On the existence of atomic models

arXiv:math/9301210v1 [math.LO] 15 Jan 1993On the existence of atomic modelsM. C. Laskowski∗Department of MathematicsUniversity of MarylandS.

ShelahDepartment of MathematicsHebrew University of JerusalemDepartment of MathematicsRutgers University †December 14, 1992AbstractWe give an example of a countable theory T such that for everycardinal λ ≥ℵ2 there is a fully indiscernible set A of power λ such thatthe principal types are dense over A, yet there is no atomic model ofT over A. In particular, T(A) is a theory of size λ where the principaltypes are dense, yet T(A) has no atomic model.If a complete theory T has an atomic model then the principal types aredense in the Stone space Sn(T) for each n ∈ω.

In [2, Theorem 1.3], [3, page168] and [5, IV 5.5], Knight, Kueker and Shelah independently showed thatthe converse holds, provided that the cardinality of the underlying languagehas size at most ℵ1.∗Partially supported by an NSF Postdoctoral Fellowship.†The authors thank the U.S. Israel Binational Science Foundation for its support ofthis project. This is item 489 in Shelah’s bibliography.1

In this paper we build an example that demonstrates that the conditionon the cardinality of the language is necessary. Specifically, we construct acomplete theory T in a countable language having a distinguished predicateV such that if A is any subset of V M for any model M of T, then theprincipal types are dense in Sn(T(A)) for each n ∈ω.

(T(A) = Th(M, a)a∈A).However, T(A) has an atomic model if and only if |A| ≤ℵ1.In fact, by modifying the construction (Theorem 0.5) we may insist thatthere is a particular non-principal, complete type p such that, for any subsetA of V M, p is realized in every model of T(A) if and only if |A| ≥ℵ2. WithTheorem 0.6 we show that the constructions can be generalized to largercardinals.We first build a countable, atomic model in a countable language havingan infinite, definable subset of (total) indiscernibles.

Let L be the languagewith unary predicate symbols U and V , a unary function symbol p, andcountable collections of binary function symbols fn and binary relation sym-bols Rn for each n ∈ω. By an abuse of notation, p and each of the fn’s willactually be partial functions.For a subset X of an L-structure M, define the closure of X in M, cl(X),to be the transitive closure of cl0(X) = {fn(b, c) : b, c ∈X, n ∈ω}.

So cl(X)is a subset of the smallest substructure of M containing X.Let K be the set of all finite L-structures A satisfying the following eightconstraints:i) U and V are disjoint sets whose union is the universe A;ii) p : U →V ;iii) each fn : U × U →U;iv) for each n, Rn(x, y) →(U(x) ∧U(y));v) the family {Rn : n ∈ω} partitions all of U2 into disjoint pieces;vi) for each n and m ≥n, Rn(x, y) →fm(x, y) = x;vii) if x′, y′ ∈cl({x, y}) and Rn(x, y), then Wj≤n Rj(x′, y′);viii) there is no cl-independent subset of U of size 3 (i.e., for all x0, x1, x2 ∈U,there is a permutation σ of {0, 1, 2} such that xσ(0) ∈cl({xσ(1), xσ(2)}). )2

It is routine to check that K is closed under substructures and isomor-phism and that K contains only countably many isomorphism types. Weclaim that K satisfies the joint embedding property and the amalgama-tion property.

As the proofs are similar, we only verify amalgamation. LetA, B, C ∈K with A ⊆B, A ⊆C and A = B∩C.

It suffices to find an elementD of K with universe B ∪C such that B ⊆D and C ⊆D. Let {b0, .

. .

, bl−1}enumerate UB, let {c0, . .

. , cm−1} enumerate UC and let k > l, m be largeenough that S{RBj : j < k} = (UB)2 and S{RCj : j < k} = (UC)2.

For eachb ∈B ∖A and c ∈C ∖A, let fj(b, c) = cj for j < m, and fj(b, c) = b forj ≥m and let Rk(b, c). Similarly, for j < l let fj(c, b) = bj and fj(c, b) = cfor all j ≥l and Rk(c, b).

It is easy to check that D ∈K.It follows (see e.g., [4, Theorem 1.5]) that there is a countable, K-genericL-structure B. That is, (*) B is the union of an increasing chain of elementsof K, (**) every element of K isomorphically embeds into B and (***) ifj : A →A′ is an isomorphism between finite substructures of B then thereis an automorphism σ of B extending j.

Such structures are also referred toas homogeneous-universal structures. Let T be the theory of B.We record the following facts about B and T: First, V B is infinite as thereare elements A of K with V A arbitrarily large; V B is a set of indiscerniblesbecause of property (***) and the fact that any two n-tuples of distinctelements from V B are universes of isomorphic substructures of B; As everyfinite subset of B is contained in an element of K, it follows that cl(X) isfinite for all finite subsets X of B and there is no cl-independent subset ofUB of size 3; Finally, B is atomic as for any tuple b from B, property (***)guarantees that the complete type of b is isolated by finitely much of theatomic diagram of the smallest substructure A of B containing b.

(If n isleast such that Wj≤n Rj(a, a′) for all a, a′ ∈UA then we need only the reductof the atomic diagram of A to Ln = {U, V, p, Rj, fj : j ≤n}. )Lemma 0.1 Let C be a model of T and let A be any subset of V C. Then theprincipal types are dense over A.Proof.Let θ(x, a) be any consistent formula, where a is a tuple of kdistinct elements from A.

Let b be any k-tuple of distinct elements fromV B. As the elements from V are indiscernible, B |= ∃xθ(x, b).

Let c from Brealize θ(x, b). Since B is atomic, there is a principal formula φ(x, y) isolatingtp(c, b).

It follows from indiscernibility that φ(x, a) is a principal formula suchthat C |= ∀x(φ(x, a) →θ(x, a)).3

Lemma 0.2 Let C be an arbitrary model of T and let A ⊆V C. If C is atomicover A then:i) |UC| ≥|A|;ii) cl(X) is finite for all finite X ⊆UC;iii) there is no cl-independent subset of size 3 in UC.Proof. (i) |UC| ≥|A| since for each a ∈A, p−1(a) is non-empty.

(ii) As C is atomic, tp(X/A) is isolated by some formula θ(x, c), wherec is a k-tuple of distinct elements from A.As V is indiscernible, θ(x, b)is principal for any k-tuple b of distinct elements from B. Choose d from Brealizing θ(x, b) and suppose that |cl(d)| = l < ω.

Then as θ(x, a) is principal,θ(x, a) implies |cl(x)| ≤l. (iii) Assume c0, c1, c2 ∈UC are cl-independent.

As C is atomic over A,tp(c) is principal, so let θ(x, a) isolate tp(c). Again choose b from V B and dfrom UB such that B |= θ(d, b).

But then d is a cl-independent subset of UB,which is a contradiction.We next record a well-known combinatorial lemma (see e.g., [1]). Anabstract closure relation on a set X is a function cl : P(X) →P(X) suchthat, for all subsets A, B of X and all b ∈X, A ⊆cl(A), cl(cl(A)) = cl(A),A ⊆B implies cl(A) ⊆cl(B) and b ∈cl(A) implies there is a finite subsetA0 of A such that b ∈cl(A0).Lemma 0.3 For all ordinals α and all n ∈ω, if |X| ≥ℵα+n and cl is aclosure relation on X such that |cl(A)| < ℵα for all finite subsets A of X.Then X contains a cl-independent subset of size n + 1.Proof.

Fix an ordinal α. We prove the lemma by induction on n. Forn = 0 this is trivial, so assume the lemma holds for n. Suppose X has size atleast ℵα+n+1.

As cl is finitely based we can find a subset Y of X, |Y | = ℵα+n,such that cl(A) ⊆Y for all A ⊆Y . Choose b ∈X ∖Y .

Define a closurerelation cl′ on Y by cl′(A) = cl(A ∪{b}) ∩Y . By induction there is a cl′-independent subset B of Y of size n. It follows that B ∪{b} is the desiredcl-independent subset of X.4

Note that by taking α = 0 in the lemma above, if cl is a locally finiteclosure relation on a set X of size ℵ2, then X contains a cl-independentsubset of size 3.Theorem 0.4 Let A be a subset of V C for an arbitrary model C of T. ThenA is a set of indiscernibles and the principal types over A are dense, but thereis an atomic model over A if and only if |A| ≤ℵ1.Proof. The principal types are dense over A by Lemma 0.1.

If |A| ≤ℵ1then there is an atomic model over A by e.g., Theorem 1.3 of [2]. How-ever, if |A| ≥ℵ2 then there cannot be an atomic model over A by Lem-mas 0.2 and 0.3.Our next goal is to modify the construction given above so that there isa non-principal complete type p that is realized in any model containing A,provided that |A| ≥ℵ2.

To do this, note that any atomic model of T(A) islocally finite and omits the type of a pair of elements from U with every Rnfailing. We shall enrich the language so as to code each of these by a single1-type.Let L′ = L ∪{W, g, h} ∪{cn : n ∈ω}, where W is a unary predicate,g and h are respectively binary and ternary function symbols, and the cn’sare new constant symbols.

Let B′ be the L′-structure with universe B ∪D,where D = {dn : n ∈ω} is disjoint from B, W is interpreted as D, each cnis interpreted as dn, g : U × U →W is given by g(a, b) = dn, where Rn(a, b)holds and h(a, b, c) = dn if and only if n = |cl({a, b, c})|. Let T ′ be the theoryof B′.Note that any automorphism σ of B extends to an automorphism σ′ ofB′, where σ′ ↾D = id.

It follows that B′ is atomic and V B′ is an indiscernibleset. Let p(x) be the non-principal type {W(x)} ∪{x ̸= cn : n ∈ω}.

Weclaim that p is complete. This follows from the fact that for any L′-formulaθ(x), if n is greater than the number of terms occurring in θ and if θ(x) isan L′n = {U, V, W, g, h, p, Rl, fl, cl : l < n}-formula then B′ |= θ(ci) ↔θ(cj)for all i, j ≥n.

(This fact can be verified by finding a back-and-forth systemS = {⟨a, b⟩: |a| = |b| < n} such that, for every ⟨a, b⟩∈S and every atomicL′n ∪{Ri, fi, ci}-formula φ(x), |x| = |a|,B′ |= φ(a) ↔φ′(b),5

where φ′(x) is the atomic L′n ∪{Rj, fj, cj}-formula generated from φ(x) byreplacing each occurrence of Ri, fi, ci by Rj, fj, cj, respectively. )Theorem 0.5 Let A be a subset of V C′ for any model C′ of T ′.

Then A isa set of indiscernibles and the principal types over A are dense. Further, if|A| ≤ℵ1 then there is an atomic model over A, while if |A| > ℵ1 then anymodel of T ′(A) realizes the complete type p.Proof.

The first two statements follow from the atomicity of B′ and theindiscernibility of V B′. If |A| ≤ℵ1 then the existence of the atomic modelover A follows from Knight’s theorem.

So suppose |A| ≥ℵ2 and let D′ be anymodel of T ′ containing A. By examining the proof of Lemma 0.3 it followsthat either there are a, b, c ∈UD′ such that cl({a, b, c}) is infinite or that thereare cl-independent elements a0, a1, a2 ∈UD′ (i.e., if cl is a closure relation onX and |X| ≥ℵ2 then either cl(x, y, z) is infinite for some x, y, z ∈X or thereis an independent subset of X of size 3).In the first case h(a, b, c) realizes p. Now assume that the closure of anytriple from UD′ is finite.

If, in addition, for every two elements a, b from UD′there were an integer n such that D′ |= Rn(a, b), then T ′ would ensure thatthere would not be any 3-element cl-independent subset of UD′. (Under theseassumptions there would be only finitely many possibilities for the diagramof a triple under the functions {fi : i ∈ω} and no independent triple existsin B′.) Consequently, there must be a pair of elements a, b from UD′ suchthat D′ |= ¬Rn(a, b) for all n ∈ω, so g(a, b) realizes p.We close with the following theorem demonstrating that the behaviorbetween ℵ1 and ℵ2 holds more generally between ℵk and ℵk+1 for all k ≥1.The theorem is stated in its most basic form to aid readability.

We leave itto the reader to verify that the strengthenings given in Theorems 0.4 and 0.5(i.e., no atomic model over a given set or the non-atomicity being witnessedby a specific complete type) can be made to hold as well.Theorem 0.6 For every k, 1 ≤k < ω there is a countable theory Tk suchthat Tk has an atomic model of size ℵα if and only if α ≤k.Proof. Fix k. Let Lk = {fn, Rn : n ∈ω}, where each fn is a (k + 1)-ary function and each Rn is a (k + 1)-ary relation.

Define cl(X) to be thetransitive closure of cl0(X) = {fn(a0, . .

. , ak) : ai ∈X} and let K be the setof all finite Lk-structures satisfying the following constraints:6

i) {Rn : n ∈ω} partitions the (k + 1)-tuples into disjoint pieces;ii) for each n and m ≥n, Rn(x0, . .

. , xk) →fm(x0, .

. .

, xk) = x0;iii) if x′0, . .

. , x′k ∈cl({x0, .

. .

, xk}) and Rn(x0, . .

. , xk), then Wj≤n Rj(x0, .

. .

, xk);iv) there is no cl-independent subset of size k + 2.As before, there is a countable, K-generic Lk-structure B.Let Tk =Th(B). Just as before, B is atomic, cl is locally finite on B, Wn∈ω Rn(b) holdsfor all (k + 1)-tuples b from B and there is no (k + 2)-element cl-independentsubset of B.

Thus, the proof that there is no atomic model of Tk of powerλ > ℵk is exactly analogous to the proof of Theorem 0.4. What remains isto prove that there is an atomic model of size ℵk.

To help us, we quote thefollowing combinatorial fact, which is a sort of converse to Lemma 0.3.Lemma 0.7 For every k, 1 ≤k < ω and every set A, |A| ≤ℵk−1, there isa family of functions {gn : Ak →A : n ∈ω} such that, letting cl denote thetransitive closure under the gn’s:i) cl is locally finite;ii) for all a ∈Ak there is an n such that gm(a) ∈a for all m ≥n;iii) there is no cl-independent subset of A of size k + 1.Proof. We prove this by induction on k. If k = 1, let {ai : i < α ≤ℵ0}enumerate A. Define gn bygn(ai) =anif n < i;aiotherwise.Now assume the lemma holds for k. Let {ai : i < α ≤ℵk} enumerateA.

Define gn : Ak+1 →A as follows: Given a = ⟨ai0, . .

. , aik⟩∈Ak+1, leti∗= max{i0, .

. .

, ik}. If i∗= il for a unique l < k + 1 then we can applythe inductive hypothesis to the set Ai∗= {aj : j < i∗} and obtain a familyof functions hn : Aki∗→Ai∗satisfying the conditions of the lemma.

Nowdefine gn(a) = hn(b), where b is the subsequence of a of length k obtained bydeleting ail from a. On the other hand, if there are j < l < k + 1 such thatij = il = i∗, then simply let gn(a) = ai∗for all n ∈ω.7

To show that there is an atomic model of Tk of size ℵk, as the principalformulas are dense and are Σ1 it suffices to show the following:(#) If A is an Lk-structure of size at most ℵk−1 such that every finitelygenerated substructure of A is an element of K and φ(x, d) (d from A)is a principal formula consistent with Tk, then there is an extension C ⊇Acontaining a witness to φ(x, d) such that every finitely generated substructureof C is in K.So choose A and φ(x, d) as above. We may assume that cl(d) = d. Weshall produce an extension C ⊇A such that C ∖A is finite, C |= ∃xφ(x, d)and every finitely generated substructure of C is in K.Since φ(x, d) is consistent with Tk there is an element D of K such thatd embeds isomorphically into D and D |= ∃xφ(x, d).

Let {b0, . .

. , bl−1} enu-merate the elements of D∖d.

We must extend the definitions of {fn : n ∈ω}and {Rn : n ∈ω} given in A and D to (k+1)-tuples a from A∪{b0, . .

. , bl−1}so that every finitely generated substructure is in K.We perform this extension by induction on i < l.So fix i < l andassume that {fn, Rn : n ∈ω} have been extended to all (k + 1)-tuples fromA ∪{bj : j < i} so that every finitely generated substructure is in K. Leta = ⟨a0, .

. .

, ak⟩be a (k + 1)-tuple from A ∪{bj : j ≤i} containing bi withat least one element not in D. If bi = as for some s > 0, then let fn(a) = a0for all n ∈ω and let R0(a) hold.On the other hand, if bi ̸= as for all s > 0, then a0 = bi, so let a′ =⟨a1, . .

. , ak⟩, apply Lemma 0.7 to A and k, and letfn(b, a′) =n gn(a′)if gn(a′) ̸∈a′;botherwise.It is easy to verify that cl is locally finite on A ∪{bj : j ≤i} and that thereis no cl-independent subset of size k + 2.

It is also routine to extend thepartition given by the Rn’s so as to preserve ii) and iii) in the definition ofK.Thus, we have succeeded in showing (#), which completes the proof ofTheorem 0.6.References8

[1] P. Erd¨os, A. Hajnal, A. Mate, P. Rado, Combinatorial Set Theory, NorthHolland, Amsterdam, 1984.

[2] J. Knight, Prime and atomic models, Journal of Symbolic Logic 43(1978) 385-393. [3] D. W. Kueker, Uniform theorems in infinitary logic, Logic Collo-quium ’77, A. Macintyre, L. Pacholski, J. Paris (eds), North Holland,1978.

[4] D. W. Kueker and M. C. Laskowski, On generic structures, Notre DameJournal of Formal Logic 33 (1992) 175-183. [5] S. Shelah, Classification Theory, North Holland, Amsterdam, 1978.9

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