Generic separable metric structures

GENERIC SEP ARABLE METRIC STR UCTURES ALEXANDER USVY A TSOV Abstract. W e compare three notions of genericity of separable metr ic structures. Our analysis provides a genera l model theoretic technique of showing that structures are generic in descriptive s et theoretic (topolo gical) sense and in measure theoretic sense. In pa rticular, it g ives a new p ersp ective on V er shik’s theo rems o n gener icity a nd ran- domness of Urysohn’s space among separa ble metric spaces. 1. Introduction There are sev eral w ay s t o define the notion of a “generic” metric structure. In this article we compare the mo del theoretic and t w o top ological approac hes to this question. This w ork w as motiv ated b y Anatoly V ershik’s results o n genericit y and randomness o f the Urysohn space among separable metric spaces, Theorems 1 a nd 2 in [V er0 2 ]. V ershik considers the collection of all separable metric spaces as a to p ological space, let us call it S . Some elemen ts of S are (isometric to) the Urysohn s pace. V ershik sho ws that this set is G δ dense in S , whic h leads to the conclusion that the Urys ohn space is in a sense “a g eneric” separable metric space. Then he sho ws that for an y “reasonable” probabilit y measure on S , the collection of metric spaces isometric to the Urysohn space is of measure 1. This leads to the conclusion that the Urysohn space is in a sense “the random” metric space. In his talk at the w orkshop on the Urysohn space at Ben-Gurion Univ ersit y (Ma y 2006), V e r shik said that his results ha d b een mot iv ated b y mo del theoretic prop erties of the (coun table) ra ndom graph, and that the theorems in [V er02] a r e in some sense the analo g ues of the appropriate facts in classical mo del theory , although the con text is differen t: instead of countable structures one deals with top ological spaces of car dina lity the con tinuum. In this pap er w e a im to sho w that the analo gy g o es mu ch farther. Indeed, c ountable discr ete structures are replaced in this con text with sep ar able m etric spaces, so classical mo del theory is not t he a ppro priate general fra mework. W e w ould lik e to con vince the reader tha t there exists a natural generalization o f discr ete first order logic to the c ontinuous context, in whic h V ershik’s r esults a r e the true analogues of classical facts, the Urysohn space is an analog ue of the random g r a ph, and discrete coun table mo dels a r e no more than a particular case of separable con tinuous structures. So from our p oin t of view, prop erties of the Urysohn space disco v ered b y V ershik are m uc h more than results inspired b y certain similarities b et w een this structure a nd the Date : Octob er 31 , 2 018. 1 2 ALEXANDER USV Y A TSOV random graph; in a sense, b o th of these are particular cases of the same mo del theoretic phenomenon, whic h w e inte nd to describe here. Con tinu ous first order logic, recen tly introduced b y Ita ¨ ı Ben-Y aaco v and the a uthor in [BU], allo ws o ne to study classes of metric spaces (ma yb e equipped with contin uous extra-structure, e.g. a collection of uniformly con tinuous functions from the spaces to R ) from mo del theoretic p oin t of view. Once workin g in this con text, man y results in classical mo del theory generalize to analytic structure s. This pa p er is dev oted to the connection b etw een mo del theory and descriptiv e set theory , whic h is v ery w ell-dev elop ed in the classical conte xt, i.e. studying Polish spaces of coun table structures for a giv en coun table signature, countable mo dels of a countable univ ersal theory , etc. W e will refer the reader to the excellen t expo sitor y pap er by Greg Hjorth, [Hj0 4]. In addition to generalizing V ershik’s theorem to a broad collection of classes of metric structures, our w ork generalizes a few ba sic concepts and results from classic a l first order mo del theory to the con tin uous contex t and pushes out the b oundaries of p ossible applications of con t in uous logic. So although we in t entionally try to make t he article accessible to non-logicians, it could also be of inte rest to mo del theorists . W orking in the contex t of contin uous first mo del theory , w e ada pt some basic facts and tec hniques fr om [Hj04 ] and show how one defines a Polish to p ology on the space of e.g. all separable mo dels of a certain univ ersal con tin uous theory . Ha ving done that, w e discus s three differen t not ions of genericit y of a structure . One is mo del theoretic, genericity of a mo del of a univ ersal theory among its p eers). The other t wo are top ological, g enericit y of a structure as an elemen t of the appropriate Polish space in tw o differen t w a ys: in the sense of Baire cat ego ry theory and in the sense of measure theory . Let us state things more precisely . Let K b e a “ r easonable” class of separable metric structures. In our con text K will normally b e the class of all separable mo dels o f a certain univ ersal con tinuous first order theory . F rom t he mo del theoretic p oin t of view, a generic structure in K is a structure in whic h “a n ything that can happ en” in K happ ens. Suc h structures a re called “existen tially closed” for K . W e will give precise definitions later. On the other hand, one can consider K as a P olish space (i.e. there is a natural top ology on K with respect to whic h K is a complete separable metric space). One can call a structure “generic” for K in top ological sense if its isomorphism class is a “big” subset of K . One natura l notion of “ big ness” in this con text is G δ dense. Another one comes f r om measure theory: one can consider natural measures on the space K and ask what are the sets of measure 1 . In this article w e ha v e sev era l primary go a ls: (i) In t r o duce the general mo del theoretic framew ork and t he relev an t notion of genericit y . (ii) Construct t he P olish space of separable metric structures. (iii) Connect the notio ns of genericit y . More precise ly , w e explain ho w a mo del the- oretic notion of genericit y giv es r ise to G δ dense sets in the appropriate P olish GENERIC SEP ARABLE METRIC STR UCTURES 3 space S and sets of measure 1 with resp ect to any “reasonable” probability mea- sure on S . In particular, this provides a p o w erful general tec hnique for sho wing that certain structures are top ologically generic and random (as it allo ws us to use w ell-dev elop ed mo del t heoretic to ols for this purpose). (iv) Conclude with some examples. In particular, we discuss mo del theory of Urysohn space and sho w that our results generalize V ershik’s theorems on its “top o logical” genericit y . Ac knowledgem ents. The author thanks the anon ymous referee for v ery helpful com- men ts a nd suggestions. 2. Preliminaries and b asics 2.1. Contin uous logic. Con tinuous first order logic w as in tro duced in [BU] and deve l- op ed further b y Ita ¨ ı Ben-Y aacov, Alexander Berenstein, C. W ard Henson and the author. W e refer the reader to [BBHU] for a detailed exp osition. W e will no w try to summarize some imp ortant basic notions, facts and no tations. Just as in classical predicate logic, o ne start s with a fixed s ignatur e (vo c abulary) τ . In this pap er, τ will b e coun table. A signature ( v o cabular y) is a collection of function sym b ols and predicate sym b ols as w ell as con tinuit y mo duli for all these sym b ols. There is a distinguished predicate sym bo l d ( x, y ), whic h will corresp ond to the metric. Giv en a v o cabulary τ , one constructs the con tinuous language L whic h corresp o nds to it, whic h consists of con tinuous first order τ - formulae . As in classical first order logic, form ulae are constructed by induction using c onne ctives and quan tifie rs . An y countable collection of contin uous functions from [0 , 1] k to [0 , 1] (for an y k ) whic h is dense in the set of all suc h contin uous functions can b e ta ken a s the set of connectiv es. W e will assume that the follow ing functions are among our connectiv es: (i) The c onstant function q for ev ery q ∈ [0 , 1] ∩ Q (ii) p ointwise minimum ([0 , 1] 2 → [0 , 1]) (iii) p ointwise maximum ([0 , 1] 2 → [0 , 1]) (iv) Multiplic ation by q , [ x 7→ x · q ], for ev ery q ∈ [0 , 1] ∩ Q ([0 , 1] → [0 , 1]) (v) ne gation , [ x 7→ 1 − x ], ([0 , 1] → [0 , 1]) (vi) dotminus or im plic ation : T runcated (at 0) min us [( x, y ) 7→ x − · y ] ( [0 , 1] 2 → [0 , 1]) (vii) T runcated (a t 1) plus [( x, y ) 7→ x + y ] ([0 , 1] 2 → [0 , 1]) (viii) ( x, y ) 7→ | x − y | ([0 , 1 ] 2 → [0 , 1]) Of course, some of the functions ab ov e can b e defined using the others, but w e a re not lo oking for “minimal” sy stems of connectiv es here. The con tin uo us quantifiers are inf x and sup x . As in classical first o rder logic, we only allo w quantification o v er elemen ts. So the follow ing are examples of formulae: • d ( x, y ) • d ( x, y ) − · d ( y , x ) 4 ALEXANDER USV Y A TSOV • inf x,y d ( x, y ) • sup x,y ,z ( d ( x, z ) − · ( d ( x, y ) + d ( y , z ))) As usual, formulae with no “free v ariables” (i.e. each v ariable is in a scop e of one of the quan tifiers) are called sentenc es . The first t wo form ulae ab o ve are not sen t ences, while the last tw o are. An L - pr e-structur e is a set M equipped with i n terpr etations for all τ -sym b ols suc h that d is inte r preted as a pseudometric, eac h predicate sym b ol is in terpreted as a function fro m (some p o wer of ) M to [0 , 1], each function sym b ol is inte r preted as a function from (some p ow er o f ) M to M , and all of them resp ect their con tinuit y mo duli with resp ect to d . In other w ords: • d M : M 2 → [0 , 1] is a pseudometric • F or every n -ary predicate sym b ol P , w e hav e P M : M n → [0 , 1] uniformly con- tin uous with resp ect to d (resp ecting the contin uit y mo dulus of P dictated by τ ) • F or ev ery n -ary function sym b ol f , w e hav e f M : M n → M uniformly contin uous with resp ect to d (resp ecting the con tinu ity mo dulus of f dictated by τ ) A structur e is a pre-structure in whic h d is a complete metric. See [BU ] or [BBHU] fo r more details (on e.g. contin uit y mo duli). F ormal definitions of these notions are not imp orta nt f o r us here; but it is crucial that the interpretation of eac h predicate sym b ol and of eac h function sym b ol is uniformly con tin uous, and uniformly so in all L -structures (t his is what w e need the con tin uity mo duli for). Uniform con tin uity allo ws us to take ultra pro ducts of L -structures and obtain e.g. compactnes s of first order con tinuous log ic. Note that given a structure M , one can easily define (b y induction) the M -v alue of ϕ for any senten ce ϕ . W e will denote this v alue b y ϕ M (it is a real n um b er in t he in terv al [0 , 1]). Note also t hat there is no particular imp ortance for the in terv al [0 , 1], but ev ery pred- icate sym b ol m ust ha v e bo unded range ( a gain, so that ultrapro ducts will work), and b y rescaling w e ma y assume it is in f a ct a lw ay s [0 , 1]. A c ondition is a statement concerning the v alue of a sen tence ϕ . F or example, ϕ ≤ ε , ϕ = 0, ϕ < ε are conditions (where ε ∈ [0 , 1]). W e will call conditions of the f orm ϕ ≤ ε , ϕ = 0, etc close d c onditions and those of the form ϕ < ε , etc op en c onditions . Note that as w e will mostly w ork with conditions of the form ϕ ≤ ε and ϕ < ε , the con tinuous quantifiers inf x and sup x can b e view ed as analo g ues of t he existen tial and the univ ersal quan tifiers resp ectiv ely . It is clear what it means for a struc t ure M to satisfy a condition α , and w e write M | = α . If ev ery structure whic h satisfies α also satisfies β , we sa y that β fol lows from α a nd write α | = β . If Λ is a collection of conditions and M is a structure, w e sa y that M is a model for (of ) Λ if M satisfies ev ery condition in Λ, and write M | = Λ. A the ory T is a collection of close d conditio ns whic h is consisten t (i.e. there is a structure M whic h satisfies a ll the conditions in T , M | = T ). W e will alw ays assume that GENERIC SEP ARABLE METRIC STR UCTURES 5 theories are closed under en ta ilmen t, i.e. if α ∈ T and α | = β , then β ∈ T . W e denote b y Mo d( T ) the class of all mo dels of T . W e encourage the r eader to ha v e a lo ok at examples of con tinuous languages and theories presen ted in [BU ] and [BBHU]. W e shall not discuss ultrapro duct cons tructions in this paper. Again, curious r eaders are referred to [BU] or [BBHU]. An imp ortant consequenc e is the Comp actness The or em for con tinuous lo gic, whic h will b e useful for us: F act 2.1. (Comp actness The or em) L et Λ b e a c ol le ction of closed c onditions whic h is finitely satisfiable (i. e . every fi n ite subset of Λ h a s a mo del). Then Λ has a mo del. Let M ⊆ N b e L -structures. W e say t hat M is an elem e ntary submo del of N ( M ≺ N ) if for ev ery L -sen tence ϕ w e hav e ϕ M = ϕ N . W e say tha t a theory T is mo del c omplete if for ev ery M , N | = T , M ⊆ N ⇒ M ≺ N . Most theories are not mo del complete; w e will discuss this not ion more later. T is mo del complete if (but not only if ) it eliminates quantifiers ; see more in [BU] o r [BBHU]. Note that contin uous first order log ic is a natural g eneralizatio n of classical first order logic. Indeed, ev ery classical first o rder theory can b e view ed as a con tin uous theory in whic h the metric is disc rete. 2.2. Polish space of sep arable con tin uous str uctures. Let τ b e a fixed coun table con tinuous v o cabulary . F or simplicit y w e assume that τ is relational (i.e. no function sym b ols). Let L be the corresponding ( coun table) con tinuous language. W e denote the space o f all L - contin uous separable structures M with a distinguished coun t a ble dense subset N ⊆ M by S . Consider the follo wing top ology on S : basic op en sets are of the form U ϕ ( ¯ x ) , ¯ a,ε = U ϕ (¯ a ) ,ε = { M ∈ S : ϕ M (¯ a ) < ε } where ϕ ( ¯ x ) is a quantifier fr e e L -fo rm ula, ¯ a ∈ N , ε ∈ [0 , 1] ∪ {∞} . Prop osition 2.2. S with the top olo gy ab ove is a Polish sp ac e. Pr o of. Let h R i : i < ω i b e an en umeration of τ , R 0 b eing the metric. Let k i b e the ar it y of R i (so k 0 = 2 ) . By section 2 of [Hj04] the pro duct space X = [0 , 1] F i N k i is P olish. W e can view S as a subspace of X via the follo wing em b edding φ : S → X : φ ( M ) = h f i : i < ω i such t ha t f i is precisely R M i on the dense subset N of M . Note that the fact that R 0 is a pseudometric and all the rest of the predicates resp ect the appropriate con t in uity mo duli with resp ect to it is a collection of closed conditions. The fact that R 0 is an actual metric can b e expressed as a collection of op en conditions. So S can b e view ed as a G δ subset of a P olish space, a nd t herefore, b y Lemma 2.2 in [Hj04], S is a Polis h space itself. qed 2 . 2 Let T be an L -theory . W e denote the space of all elemen ts of S whic h are mo dels of T b y S T . So S = S ∅ . 6 ALEXANDER USV Y A TSOV 2.3. Univ er sal theories and existentially closed mo dels. Definition 2.3. (i) W e call a theory universal if it is (the closure under en tailmen t of ) a collection of conditions of the form [sup ¯ x ϕ ( ¯ x ) = 0] where ϕ is quan tifier free. (ii) Let K be a class of L -structures. W e call M ∈ K existential ly close d for K if the following holds: for ev ery M ⊆ N ∈ K , a quan tifier free for mula ϕ ( x, ¯ y ) a nd a tuple ¯ b ∈ M , we hav e inf M x ϕ ( x, ¯ b ) = inf N x ϕ ( x, ¯ b ). (iii) If T is a univ ersal theory we sa y that M | = T is existen tially closed f or T if it is existen tially closed for K = Mo d( T ). When T is clear from the con text w e omit it and sa y “ M is existen tially closed” or “ M is an e.c. s tr ucture” or “ M is an e.c. model”. R emark 2.4 . F or M ∈ K , to b e existen tia lly closed for K means in a sense that an ything whic h is quantifie r free definable with parameters in M , that can happ en in some mo del in K , happ ens already in M . In this sense, existen tially closed models are “generic” among structures in K . Example 2.5 . (i) A tomless probabilit y algebras are existen tially closed among all probabilit y a lgebras (see [BU] or [BBHU ]). (ii) Hilb ert spaces equipp ed with a unitary op erator U with full sp ectrum ( S pec ( U ) = S 1 ) are e.c. among all Hilb ert spaces equipp ed with a unitary op erator, see [BUZ]. (iii) Atomle ss probability algebras with an ap erio dic automor phism are e.c. a mong probabilit y a lgebras equipp ed with an automo r phism, see [BH]. Let T b e a univers a l theory , K = Mo d( T ), a nd K ec b e the class o f e.c. mo dels o f T . W e call K ec the c ontinuous R o binson the ory of T . One ma y ask: is K ec elemen tary (i.e. is there a con tinuous theory T ∗ suc h that K ec = Mo d( T ∗ ))? The answ er is not alw ay s p ositiv e, eve n in the classical (discrete) con text. F or example, the Robinson theory of groups (i.e. T is collection of first o rder sen tences whic h ar e true in all groups, K is the class o f all gr oups, and K ec consists of all gr oups whic h are existen tially closed) is not elemen ta ry . But often the answ er is y es; in this case w e sa y that T admits a model companion and call T ∗ the mo del c o m p anion of T . It is easy to see that in this case T ∗ is mo del c omple te : if M , N | = T ∗ and M ⊆ N , then M ≺ N . It do es not necessarily eliminate quan tifiers; if it do es, w e call it the mo del c ompletion of T . R emark 2.6 . In Example 2.5 ab o v e, the classes of e.c. models are in fact elemen tar y , and the appropriate t heories are the mo del companions, and ev en the mo del completions of the univ ersal theories. Observ ation 2.7. Let T b e a unive rsal theory . Then S T is a closed subset of S , and therefore a P olish space. GENERIC SEP ARABLE METRIC STR UCTURES 7 Pr o of. Clear. qed 2 . 7 The following fact is w ell-kno wn, but the author is not a w a re o f a written reference. Although the pro of is iden t ical to that of t he classical (discrete) analogue, w e include it for completeness . In order not t o scare the reader, w e only deal with separable structures, whic h is all w e need in this article (the pro of for an arbitrary infinite cardinalit y is essen tially the same). F act 2.8. L et T b e a univers a l the ory, M | = T sep ar able. Then ther e exists a sep ar able N ⊇ M , N | = T , N is e . c . for T . Pr o of. The pro of is standard and resem bles v ery m uc h the construction of the algebraic closure of a giv en field. Let M 0 = M . W e cons t ruct separable M i | = T for i < ω b y induc tion as follows: Giv en M i let h ϕ α ( ¯ x α ) : α < ω i be an en umeration of all quan tifier free form ulae ϕ ( ¯ x ) with parameters in M i . No w define a sequenc e M α i of separable mo dels of T b y induction on α < ω as follows: • M 0 i = M i • Give n M α i , if there is no ¯ a ∈ M i satisfying [ ϕ α (¯ a ) = 0], but there exists M ′ ⊇ M α i , M ′ | = T where suc h ¯ a exis ts, let M α +1 i b e an y suc h separable M ′ (for the cardinalit y preserv ation one can use e.g. Prop osition 7.3 in [BBHU]). Otherwise let M α +1 i = M α i . No w define M i +1 = ∪ α<λ M α i . Note that M i +1 has the following prop erty : if there exis t s a quantifie r free formula ϕ ( ¯ x ) with parameters in M i , an extension M ′ | = T of M i +1 and ¯ a ∈ M ′ satisfying [ ϕ (¯ a ) = 0], then suc h ¯ a exists already in M i +1 . Finally , let N = ∪ i<ω M i ; it is easy to c hec k that it is existen tially closed. qed 2 . 8 3. Inductive theories Recall that w e assume that theories are closed under en tailmen t, i.e. ev ery closed condition whic h follow s from T is already in T . W e denote by T o the collection of all op en conditions whic h follo w f r om T . Let T oc = T ∪ T o . Let T b e an L -theory and let ∆ b e collection o f conditions (open or/and closed). W e denote b y T ∆ the ∆- p art of T . So T ∆ = T oc ∩ ∆ F o r an L -structure M , w e denote the ∆-part of Th( M ) b y T h ∆ ( M ). As usual, w e define Σ n and Π n form ulae b y induction on n : • Σ 0 = Π 0 = quan tifier free formulae • Σ n +1 is the collection of f o rm ulae of the form inf ¯ x ϕ ( ¯ x, ¯ y ) where ϕ ( ¯ x, ¯ y ) ∈ Π n • Π n +1 is the collection of formulae of the fo r m sup ¯ x ϕ ( ¯ x, ¯ y ) where ϕ ( ¯ x, ¯ y ) ∈ Σ n R emark 3.1 . So Σ 1 is the collection of a ll the existen tial fo r mulae, Π 1 is the collection of all the univ ersal form ulae. 8 ALEXANDER USV Y A TSOV Definition 3.2. (i) F or Λ ⊆ L , w e denote b y Λ o the collection of all op en conditions of the form ϕ < ε for ϕ ∈ Λ, ε > 0 . (ii) F or Λ ⊆ L , w e denote by Λ c the collection of all closed conditions of the f o rm ϕ ≤ ε for ϕ ∈ Λ, ε > 0. (iii) Let ∆ b e a collection o f conditions. W e call a theory T a ∆- the ory if T ∆ | = T . (iv) W e call a theory T inductive if it is axiomatizable by op en conditions of the form sup ¯ x inf ¯ y ϕ ( ¯ x, ¯ y ) < ε , where ϕ is quan tifier-free. So T is inductiv e if it is a Π o 2 -theory . R emark 3.3 . (i) So a theory T is unive r sal iff it is a ∆- theory for ∆ = Π c 1 . (ii) May b e the reader would exp ect us to work with Π c 2 -theories instead of Π o 2 . Not e that if T is Π c 2 then it is inductiv e, and for complete theories t he notions are equiv a len t; but as w e w an t Theorem 3.6 to hold for all theories, not necessarily complete, a nd a s w e w an t our theories to define G δ subsets of S , the natural c hoice is open conditio ns. Lemma 3.4. L et ∆ = Σ o n or ∆ = Π o n for some n (or just ∆ is a c ol le ction of op en c ondi- tions cl o se d under r e s c aling , i.e. multiplic ation by sc alars and the “p ointwise min imum” c onne ctive). L et T b e a the ory and supp ose that for every two L -structur e s M , N such that M | = T and Th ∆ ( M ) ⊆ Th ∆ ( N ) , we have N | = T . T hen T is a ∆ -the ory. Pr o of. Supp ose not; so there exists N | = T ∆ , N 6| = T . By the a ssumption, for no M | = T do we ha ve Th ∆ ( M ) ⊆ Th ∆ ( N ). In other w ords, fo r ev ery M | = T t here exists a formu la ϕ M with [ ϕ M < ε ] ∈ ∆ suc h that ϕ M M < ε , ϕ N M ≥ ε . By rescaling we ma y assume ε = 1 2 . So the set T ∪ { [ ϕ M ≥ 1 2 ] : M | = T } is inconsisten t. By compactness, [min k i =1 ϕ i < 1 2 ] ∈ T o for some finite collection of suc h ϕ i . But ∆ is closed under t a king minima, and ev ery one of the conditions [ ϕ i < 1 2 ] is in ∆, so (as N | = T ∆ ), (min k i =1 ϕ i ) N < 1 2 , so for some i ϕ N i < ε , a con tra diction. qed 3 . 4 Lemma 3.5. L et T b e a c omplete L -the ory and M an L -structur e with Th Σ c n ( M ) ⊆ T Σ c n . Then ther e exists M ′ | = T an d a Σ n -elementary emb e d ding f : M → M ′ . Pr o of. Let M = h a α : α < λ i , M ′ a λ -satura t ed mo del of T . Construct f α : A α = { a β : β < α } → M ′ an increasing con tin uous sequence of Σ n -elemen tary em b eddings. Giv en f α , consider the Σ n -t yp e in M of a α o ve r A α , call it π ( x ). By the assumption that Th Σ c n ⊆ T , f ( π ( x )) is a ty p e in M ′ o ve r f ( A α ) (as Σ n is closed under “inf ”), and use the saturation of M ′ . qed 3 . 5 Theorem 3.6. L et T b e a n L -the ory such that Mo d( T ) is pr e s e rve d under unions of chains, that is if h M i : i < ω i is an incr e asing chain of mo dels of T , then the closur e of the union M = S i M i is a ls o a mo del of T . Then T is inductive. Pr o of. Supp ose Mo d( T ) is closed under unions of c hains. W e w o uld lik e to sho w that T Π o 2 | = T . GENERIC SEP ARABLE METRIC STR UCTURES 9 Let M | = T and T ′ a complete L - theory exte nding T Π o 2 , N | = T ′ with Th Π o 2 ( M ) ⊆ T ′ W e will sho w that N | = T , which clearly suffices by Lemma 3.4. Construct a c hain N 0 ⊆ M 1 ⊆ N 1 ⊆ . . . suc h t ha t: (i) N 0 = N (ii) M i | = T , N i | = T ′ (iii) N i ≺ N i +1 If t he construction is p ossible, w e are done: Let M = S i M i = S i N i . M | = T by the assumption on T and (ii) ab o v e. On the other hand, clearly N 0 ≺ M (as N 0 ≺ N i for all i by (iii) ab o ve ), so N = N 0 | = T , and w e a re do ne. Wh y is the construction p o ssible? Let N 0 = N . As Th Π o 2 ( M ) ⊆ Th( N ) = T ′ , w e hav e Th Σ c 2 ( N ) ⊆ Th( M ), so b y Lemma 3.5, there exists M 0 | = Th ( M ) and a Σ c 2 -em b edding of N 0 in to M 0 . Let N 0 = h a α : α < λ i . Enric h the v o cabulary τ with λ -many constan t sym b ols, call the new language L ′ . Claim 3.6.1 . Th Σ c 1 ( M 0 , h a α : α < λ i ) ⊆ Th( N 0 , h a α : α < λ i ) as L ′ -theories. Pr o of. Clearly (as N 0 is a Σ c 2 -elemen tary submo del of M 0 ), Th Σ c 2 ( N 0 , h a α : α < λ i ) ⊆ Th( M 0 , h a α : α < λ i ) as L ′ -theories, and therefore Th Π o 2 ( M 0 , h a α : α < λ i ) ⊆ Th( N 0 , h a α : α < λ i ) as L ′ -theories, in particular (1) Th Σ o 1 ( M 0 , h a α : α < λ i ) ⊆ Th( N 0 , h a α : α < λ i ) as L ′ -theories. Let [inf ¯ x ψ ( ¯ x, ¯ a ) ≤ ε ] b e a closed existen tial conditio n satisfied b y M 0 with parameters ¯ a ∈ N 0 (i.e. ¯ a = a α 1 , . . . , a α k for some α 1 , . . . , α k < λ ). Then M 0 | = [inf ¯ x ψ ( ¯ x, ¯ a ) < ε ′ ] for ev ery ε ′ > ε . So this is true in N 0 (b y (1 ) a b ov e), whic h completes the pro of of the claim. qed 3 . 6 By the Claim ab o ve and Lemma 3.5, t here exists N 1 | = T ′ in to whic h M 0 is Σ c 1 - em b edded in the lang uage L ′ . Clearly , this means that N 0 ≺ N 1 . The rest of the construction is similar. qed 3 . 6 W e obtain the a nalogue of a w ell- known Robinson’s theorem in the con tin uo us con text: Corollary 3.7. If T is mo del c omplete, then it is inductive. Corollary 3.8. L et T b e a universal the ory which has a mo del c omp anion T ′ . Then T ′ is inductive. Pr o of. T ′ is mo del complete. qed 3 . 8 10 ALEXANDER USV Y A TSOV 4. Generic and random models 4.1. Mo del completions and top ological genericity. Observ ation 4.1. Let T ′ b e an inductiv e theory . Then S T ′ is a G δ subset of S . Pr o of. F or ev ery quantifier free formula ϕ ( ¯ x ), ¯ a, ¯ b ∈ N and ε > 0, the op en conditio n [ ϕ (¯ a, ¯ b ) < ε ] defines an op en subset of S , whic h w e called U ϕ (¯ a , ¯ b ) ,ε . The op en con- dition [inf ¯ x ϕ ( ¯ x, ¯ b ) < ε ] corr esp o nds, therefore, to an op en subset of S , whic h equals S ¯ a ∈ N U ϕ (¯ a , ¯ b ) ,ε . A Π o 2 condition defines a subset of S which is a (coun ta ble) interse ction (o ve r all p ossible ¯ b ∈ N ) of sets a s ab ov e; therefore it is a G δ set. Clearly , a coun table collection of Π o 2 conditions still corresp onds t o a G δ set. qed 4 . 1 F act 4.2. L et T b e a universal the ory. Then the c ol le ction of se p ar able e.c. mo dels is dense in S T . Pr o of. By F act 2.8 ev ery separable M | = T can b e extended to a separable e.c. mo del M ′ | = T . No w it is easy to see that one can rename the elemen ts of M ′ suc h tha t a certain finite ¯ a ∈ N remains unc hanged (and so M ′ is indeed in a sp ecified op en neighborho o d of M in S T ). qed 4 . 2 Corollary 4.3. L et T b e a universal the ory which h a s a mo del c omp anion T ′ . Then S T ′ is a G δ dense subset of S T . Recall that a theory T is called ℵ 0 - c ate goric al if an y t wo separable mo dels of T are isomorphic. Definition 4.4. Let T b e a univ ersal t heory . W e call M ∈ S T generic if the isomorphism class of M is G δ dense in S T . Corollary 4.5. L et T b e a unive rsal the ory whic h has a mo del c omp anion T ′ , and assume T ′ is ℵ 0 -c ate goric al. Then (any) existential ly c l o s e d mo del o f T is ge neric in S T . Pr o of. By Corolla ry 4.3, S T ′ is G δ dense in S T . By ℵ 0 - categoricity of T ′ , S T ′ is the isomorphism class of any e.c. mo del of T (whic h is in S T ). qed 4 . 5 4.2. Random struct ures. Once w e ha v e sho wn that the class of existen tially closed mo dels in S is “big ” in the sense of Baire category theory , a natural question is: is there a similar measure-theoretic result? In [V er02] V ershik sho ws that the Urysohn space is in a sense the random metric space. W e know t ha t t he model companio n of the univ ersal theory of gra phs is the random graph. Are these facts particular cases of a mo del theoretic phenomenon? Let T b e a unive rsal theory , µ a probabilit y measure on S T satisfying Assumption 4.6 . (i) No nonempt y op en set has probability 0. (ii) µ is in v ariant under the actio n of S ∞ on S T . In other w ords, for ev ery form ula ϕ ( ¯ x ), ε > 0 and ¯ a, ¯ b ∈ N , w e ha ve t he equalit y µ ( U ϕ (¯ a ) ,ε ) = µ ( U ϕ ( ¯ b ) ,ε ). So µ ( U ϕ (¯ a ) ,ε ) do es not dep end on ¯ a . GENERIC SEP ARABLE METRIC STR UCTURES 11 Clearly , these are v ery natural assumptions on a measure on S T , once w e a re in terested in “random structures”: first, w e assume that if a certain op en ev ent o ccurs in some mo del of T , then its proba bility is p o sitiv e. Second, w e assume that in a sense isomorphic mo dels “o ccur” with equal pr o babilit y . Lemma 4.7. L et µ b e as ab ove. Then the se t of al l existential ly close d s tructur es in S T has pr ob ability 1 . In other wor ds, if we pick a structur e “r andomly”, it is going to b e existential ly c lose d alm ost sur ely. Pr o of. Let M be a randomly c hosen structure. W e aim to sho w that with probabilit y 1 it is existen tia lly closed. Let ϕ ( ¯ x ) b e a fo rm ula, and supp ose that in some M ⊆ N | = T w e hav e inf N ¯ x ϕ ( ¯ x ) ≤ ε . Let ε ′ > ε . So there exists ¯ a ∈ N suc h that ϕ N (¯ a ) < ε ′ , and therefore µ ( U ϕ (¯ a ) ,ε ′ ) = δ > 0. By the in v ariance of µ , µ ( U ϕ ( ¯ b ) ,ε ′ ) = δ for ev ery ¯ b ∈ N , and so the probability that in a randomly c ho sen structure M w e ha ve ϕ M ( ¯ b ) ≥ ε ′ is b ounded a wa y from 1 for each ¯ b ∈ M . No w clearly with probabilit y 1 for some ¯ b ∈ M w e hav e ϕ M ( ¯ b ) < ε ′ , therefore inf M ¯ x ϕ ( ¯ x ) ≤ ε almost surely , and we are done. qed 4 . 7 R emark 4.8 . Note that w e did not really use the inv aria nce of µ . W e only need that the probabilit y of the ev en t U ϕ (¯ a ) ,ε is either 0 for all ¯ a or b o unded a wa y from 0 fo r all ¯ a . Corollary 4.9. L et T b e a universal the ory which h a s a mo del c omp anion T ′ . Then S T ′ is a set of pr ob ability 1 in S T . Definition 4.10. W e call a separable mo del of a unive rsal theory T r andom if the measure of its isomorphism class in S T is 1 with resp ect to an y pr o babilit y measure µ as in Assumption 4.6. In other w ords, M is a random mo del of T if for ev ery µ as ab o v e, a randomly c hosen structure in S T is almost surely isomorphic to M . Just lik e in Corollar y 4.5 w e obtain: Corollary 4.11. L et T b e a universal the ory which has a mo del c omp anion T ′ . Assume furthermor e that T ′ is ℵ 0 -c ate goric al. Then any sep ar able mo del of T ′ is a r andom mo del of T . Clearly , this generalizes the “ randomness” of t he countable random g raph; see more in the following subsection. 4.3. Concluding remarks on genericit y. In t his section w e ha ve sho wn that the mo del theoretic notion of genericit y g iv es rise to b oth Baire catego r y theoretical and measure theoretical notions of g enericit y in the space S . In other words, w e hav e sho wn: Corollary 4.12. L et T b e a universal the ory which admits a mo del c omp anion T ′ . Then S T ′ is b oth G δ dense in S and of me asur e 1 with r esp e ct to any r e asonable me asur e on S ( i.e. any me asur e satisfying Assumption 4.6). In particular, w e ha v e the following: 12 ALEXANDER USV Y A TSOV Corollary 4.13. Supp ose T is a univers a l the ory whic h has a mo del c omp anion T ′ , an d assume furthermor e that T ′ is ℵ 0 -c ate goric al. Then the (uniq ue up to isomorphism) mo del of T ′ is b oth the generic and the r andom mo del of T . Example 4.14 . The atomless separable probability algebra is b oth the generic and the random separable probability algebra. Pr o of. The theory of a t o mless proba bilit y algebras is t he mo del companion of the uni- v ersal theory of probabilit y algebras by [BU]. It is a lso ℵ 0 -categorical, so apply Corollary 4.13. qed 4 . 14 As w e hav e already men tioned, ev ery classical first order theory is a con tin uous first order theory with discrete metric. W e can therefore apply our analysis to e.g. the theory of t he random graph. Recall that t he theory of the r a ndom graph is the mo del completion of the univ ersal theory of graphs. Example 4.15 . The random graph is the generic countable graph. Pr o of. The (classical) fir st order theory of the random gra ph is t he mo del completion (and therefore the mo del companion) of the univ ersal theory o f graphs. It us also ℵ 0 - categorical. So the desired conclus io n follo ws from Corolla r y 4.13. qed 4 . 15 Similarly , the unique coun ta ble mo del of the mo del completion of the univ ersal theory of graphs is the random graph in the sense defined here in Definition 4.10. W ell, no surprise here: w e’re just say ing that the random graph is, w ell, random. In the follow ing section w e will show that the con tinuous first order theory of t he Urysohn space ha s similar prop erties, and therefore Corollary 4.1 3 applies to it as w ell. One can think of this theory as the contin uous analog ue o f the t heory of the random graph: instead of the discrete predicate R ( x, y ) in the theory of graphs which can b e either true or fa lse, w e hav e a metric whic h can tak e any v alue b et w een 0 a nd 1. 5. Ur ysohn sp a ce Man y results on the mo del theory of the Urysohn space here are “folklore”, but the author is not aw are of any written references. In order to follow the pro ofs, the reader should b e familiar with basics o f con tinu ous mo del theory slightly b ey o nd what is sk etc hed in section 2 of the ar ticle. W e remind the reader that the Urysohn sp ac e is the unive r sal complete separable metric space, first constructed b y Pa v el Urysohn. Due to the limitations of the genre, w e will consider the b ounde d Urysohn space, i.e. Urysohn space of diameter 1. W e denote it b y U . Denote by E n the collection of all p o ssible distance configuratio ns on n p oin t s of diameter 1. It will be con ve nient for us to think ab out it in the follow ing wa y: GENERIC SEP ARABLE METRIC STR UCTURES 13 ϑ ( x 1 , . . . , x n ) ∈ E n if ϑ ( x 1 , . . . , x n ) is a formula of the f orm _ 1 ≤ i,j ≤ k | d ( x i , x j ) − r ij | where the matrix ( r ij ) 1 ≤ i,j ≤ k is a distance matrix of some finite metric space of diameter 1, and W stands for the lattice op eration of p oint wise maxim um. Let us intro duce the follo wing notation: fo r ϑ ∈ E n +1 , let ϑ ↾ n be the restriction of ϑ to the first n v ariables. Clearly , for ev ery ϑ ∈ E n +1 , fo r ev ery ε > 0 there exists a δ = δ ( ε ) > 0 such that if a 1 , . . . , a n ∈ U satisfy ϑ ↾ n ( a 1 , . . . , a n ) < δ , then there exists a n +1 ∈ U suc h that ϑ ( a 1 , . . . , a n , a n +1 ) ≤ ε . Let T U b e the collection of all the conditions o f the form  sup x 1 ,...,x n inf y  ε 1 − δ (1 − ϑ ↾ n ( x 1 , . . . , x n )) ^ ϑ ( x 1 , . . . , x n , y )  ≤ ε  whic h is j ust one w ay of stating ∀ x 1 , . . . x n ∃ y ( ϑ ↾ n ( x 1 , . . . , x n ) < δ → ϑ ( x 1 , . . . , x n , y ) ≤ ε ) Note t ha t V stands for the latt ice o p eration of p o int wise minimum. The follo wing follo ws from the standard Urysohn’s arg umen t: F act 5.1. The only sep ar able c omplete mo del of T U is U . Corollary 5.2. T U is ℵ 0 -c ate goric al, and ther efor e a c omplete c ontinuous the ory. Pr o of. By (the contin uous v ersion of ) V a ugh t’s test. qed 5 . 2 Prop osition 5.3. T U eliminates quantifiers. Pr o of. By the classical back -and-forth a rgumen t (see Theorem 4.16 in [BU]) using the axioms of T U . qed 5 . 3 Corollary 5.4. T U is the mo del c ompletion (and ther ef o r e the mo del c o mp anion) of the “empty” c ontinuous universal the ory (the univers a l the ory of a metric s p ac e with no extr a-structur e). U is (the only) existential ly close d metric s p ac e. A nat ural conclusion fr o m o ur analysis is the following fo rm of V ershik’s theorems: Corollary 5.5. The Urysohn sp ac e (of diameter 1 ) is the generic and the r andom metric sp ac e (of diameter 1). Pr o of. The theory of the Urysohn space is the model companion of the unive r sal theory of metric spaces and is ℵ 0 -categorical, so the result follo ws immediately f r o m Corollary 4.13. qed 5 . 5 14 ALEXANDER USV Y A TSOV Reference s [BBHU] I. Ben-Y aacov, A. J. Berenstein, C. W. Henson, A. Usvy atsov, Mo del The ory for Metric Structures , T o app ear in a Newton Institute MAA Pro gramme pro cee dings volume. [BH] A. J. Berenstein, C. W. Henson, Mo del the ory of Pr ob ability sp ac es with an automorphism , submitted. [BU] I. Ben-Y a acov and A. Usvyatso v , L o c al stability in c ontinuous first or der lo gic , submitted [BUZ] I. Ben-Y a acov, A. Usvyatsov, M. Zadk a, Generic automorphism of a Hilb ert sp ac e , preprint. [Hj04] G. Hjorth, Gr oup actions and c ount able mo dels , Lo gic Collo quium ’99, 3 –29, Lect. Notes Log., 17, Asso c. Symbol. Log ic, Urbana, IL, 2 004. [V er0 2] A. M. V ershik, R andom metric sp ac es and the un iversal Urysohn sp ac e , F undamen tal Mathe- matics T o day , 10th Anniversary of the Indep endent Moscow Univ er sity , MCCME Publisher s (2002). Alexander U svy a tsov, University of Cal if o rnia – Los Angeles, Ma thema tics Dep ar t- ment, Box 951555, Los A ngeles, CA 90095 -1555, USA URL : http: //www. math.ucla.edu/~alexus