n .Define ψ n ( x, a Let A ⊆ M where M is strongly ( | A | + ω ) + -homogeneous, and p ∈ S ϕ ( A ) .Then Aut( M/A ) acts transitively on the extensions of p to S ϕ (acl( A )) .Proof. Follows from (and is in fact equivalent to) Lemma 6.7. (cid:4) Local stability Here we answer C. Ward Henson’s question mentioned in the introduction. Throughoutthis section T is a fixed continuous theory (not necessarily complete) in a signature L . Definition 7.1. (i) We say that a formula ϕ ( x, y ) is ε -stable for a real number ε > T there is no infinite sequence ( a i b i : i < ω ) satisfying for all i < j : | ϕ ( a i , b j ) − ϕ ( a j , b i ) | ≥ ε . BEN YAACOV AND ALEXANDER USVYATSOV (ii) We say that ϕ ( x, y ) is stable if it is ε -stable for all ε > Lemma 7.2. Let ϕ ( x, y ) be a formula, ε > . Then the following are equivalent: (i) The formula ϕ is ε -stable. (ii) It is impossible to find ≤ r < s ≤ and an infinite sequence ( a i b i : i < ω ) suchthat r ≤ s − ε and for all i < j : ϕ ( a i , b j ) ≤ r , ϕ ( a j , b i ) ≥ s . (iii) There exists a natural number N such that in model of T there is no finitesequence ( a i b i : i < N ) satisfying:for all i < j < k : | ϕ ( a j , b i ) − ϕ ( a j , b k ) | ≥ ε. ( ∗ ) Proof. (i) ⇐⇒ (ii). Left to right is immediate. For the converse assume ϕ is not ε -stable, and let the sequence ( a i b i : i < ω ) witness this. For every δ > a ′ i b ′ i : i < N ) such that in addition:If i < j and i ′ < j ′ then: | ϕ ( a i , b j ) − ϕ ( a i ′ , b j ′ ) | , | ϕ ( a j , b i ) − ϕ ( a j ′ , b i ′ ) | ≤ δ. (For this we use the classical finite Ramsey’s Theorem. We could also use the infiniteversion to obtain a single infinite sequence with the same properties.)By compactness we can find an infinite sequence ( c i d i : i < ω ) witnessing ε -instabilitysuch that in addition, for i < j , ϕ ( c i , d j ) = r and ϕ ( c j , d i ) = s do not depend on i, j .Thus | r − s | ≥ ε . If r < s we are done. If r > s we can reverse the ordering on all thefinite subsequences obtained above, thus exchanging r and s , and conclude in the samemanner.(ii) ⇐⇒ (iii). Now right to left is immediate. For left to write, we argue as above, us-ing Ramsey’s Theorem and compactness, that there exists an infinite sequence ( a i b i : i <ω ) such that for i < j < k we have | ϕ ( a j , b i ) − ϕ ( a j , b k ) | ≥ ε and ϕ ( a i , b j ) = r and ϕ ( a j , b i ) = s do not depend on i, j . Then again | r − s | ≥ ε and we conclude as above. (cid:4) It follows that stability is a symmetric property: define ˜ ϕ ( y, x ) def = ϕ ( x, y ); then ϕ is( ε -)stable if and only if ˜ ϕ is. Notation 7.3. If ϕ is ε -stable we define N ( ϕ, ε ) to be the minimal N such that nosequence ( a i b i : i < N + 1) exists satisfying ( ∗ ).Let us define the median value connective med n : [0 , n − → [0 , n ( t < n − ) = ^ w ∈ [2 n − n _ i ∈ w t i = _ w ∈ [2 n − n ^ i ∈ w t i . If ϕ ( x, y ) is ε -stable define: d ε ϕ ( y, x < N ( ϕ,ε ) − ) = med N ( ϕ,ε ) (cid:0) ϕ ( x i , y ) : i < N ( ϕ, ε ) − (cid:1) . Lemma 7.4. Let M be a model and p ∈ S ϕ ( M ) . Then there exist c ε< N ( ϕ,ε ) − ∈ M suchthat, for every b ∈ M : (cid:12)(cid:12) ϕ ( x, b ) p − d ε ϕ ( b, c ε< N ( ϕ,ε ) − ) (cid:12)(cid:12) ≤ ε. ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 37 Proof. We argue as in the proof of [Pil96, Lemma 2.2]. Choose a realisation c (cid:15) p in themonster model: c ∈ M (cid:23) M . We construct by induction on n , tuples c n ∈ M in the sortof x , an increasing sequence of sets K ( n ) ⊆ P ( ω ), and tuples a w ∈ M in the sort of y foreach w ∈ K ( n ), as follows.At the n th step we assume we have already chosen c For all n and w ∈ K ( n ): | w | < N ( ϕ, ε ). Proof of claim. If not there is w = { m < . . . < m N − } ∈ K ( n ) where N ≥ N ( ϕ, ε ).Define m N = n (so m N − < m N ), and for j < N , let w j = { m i : i < j } . Then for all i < j < k ≤ N we have m i ∈ w j ∈ K ( m k ), whereby: (cid:12)(cid:12) ϕ ( c m i , a w j ) − ϕ ( c m k , a w j ) (cid:12)(cid:12) > ε. Thus the sequence ( c m i , a w i : i < N + 1) contradicts the choice of N ( ϕ, ε ). (cid:4) Claim It follows that for every w ∈ [2 N ( ϕ, ε ) − N ( ϕ,ε ) and a ∈ M : ^ i ∈ w ϕ ( c i , a ) − ε ≤ ϕ ( c, a ) ≤ _ i ∈ w ϕ ( c i , a ) + ε Whereby | ϕ ( c, a ) − d ε ϕ ( a, c < N ( ϕ,ε ) − ) | ≤ ε , as required. (cid:4) Definition 7.5. Let p ( x ) ∈ S ϕ ( M ). A definition for p is an M -definable predicate ψ ( y )satisfying ϕ ( x, b ) p = ψ M ( b ) for all b ∈ M . If such a definable predicate exists thenit is unique (any two such definable predicates coincide on M , and therefore on everyelementary extension of M ), and is denoted d p ϕ ( y ). BEN YAACOV AND ALEXANDER USVYATSOV Assume now that ϕ ( x, y ) is stable, and let: X = ( x ni : n < ω, i < N ( ϕ, − n ) − ,dϕ ( y, X ) = F lim n d − n ϕ ( y, x n< N ( ϕ, − n ) − ) . Proposition 7.6. Let M be a model, and p ∈ S ϕ ( M ) . Then there are parameters C ⊆ M such that dϕ ( y, C ) = d p ϕ ( y ) (so in particular, a definition d p ϕ exists). Moreover, d p ϕ isan M -definable ˜ ϕ -predicate.Proof. For each n < ω choose c n< N ( ϕ, − n ) − as in Lemma 7.4, and let C = ( c ni : n < ω, i < N ( ϕ, − n ) − ξ : M → [0 , 1] be defined as b ϕ ( x, b ) p . Then | d − n ϕ ( y, c n,< N ( ϕ, − n ) − ) M − ξ | ≤ − n , whereby: ξ = F lim n d − n ϕ ( y, c n< N ( ϕ, − n ) − ) M = dϕ ( y, C ) M . This precisely means that dϕ ( y, C ) = d p ϕ .That dϕ ( x, C ) is a ˜ ϕ -predicate follows from its construction. (cid:4) From this point onwards we assume that L has a sort for the canonical parameters ofinstances of dϕ ( y, X ) for every stable formula ϕ ( x, y ) ∈ L . If not, we add these sortsas in Section 5. It should be pointed out that if M is an L -structure and k M k ≥ |L| ,the addition of the new sorts does not change k M k : this can be seen directly from theconstruction, or using the Downward L¨owenheim-Skolem Theorem (Fact 2.19) and thefact that we do not change |L| .For every stable formula and type p ∈ S ϕ ( M ) we define Cb ϕ ( p ) as the canonicalparameter of d p ϕ ( y ). With the convention above we have Cb ϕ ( p ) ∈ M . Notice thatif p, q ∈ S ϕ ( M ), c = Cb ϕ ( p ) and c ′ = Cb ϕ ( q ), then d ( c, c ′ ) (in the sense of the sort ofcanonical parameters for dϕ ) is equal to d ( p, q ) in S ϕ ( M ).As with structures, we will measure the size of a type space S ϕ ( M ) by its metric densitycharacter k S ϕ ( M ) k . Proposition 7.7. The following are equivalent for a formula ϕ ( x, y ) : (i) ϕ is stable. (ii) For every M (cid:15) T , every p ∈ S ϕ ( M ) is definable. (iii) For every M (cid:15) T , k S ϕ ( M ) k ≤ k M k . (iv) There exists λ ≥ | T | such that whenever M (cid:15) T and k M k ≤ λ then k S ϕ ( M ) k ≤ λ as well.Proof. (i) = ⇒ (ii). By Proposition 7.6.(ii) = ⇒ (iii). Let D ⊆ M dϕ be the family of canonical parameters of instances of dϕ ( y, X ) which actually arise as definitions of ϕ -types over M . Then k D k ≤ k M k , and D is isometric to S ϕ ( M ).(iii) = ⇒ (iv). Immediate. ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 39 (iv) = ⇒ (i). Let λ ≥ | T | be any cardinal and assume ϕ is unstable. It is a classicalfact that there exists a linear order ( I, < ) of cardinality λ admitting > λ initial segments:for example, let µ be the least cardinal such that 2 µ > λ and let I = { , } <µ equippedwith the lexicographic ordering.Assuming ϕ is unstable then we can find (using Lemma 7.2 and compactness) 0 ≤ r Let ( X, d ) be a metric space. The diameter of a subset C ⊆ X is definedas diam( C ) = sup { d ( x, y ) : x, y ∈ C } We say that a subset C ⊆ X is ε -finite if it can be written as C = S i Let X be a compact topometric space.For a fixed ε > 0, we define a decreasing sequence of closed subsets X ε,α by induction: X ε, = XX ε,α = \ β<α X ε,β ( α a limit ordinal) X ε,α +1 = \ { F ⊆ X ε,α : F is closed and diam( X ε,α r F ) ≤ ε } X ε, ∞ = \ α X ε,α Finally, for any non-empty subset C ⊆ X we define its ε -Cantor-Bendixson rank in X as: CB X,ε ( C ) = sup { α : C ∩ X ε,α = ∅ } ∈ Ord ∪ {∞} If CB X,ε ( C ) < ∞ we also define CBm X,ε ( C ) = C ∩ X ε, CB X,ε ( C ) , i.e., the set of points ofmaximal rank.It is worthwhile to point out that either X ε,α = ∅ for every α (and eventually stabilisesto X ε, ∞ ) or there is a maximal α such that X ε,α = ∅ . The same holds for the sequence { C ∩ X ε,α : α ∈ Ord } if C ⊆ X is closed. BEN YAACOV AND ALEXANDER USVYATSOV Assume that C ⊆ X is closed and α = CB X,ε ( C ) < ∞ . Then by the previous para-graph C contains points of maximal rank, i.e., CBm X,ε ( C ) = ∅ . Moreover, CBm X,ε ( C )is compact and admits in X ε,α an open covering by sets of diameter ≤ ε . By compact-ness, it can be covered by finitely many such, and is therefore ε -finite. This need notnecessarily hold in case C is not closed.We will use this definition for X = S ϕ ( M ), where M is at least ω -saturated. In thiscase we may write CB ϕ,M,ε instead of CB S ϕ ( M ) ,ε , etc. Remark . In the definition of the ε -Cantor-Bendixson rank we defined X ε,α +1 byremoving from X ε,α all its “small open subsets”, i.e., its open subsets of diameter ≤ ε . There exist other possible definitions for the ε -Cantor-Bendixson derivative, usingdifferent notions of smallness. Such notions are studied in detail in [Bend] where it isshown that in the end they all boil down to the same thing. Proposition 7.11. ϕ is stable if and only if for one (any) ω -saturated model M (cid:15) T : CB ϕ,M,ε (S ϕ ( M )) < ∞ for all ε .Proof. If not, let Y = { p ∈ S ϕ ( M ) : CB ϕ,M,ε = ∞} . Then Y is compact, and if U ⊆ Y is relatively open and non-empty then diam( U ) > ε . We can therefore find non-emptyopen sets U , U such that ¯ U , ¯ U ⊆ U and d ( U , U ) > ε . Proceed by induction. Thiswould contradict stability of ϕ in a countable fragment of the theory.The converse is not really important, and is pretty standard. (cid:4) From now on we assume that ϕ is stable. Definition 7.12. Let M (cid:15) T and A ⊆ M , and assume that M is ( | A | + ω ) + -saturatedand strongly homogeneous. A subset F ⊆ S ϕ ( M ) is A -good if it is:(i) Metrically compact.(ii) Invariant under automorphisms of M fixing A .Recall the notions of algebraicity and algebraic closure from Definition 4.10. Lemma 7.13. Assume that F ⊆ S ϕ ( M ) is A -good. Then every p ∈ F is definable over acl( A ) .Proof. We know that p is definable, so let dϕ ( y, C ) be its definition, and c = Cb ϕ ( p ) thecanonical parameter of the definition. We may write dϕ ( x, C ) as d p ϕ ( y, c ).Assume that c / ∈ acl( A ). Then there exists an infinite sequence ( c i : i < ω ) in tp( c/A )such that d ( c i , c j ) ≥ ε > i < j , and we can realise this sequence in M by thesaturation assumption. By the homogeneity assumption, each d ϕ p ( y, c i ) defines a type p i which is an A -conjugate of p . Therefore p i ∈ F for all i < ω , and d ( p i , p j ) ≥ ε for all i < j < ω , contradicting metric compactness. (cid:4) Lemma 7.14. Assume that A ⊆ M (cid:15) T , M is ( | A | + ω ) + -saturated and strongly homo-geneous, and F ⊆ S ϕ ( M ) is closed, non-empty, and invariant under Aut( M/A ) . Then F contains an A -good subset. ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 41 Proof. Define by induction on n : F = F , and F n +1 = CBm ϕ,M, − n ( F n ). Then ( F n : n <ω ) is a decreasing sequence of non-empty closed subsets of S ϕ ( M ), and so F ω = T n F n = ∅ . The limit set F ω is ε -finite for every ε > 0, i.e., it is totally bounded. Since the metricrefines the topology, F ω is also metrically closed in S ϕ ( M ) and thus complete. We seethat f ω is a totally bounded complete metric space and therefore metrically compact.Also, each of the F n is invariant under Aut( M/A ), and so is F ω . (cid:4) Proposition 7.15. Let A ⊆ M (cid:15) T , and let p ∈ S ϕ ( A ) . Then there exists q ∈ S ϕ ( M ) extending p which is definable over acl( A ) .Proof. We may replace M with a larger model, so we might as well assume that M is( | A | + ω ) + -saturated and strongly homogeneous. Let P = { q ∈ S ϕ ( M ) : p ⊆ q } . ByLemma 7.14, there is an A -good subset Q ⊆ P , which is non-empty by definition. ByLemma 7.13, any q ∈ Q is an acl( A )-definable extension of p . (cid:4) Proposition 7.16. Let M (cid:15) T , p ( x ) ∈ S ϕ ( M ) and q ( y ) ∈ S ˜ ϕ ( M ) . Let d p ϕ ( y ) and d q ˜ ϕ ( x ) be their respective definitions, and recall that these are a ˜ ϕ - and a ϕ -predicate,respectively. Then d p ϕ ( y ) q = d q ˜ ϕ ( x ) p .Proof. Let M = M . Given M n (cid:23) M , obtain p n ∈ S ϕ ( M n ) and q n ∈ S ˜ ϕ ( M n ) by applyingthe definition of p and q , respectively, to M n (these are indeed complete satisfiable ϕ -and ˜ ϕ -types). Realise them by a n and b n , respectively, in some extension M n +1 (cid:23) M n .Repeat this for all n < ω .We now have for all i < j : ϕ ( a j , b i ) = d p ϕ ( y ) q and ϕ ( a i , b j ) = d q ˜ ϕ ( x ) p , and if thesediffer we get a contradiction to the stability of ϕ . (cid:4) Proposition 7.17. Assume that A ⊆ M is algebraically closed, p, p ′ ∈ S ϕ ( M ) are bothdefinable over A , and p ↾ A = p ′ ↾ A . Then p = p ′ .Proof. Let b ∈ M , q = tp ˜ ϕ ( b/A ). By Proposition 7.15 there is ˆ q ∈ S ˜ ϕ ( M ) extending q which is definable over acl( A ) = A , and let d ˆ q ˜ ϕ ( x ) be this definition. Recalling that d p ϕ and d p ′ ϕ are ˜ ϕ -predicates, d ˆ q ˜ ϕ is a ϕ -predicate, and all of them are over A , we have: ϕ ( x, b ) p = d p ϕ ( b ) = d p ϕ ( y ) q = d p ϕ ( y ) ˆ q = d ˆ q ˜ ϕ ( x ) p = d ˆ q ˜ ϕ ( x ) p ′ = d p ′ ϕ ( y ) ˆ q = d p ′ ϕ ( y ) q = d p ′ ϕ ( b )= ϕ ( x, b ) p ′ . Therefore p = p ′ . (cid:4) Given A ⊆ M (cid:15) T and p ∈ S ϕ (acl( A )), we denote the unique acl( A )-definable extensionof p to M by p ↾ M . The definition of p ↾ M is an acl( A )-definable ˜ ϕ -predicate which doesnot depend on M , and we may therefore refer to it unambiguously as d p ϕ ( y ) (so far weonly used the notation d p ϕ ( y ) when p was a ϕ -type over a model). BEN YAACOV AND ALEXANDER USVYATSOV If p ∈ S ϕ ( A ), we define (with some abuse of notation) p ↾ M = { q ↾ M : p ⊆ q ∈ S ϕ (acl( A )) } . Proposition 7.18. If A ⊆ M (cid:15) T and p ∈ S ϕ ( A ) as above, then p ↾ M is closed in S ϕ ( M ) .Assume moreover that M is ( | A | + ω ) + -saturated and strongly homogeneous, and let P = { q ∈ S ϕ ( M ) : p ⊆ q } . Then p ↾ M is the unique A -good set contained in P .Also, we have p ↾ M = T ε> CBm ϕ,M,ε ( P ) , and in fact p ↾ M = T ε ∈ E CBm ϕ,M,ε ( P ) forany E ⊆ (0 , ∞ ) such that inf E = 0 . In other words, q ∈ p ↾ M if and only if q ∈ P , andit has maximal CB ϕ,M,ε -rank as such for every ε > .Proof. If M (cid:22) M ′ , then p ↾ M = { q ↾ M : q ∈ p ↾ M ′ } , so we may assume that M is ( | A | + ω ) + -saturated and strongly homogeneous.Let Q ⊆ P be any A -good subset. If q ∈ Q , then q = ( q ↾ acl( A ) ) ↾ M ∈ p ↾ M . ByLemma 6.8 it follows that Q = p ↾ M . Therefore p ↾ M is closed.It follows by Lemma 7.14 that p ↾ M ⊆ CBm ϕ,M,ε ( P ) for every ε > 0, whereby T ε> CBm ϕ,M,ε ( P ) = ∅ . The other requirements for T ε> CBm ϕ,M,ε ( P ) to be A -goodfollow directly from its definition, and we conclude that p ↾ M = T ε> CBm ϕ,M,ε ( P ). (cid:4) Proposition 7.19. Assume that M (cid:22) N (cid:15) T are both ω -saturated. Let p ∈ S ϕ ( M ) , andlet q ∈ S ϕ ( N ) extend it. Then CB ϕ,M,ε ( p ) ≥ CB ϕ,N,ε ( q ) , and equality holds for all ε > if and only if q = p ↾ N .Proof. Assume first that N is ω -saturated and strongly homogeneous. Let: X ε,α = { p ∈ S ϕ ( M ) : CB ϕ,M,ε ( p ) ≥ α } Y ε,α = { q ∈ S ϕ ( N ) : CB ϕ,N,ε ( q ) ≥ α } We first prove by induction on α that if CB ϕ,M,ε ( p ) ≤ α and p ⊆ q ∈ S ϕ ( N ) thenCB ϕ,N,ε ( q ) ≤ α . Given a ϕ -predicate ψ ( x, a ) with parameters a ∈ M and r ∈ [0 , ψ ( x, a ) < r ] M = [ ψ ( x, a ) < r ] S ϕ ( M ) = { p ′ ∈ S ϕ ( M ) : ψ ( x, a ) p ′ < r } . Sets of this form form a basis of open sets for S ϕ ( M ). Since CB ϕ,M,ε ( p ) ≤ α , there aresuch ψ ( x, a ) and r such that p ∈ [ ψ ( x, a ) < r ] M and diam([ ψ ( x, a ) < r ] M ∩ X ε,α ) ≤ ε .Clearly, q ∈ [ ψ ( x, a ) < r ] N , so we’ll be done if we prove that diam([ ψ ( x, a ) < r ] N ∩ Y ε,α ) ≤ ε as well. Indeed, assume that there are q ′ , q ′′ ∈ [ ψ ( x, a ) < r ] N such that d ( q ′ , q ′′ ) > ε . Let dϕ ( y, e ′ ) and dϕ ( y, e ′′ ) be their respective definitions, where e ′ and e ′′ are the canonical parameters. Then we can find f ′ , f ′′ , b ∈ M such that f ′ f ′′ b ≡ e ′ e ′′ a and d ( a, b ) is as small as we want (in fact using ω -saturation we can actually have a = b ,but the argument goes through even if we can only have b arbitrarily close to a ; thereforethe result is true even if M is merely approximately ω -saturated , as defined in [Ben05] or[BU07]).Let p ′ , p ′′ ∈ S ϕ ( M ) be defined by dϕ ( y, f ′ ) and dϕ ( y, f ′′ ), respectively. Then p ′ , p ′′ ∈ [ ψ ( x, b ) < r ] M , and having made sure that b is close enough to a , we can get p ′ , p ′′ ∈ [ ψ ( x, a ) < r ] M . Also, we still have d ( p ′ , p ′′ ) > ε . Therefore at least one of p ′ / ∈ X ε,α or ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 43 p ′′ / ∈ X ε,α must hold, so let’s say it’s the former. In this case, by the induction hypothesis, p ′ ↾ N / ∈ Y ε,α . Since N is ω -strongly homogeneous, p ′ ↾ N and q ′ are conjugates by Aut( N ),so q ′ / ∈ Y ε,α , as required.Now let q = p ↾ N , and let dϕ ( y, e ) be the common definition. Assume that CB ϕ,N,ε ( q ) ≤ α , so there are a ϕ -predicate ψ ( x, a ) with a ∈ N , and r , such that q ∈ [ ψ ( x, a ) < r ] N anddiam([ ψ ( x, a ) < r ] N ∩ Y α,ε ) ≤ ε . We may find b, f ∈ M such that bf ≡ ae , and such that f is as close as we want to e (again: if we take the plain definition of ω -saturation we caneven have e = f , but we want an argument that goes through if M is only approximately ω -saturated). If p ′ ∈ S ϕ ( M ) is defined by dϕ ( y, f ), then p ′ ∈ [ ψ ( x, b ) < r ] M , andassuming f and e are close enough we also have p ∈ [ ψ ( x, b ) < r ] M . By the homogeneityassumption for N we get diam([ ψ ( x, b ) < r ] N ∩ Y α,ε ) ≤ ε .Assume now that p ′′ , p ′′′ ∈ [ ψ ( x, b ) < r ] M , and d ( p ′′ , p ′′′ ) > ε . Let q ′′ = p ′′ ↾ N and q ′′′ = p ′′′ ↾ N . Then d ( q ′′ , q ′′′ ) > ε , so either q ′′ / ∈ Y ε,α or q ′′′ / ∈ Y ε,α (or both), so let’s say it’sthe former. By the induction hypothesis we get p ′′ / ∈ X ε,α , showing that diam([ ψ ( x, b ) Again, let M (cid:22) N be ω -saturated. Let X ⊆ S ϕ ( M ) be any set of ϕ -types (without any further assumptions), and let Y ⊆ S ϕ ( N ) be its pre-image under therestriction mapping. Then CB ϕ,M,ε ( X ) = CB ϕ,N,ε ( Y ) . In particular, this gives us an absolute notion of CB ϕ,ε ( p ) where p is a partial ϕ -type,without specifying over which ( ω -saturated) model we work: just calculate it in any ω -saturated model containing the parameters for p .For example, assume that A ⊆ B ⊆ M (cid:15) T , p ∈ S ϕ ( A ) and p ⊆ q ∈ S ϕ ( B ). Then q isdefinable over acl( A ) if and only if CB ϕ,ε ( p ) = CB ϕ,ε ( q ) for all ε > Global stability and independence In the previous section we only considered local stability, i.e., stability of a singleformula ϕ ( x, y ). In this section we will use those results to deduce a global stabilitytheory.8.1. Gluing local types. Let A be an algebraically closed set, and say A ⊆ M . Let ϕ ( x, y ) and ψ ( x, z ) be two stable formulae, p ϕ ∈ S ϕ ( A ), p ψ ∈ S ψ ( A ). Then we know thateach of p ϕ and p ψ have unique extensions q ϕ ∈ S ϕ ( M ) and q ψ ∈ S ψ ( M ), respectively,which are A -definable.Assume now that p ϕ and p ψ are compatible, i.e., that p ϕ ( x ) ∪ p ψ ( x ) is satisfiable. Wewould like to show that q ϕ and q ψ are compatible as well. For this purpose there is no BEN YAACOV AND ALEXANDER USVYATSOV harm in assuming that M is strongly ( | A | + ω ) + -homogeneous, or even that M = M isour monster model.Let t, w be any variables in a single sort, say the home sort, and e = e ′ ∈ M in thatsort. We may assume that d ( e, e ′ ) = 1: even if not, everything we do below would workwhen we replace d ( t, w ) with d ( t, w ) ∔ . . . ∔ d ( t, w ). Define: χ ϕ,ψ ( x, yztw ) = ϕ ( x, y ) ∧ d ( t, w ) ∔ ψ ( x, z ) ∧ ¬ d ( t, w ) . Since we assume that ϕ and ψ are stable so is χ ϕ,ψ ( x, yztw ) by the following easyresult: Lemma 8.1. Assume ϕ i ( x, y ) are stable formulae for i < n and f is an n -ary continuousconnective. Then ( f ◦ ϕ A theory T is stable if all formulae are stable in T . ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 45 Definition 8.3. A theory T is λ -stable if for all n < ω and all sets A such that | A | ≤ λ : k S n ( A ) k ≤ λ . Definition 8.4. Let M be a model and p ∈ S n ( M ). We say that p is definable if p ↾ ϕ is forevery formula of the form ϕ ( x The following are equivalent for a theory T : (i) T is stable. (ii) All types over models are definable. (iii) T is λ -stable for all λ such that λ = λ | T | . (iv) T is λ -stable for some λ ≥ | T | .Proof. (i) ⇐⇒ (ii). By Proposition 7.7.(i) = ⇒ (iii). Assume T is stable λ = λ | T | , n < ω , and | A | ≤ λ . Then by DownwardL¨owenheim-Skolem we can find M ⊇ A such that k M k ≤ λ . Let ϕ ( x From now on we assume T is stable. Proposition 8.7. Let A ⊆ M , where A is algebraically closed, and let p ( x ) ∈ S x ( A ) .Then p has a unique extension to M , denoted p ↾ M , which is A -definable. Moreover, the A -definable definitions of such extensions do not depend on M , and will be denoted asusual by d p ϕ .Proof. For every formula ϕ ( x, y ) let d p ϕ = d p ↾ ϕ ϕ . Then uniqueness and moreover partare already a consequence of Proposition 7.17. Thus all that is left to show is that thefollowing set of conditions is satisfiable (and therefore a complete type): p ↾ M = { ϕ ( x, b ) = d p ϕ ( x, b ) M : ϕ ( x, y ) ∈ L , b ∈ M in the sort of y } = [ ϕ ( x,y ) ∈L ( p ↾ ϕ ) ↾ M . BEN YAACOV AND ALEXANDER USVYATSOV (Here x is fixed but y varies with ϕ .)By compactness it suffices to show this for unions over finitely many formulae ϕ . Fortwo formulae this was proved is the previous subsection, by coding both formulae in asingle one. But we can repeat this process encoding any finite set of formulae in a singleone, whence the required result. (cid:4) Definition 8.8. Let A ⊆ B , p ∈ S( B ). We say that p does not fork over A if there existsan extension p ⊆ q ∈ S n (acl( B )) such that all the definitions d q ϕ are over acl( A ).If ¯ a is a tuple, A and B sets, and tp(¯ a/AB ) does not fork over A , we say ¯ a is independent from B over A , in symbols ¯ a | ⌣ A B . Corollary 8.9. Let A ⊆ B , where A is algebraically closed, and let p ∈ S n ( A ) . Thenthere exists a unique q ∈ S n ( B ) extending p and non-forking over A . This unique non-forking extension is denoted p ↾ B , and is given explicitly as p ↾ B = { ϕ (¯ x, ¯ b ) = d p ϕ (¯ b ) : ϕ (¯ x, ¯ y ) ∈ L , ¯ b ∈ B } . Proof. Let M be any model such that B ⊆ M . Then acl( B ) ⊆ M , and ( p ↾ M ) ↾ acl( B ) = p ↾ acl( B ) is A -definable, so p ↾ B is a non-forking extension of p .Conversely, let q ∈ S n ( B ) be a non-forking extension of p . Then there exists q ′ ∈ S n (acl( B )) which is A -definable. Then q ′ ↾ M is an A -definable extension of p , so q ′ ↾ M = p ↾ M , whereby q = p ↾ B . (cid:4) Corollary 8.10. Let A and B be sets, ¯ a a tuple, and let p = tp(¯ a/ acl( A )) and q =tp(¯ a/ acl( AB )) . Then ¯ a | ⌣ A B if and only if d p ϕ = d q ϕ for every formula ϕ (¯ x, ¯ y ) .Proof. Right to left is immediate from the definition. So assume ¯ a | ⌣ A B . This meansthere is a type q ′ ∈ S n (acl( AB )) extending tp(¯ a/AB ) such that d q ′ ϕ is acl( A )-definablefor all ϕ . Then q and q ′ are conjugates by an automorphism fixing AB . Such an auto-morphism would fix acl( A ) setwise, so d q ϕ is acl( A )-definable for all ϕ . Now let M ⊇ AB be a model, and r = q ↾ M . Then d q ϕ = d r ϕ by definition, and r is acl( A )-definable andextends p , whereby d p ϕ = d r ϕ . (cid:4) We conclude: Theorem 8.11. Assume T is stable. Then: (i) Invariance: The relation | ⌣ is automorphism-invariant. (ii) Symmetry: ¯ a | ⌣ A ¯ b ⇐⇒ ¯ b | ⌣ A ¯ a (iii) Transitivity: ¯ a | ⌣ A BC if and only if ¯ a | ⌣ A B and ¯ a | ⌣ AB C . (iv) Existence: For all ¯ a , A and B there is ¯ b ≡ A ¯ a such that ¯ b | ⌣ A B . (v) Finite character: ¯ a | ⌣ A B if and only if ¯ a | ⌣ A ¯ b for all finite tuples ¯ b ∈ B . (vi) Local character: For all ¯ a and A there is A ⊆ A such that | A | ≤ | T | and ¯ a | ⌣ A A . ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 47 (vii) Stationarity: Assume A is algebraically closed, and B ⊇ A . If ¯ a ≡ A ¯ b and ¯ a | ⌣ A B , ¯ b | ⌣ A B then ¯ a ≡ AB ¯ b .Proof. Invariance is clear.Symmetry follows from Proposition 7.16.Transitivity is immediate from Corollary 8.10.For existence, we may replace B with any model containing B . Then let p ′ be anyextension of p = tp(¯ a/A ) to acl( A ), and then let ¯ b realise the unique non-forking extensionof p ′ to M .For finite character, we may replace A with acl( A ) without changing the statement.But then ¯ a | ⌣ A B if and only if tp(¯ a/AB ) = tp(¯ a/A ) ↾ AB , and if this fails it is due tosome finite tuple in B .Let p = tp(¯ a/ acl( A )). Recall we defined Cb ϕ ( p ) as the canonical parameter of d p ϕ .Let Cb( p ) = { Cb ϕ ( p ) : ϕ (¯ x, . . . ) ∈ L} be the canonical base of p . Then | Cb( p ) | = | T | ,and p is definable over its canonical base so ¯ a | ⌣ Cb( p ) acl( A ). For each c ∈ Cb( p ) we knowthat tp( c/A ) is algebraic, and going back to the definition of algebraicity in Lemma 4.9we see that countably many parameters in A suffice to witness this. Let A be the unionof all these witness sets for all c ∈ Cb( p ). Then A ⊆ A , | A | ≤ | T | , and Cb( p ) ⊆ acl( A ),so ¯ a | ⌣ acl( A ) acl( A ), or equivalently, ¯ a | ⌣ A A .Stationarity is just Corollary 8.9. (cid:4) Appendix A. A remark on continuity moduli on bounded spaces The usual definition of (uniform) continuity in the metric setting goes “for all ε > δ > δ : (0 , ∞ ) → (0 , ∞ ), mapping each ε to a corresponding δ . We would like to present herean alternative definition, which rather goes the other way around.In the cases which interest us all metric spaces (structures and type spaces) arebounded, usually of diameter ≤ 1. We may therefore allow ourselves the following sim-plification: Convention A.1. Hereafter, all metric spaces are bounded of diameter ≤ Definition A.2. An inverse continuity modulus is a continuous monotone function u : [0 , → [0 , 1] such that u (0) = 0.We say that a mapping f : ( X, d ) → ( X ′ , d ′ ) respects u , or that it is uniformly contin-uous with respect to u , if for every x, y ∈ X : d ′ ( f ( x ) , f ( y )) ≤ u ( d ( x, y )) . In other words, an inverse uniform continuous modulus maps a δ to an ε . (In case thatthe destination space is not bounded we may still consider inverse continuity moduli, butthen we need to allow the range of u to be [0 , ∞ ].) BEN YAACOV AND ALEXANDER USVYATSOV Lemma A.3. Let u be an inverse continuity modulus. For ε > define δ ( ε ) = sup { t ∈ [0 , 1] : u ( t ) ≤ ε } . Then δ is a continuity modulus, and every function which respects u (asan inverse uniform continuity modulus) respects δ (as a uniform continuity modulus).Proof. That ε > ⇒ δ ( ε ) > u ( t ) → t → 0. Assumenow that f : ( X, d ) → ( X ′ , d ′ ) respects u , ε > 0, and d ( x, y ) < δ ( ε ). By monotonicity of u and definition of δ : d ′ ( f ( x ) , f ( y )) ≤ u ( d ( x, y )) ≤ ε. (cid:4) A.3 The converse is not much more difficult: Lemma A.4. Let δ be a continuity modulus. Then there exists an inverse continuitymodulus u such that every function respecting u respects δ .Proof. For r, r ′ ∈ [0 , 1] define: u ( r ) = inf { ε > δ ( ε ) > r } (inf ∅ = 1) u ( r, r ′ ) = u ( r ′ ) r ≥ r ′ u ( r ′ ) · (cid:0) rr ′ − (cid:1) r ′ ≤ r < r ′ r < r ′ u ( r ) = sup r ′ ∈ [0 , u ( r, r ′ ) . Then u : R + → R + is an increasing function, not necessarily continuous. It is howevercontinuous at 0: lim r → + u ( r ) = 0 = u (0) (since for every ε > u ( δ ( ε )) ≤ ε ). Forevery r > 0, the family of function r u ( r, r ′ ), indexed by r ′ , is equally continuouson [ r , u is continuous on (0 , r ≤ / ⇒ u ( r ) ≤ u (2 r ) (sincefor r ′ ≥ r , u ( r, r ′ ) contributes nothing to u ( r )), whereby lim r → + u ( r ) = 0 = u (0).Therefore u is continuous on [0 , u ( r ) ≥ u ( r, r ) = u ( r ) ≥ sup { ε ≤ ∀ < ε ′ < ε )( δ ( ε ′ ) ≤ r ) } . Assume now that f : ( X, d ) → ( X ′ , d ′ ) respects δ . If x, y ∈ X and ε > d ′ ( f ( x ) , f ( y )) > ε , Then d ( x, y ) ≥ δ ( ε ′ ) for all 0 < ε ′ < ε , whereby u ( d ( x, y )) ≥ ε .Therefore d ′ ( f ( x ) , f ( y )) ≤ u ( d ( x, y )), and f respects u , as required. (cid:4) A.4 Together we obtain: Theorem A.5. A mapping between bounded metric spaces f : ( X, d ) → ( X ′ , d ′ ) is uni-formly continuous with respect to a (standard) continuity modulus if and only if it isuniformly continuous with respect to an inverse one. In other words, the two distinctdefinitions of continuity moduli give rise to the same notion of uniform continuity. ONTINUOUS FIRST ORDER LOGIC AND LOCAL STABILITY 49 Inverse continuity moduli give us (continuously) a direct answer to the question “howmuch can the value of f change from x to y ?” For example, if we attached to symbolsin a signature inverse continuity moduli, rather than usual ones, the axiom scheme UC L would take the more elegant form:sup x
This appendix answers a question posed by C. Ward Henson to the first author con-cerning stability of a formula inside a specific structure (in contrast with stability of aformula in all models of a theory, discussed in Section 7 above). The notion of stabilityinside a model appears for example in the work of Krivine and Maurey on stable Banachspaces [KM81]: a Banach space E is stable in the sense of Krivine and Maurey preciselyif the formula k x + y k is stable in the unit ball of E (viewed as a continuous structure inan appropriate language) in the sense defined below. Definition B.1. Let M be a structure, ϕ ( x, y ) be a formula and ε > 0. We say that ϕ is ε -stable in M if there is no sequence ( a i b i : i < ω ) in M such that | ϕ ( a i , b j ) − ϕ ( a j , b i ) | ≥ ε for all i < j < ω . We say that ϕ is stable in M if it is ε -stable in M for all ε > ϕ ( x, y ) is ( ε -)stable in M if and only if ˜ ϕ ( y, x ) is. Also, ϕ is ( ε -)stable in atheory T if and only if it is in every model of T . Lemma B.2. Assume that ϕ ( x, y ) is ε -stable in M . Then for every p ∈ S ϕ ( M ) thereexists a finite sequence ( c i : i < n ) in M such that for all a, b ∈ M : ( ∀ i < n )( ϕ ( c i , a ) ≤ ϕ ( c i , b ) + ε ) = ⇒ ϕ ( x, a ) p ≤ ϕ ( x, b ) p + 3 ε. Proof. Assume not. We will choose by induction on n elements a n , b n , c n ∈ M and r n , s n ∈ [0 , 1] as follows. At each step, there are by assumption a n , b n ∈ M such that ϕ ( c i , a n ) ≤ ϕ ( c i , b n ) + ε for all i < n , and yet ϕ ( x, a n ) p > ϕ ( x, b n ) p + 3 ε . Choose r n , s n such that ϕ ( x, b n ) p < r n < r n + 3 ε < s n < ϕ ( x, a n ) p . Once these choices are made wehave ϕ ( x, a i ) p > s i and ϕ ( x, b i ) p < r i for all i ≤ n , and we may therefore find c n ∈ M such that ϕ ( c n , a i ) > s i and ϕ ( c n , b i ) < r i for all i ≤ n . BEN YAACOV AND ALEXANDER USVYATSOV Once the construction is complete, for every i < j < ω colour the pair { i, j } as follows:if s i > ϕ ( c i , a j )+ ε , colour the pair { i, j } yellowish maroon; otherwise, colour it fluorescentpink. Notice that if { i, j } is fluorescent pink then ϕ ( c i , b j ) − ε > r i . By Ramsey’s Theoremthere is an infinite monochromatic subset I ⊆ ω , and without loss of generality I = ω .If all pairs are fluorescent pink then we have for all i < j < ω : ϕ ( c j , a i ) − ϕ ( c i , a j ) >s i − ( s i − ε ) = ε . If all are yellowish maroon we get ϕ ( c i , b j ) − ϕ ( c j , b i ) > ( r i + ε ) − r i = ε .Either way, we get a contradiction to ε -stability in M . (cid:4) B.2 Lemma B.3. Assume that ϕ ( x, y ) is ε -stable in M . Then for every p ∈ S ϕ ( M ) thereexists a finite sequence ( c i : i < n ) in M and a continuous increasing function f : [0 , n → [0 , , such that for all a ∈ M : | ϕ ( x, a ) p − f ( ϕ ( c i , a ) : i < n ) | ≤ ε. Proof. Let ( c i : i < n ) be chosen as in the previous Lemma. As a first approximation, let: f (¯ u ) = sup { ϕ ( x, a ) p : a ∈ M and ϕ ( c i , a ) ≤ u i for all i < n } . This function is increasing, but not necessarily continuous. We define a family of auxiliaryfunctions h ¯ u : [0 , n → [0 , 1] for ¯ u ∈ [0 , n : h ¯ u (¯ v ) = 1 ε ^ i Theorem B.4. Assume ϕ is stable in M . Then every p ∈ S ϕ ( M ) is definable. Moreover,for every such p there is a sequence ( c i : i < ω ) and a continuous increasing function f : [0 , ω > [0 , such that d p ϕ ( y ) = f ◦ ( ϕ ( c i , y ) : i < ω ) .Proof. For all m < ω choose a sequence ( c m,i : i < n m ) and function f m : [0 , n m → [0 , ε = 2 − m − . Let: N m = X k Notice that we get almost the same result as for a formula which is stable in a theory:the definition is still a limit of positive (i.e., increasing) continuous combinations ofinstances of ϕ with parameters in M . However, these combinations are not necessarilythe particularly elegant median value as in Section 7. References [BBH] Ita¨ı Ben Yaacov, Alexander Berenstein, and C. Ward Henson, Model-theoretic independence inthe Banach lattices L p ( µ ), submitted.[Bena] Ita¨ı Ben Yaacov, Definability of groups in ℵ -stable metric structures , submitted.[Benb] , Modular functionals and perturbations of Nakano spaces , submitted.[Benc] , On perturbations of continuous structures , submitted.[Bend] , Topometric spaces and perturbations of metric structures , submitted.[Ben03] , Positive model theory and compact abstract theories , Journal of Mathematical Logic (2003), no. 1, 85–118.[Ben05] , Uncountable dense categoricity in cats , Journal of Symbolic Logic (2005), no. 3,829–860.[BU07] Ita¨ı Ben Yaacov and Alexander Usvyatsov, On d -finiteness in continuous structures , Funda-menta Mathematicæ (2007), 67–88.[CK66] C. C. Chang and H. Jerome Keisler, Continuous model theory , Princeton University Press, 1966. BEN YAACOV AND ALEXANDER USVYATSOV [Fre04] D. H. Fremlin, Measure theory volumne 3: Measure algebras Ultraproducts in analysis , Analysis and Logic (CatherineFinet and Christian Michaux, eds.), London Mathematical Society Lecture Notes Series, no.262, Cambridge University Press, 2002.[KM81] Jean-Louis Krivine and Bernard Maurey, Espaces de Banach stables , Israel Journal of Mathe-matics (1981), no. 4, 273–295.[Mac73] Hilton Vieira Machado, A characterization of convex subsets of normed spaces , K¯odai Mathe-matical Seminar Reports (1973), 307–320.[Pil96] Anand Pillay, Geometric stability theory , Clarendon Press, 1996.[SU] Saharon Shelah and Alexander Usvyatsov, Model theoretic stability and categoricity for completemetric spaces , preprint. Ita¨ı Ben Yaacov, Universit´e de Lyon, Universit´e Lyon 1, Institut Camille Jordan, UMR5208 CNRS, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France URL : http://math.univ-lyon1.fr/~begnac/ Alexander Usvyatsov, UCLA Mathematics Department, Box 951555, Los Angeles, CA90095-1555, USA URL :: λ and concluding the proof. (cid:4) Definition 7.8.