Bounds on Weak Scattering

Transcription

1 Notre Dame Journal of Formal Logic Volume 48, Number, 27 Bounds on Weak Scattering Gerald E. Sacks In Memory of Jon Barwise Abstract The notion of a weakly scattered theory T is defined. T need not be scattered. For each A a model of T, let sr(a) be the Scott rank of A. Assume sr(a) ω A for all A a model of T. Let σ 2 T be the least 2 admissible ordinal relative to T. If T admits effective k-splitting as defined in this paper, then θ < σ2 T such that sr(a) < θ for all A a model of T. Introduction This paper has two themes less disparate than they seem at first reading:. extending classical descriptive set theoretic results that impose bounds on suitably defined functions from ω ω into ω ; 2. extending and clarifying some early results on Scott ranks of countable structures sketched in [5]. Let F be a function, possibly partial, from ω ω into ω. A typical classical bounding theorem says the range of F is bounded by a countable ordinal if the graph of F has a suitable definition. For example, the graph of F is with real parameter p; in this formulation the graph of F is viewed as a subset of ω ω ω by requiring each value of F to be a well ordering of ω. Let F(X) ambiguously denote the well ordering and also the ordinal represented by the well ordering. For each X, F(X) is the unique solution of a formula with parameters p, X. Consequently F(X) (the well ordering) is hyperarithmetic in p, X, and so F(X) < ω p,x, (.) Received August 8, 25; accepted November 4, 25; printed February 26, 27 2 Mathematics Subect Classification: Primary, 3C7, 3D6 Keywords: weakly scattered theories, bounds on Scott rank c 27 University of Notre Dame 5

2 6 Gerald E. Sacks the least ordinal not recursive in p, X. The effective version of the theorem says that the bound on the range of F is an ordinal less than ω p. A recursion-theoretic approach to the effective bound originated by Kleene is as follows. (See Sacks [6] for details.) Suppose ( γ < ω p )( X)[F(X) > γ ]. (.2) Let R p e be the linear ordering of ω recursive in p via index e. Define W p to be the set of all e such that R p e is a well ordering. Then e W p X g[g is a - order-preserving map of R p e into F(X)]. (.3) But then W p is with parameter p. This last is false according to a Kleene hierarchy result that says W p is universal with parameter p, hence not with parameter p. A model theoretic approach to effective bounds is the path taken in this paper. A sketch may help to clarify later sections. Let A(p) be the least admissible set with p as a member. Let Z be a A(p) definable set of sentences of L ω,ω coded by elements of A(p) such that every model M of Z has the following properties:. The ordinals recursive in p form a proper initial segment of the ordinals in the sense of M. 2. There is an X M such that for all γ < ω p, F(X ) > γ. 3. p M and M is a admissible structure. Assume the range of F is not bounded by an ordinal below ω p. Then each A(p)- finite subset of Z (i.e., each subset of Z that is a member of A(p)) is consistent, and so Z has a model by Barwise Compactness. With the addition of effective type omitting, as in Grilliot [5] or Keisler [7], Z has a model M that omits ω p but has nonstandard ordinals greater than all standard ordinals less than ω p. Then ω p,x ω p ; (.4) otherwise, ω p is recursive in p, X and so ω p M. But then ω p,x = ω p and F(X ) ω p,x by property (2) of Z, which contradicts (.). The search for a bounding theorem that extends the classical result seems hopeless at first. An extension has to talk about an F that allows F(X) ω X,p, but ω X,p, as a function of X, is unbounded. Model theory comes to the rescue. Every countable structure A has a Scott rank [7], sr(a), an ordinal that can be as high as ω A + (see Section 2 for elaboration). Let T be a countable theory. A reasonable starting assumption on T is A[A T sr(a) ω A ]. (.5) An ingenious example (MA) devised by Makkai [] shows that (.5) is not enough. Examination of (MA) and its illuminative extensions in Knight and Young [8] leads to two further assumptions on T. The first, effective k-splitting, is technical and perhaps peripheral and is discussed further in Sections 9 and. The second, weakly scattered, is central. The theory T M associated with (MA) satisfies (.5) and has properties similar to effective k-splitting. In addition for every admissible countable α, T M has a model A such that ω A = α = sr(a). (.6)

3 Bounds on Weak Scattering 7 Corollary 9.5 says if T is weakly scattered, satisfies (.5), and has effective k- splitting, then there is a countable bound on the Scott ranks of the countable models of T ; the effective version provides a bound less than the first 2 admissible ordinal relative to T in contrast to the classical case above, where the effective bound on the range of F is less than ω p, the first admissible ordinal relative to p. The notion of weakly scattered is inspired by Morley s concept of scattered. Let L be a countable first-order language, L a countable fragment of L ω,ω, and T L a theory (i.e., a set of sentences) with a model. For (a) and (b) below, let L be any countable fragment of L ω,ω extending L, and T any finitarily consistent, ω-complete theory contained in L and extending T. (The notions of finitary consistency and ω-completeness for fragments are reviewed at the beginning of Section 4.) T is said to be scattered if and only if (a) and (b) hold. (a) For all n > and all T, S n T, the set of all n-types over T, is countable. (b) For all L, the set {T T L } is countable. The above definition of scattered is equivalent to the one in Morley s groundbreaking [3]. T is said to be weakly scattered if and only if (a) holds. By [3], a scattered theory can have at most ω many countable models. In contrast, a weakly scattered theory can have 2 ω many countable models. Knight [9] has announced a counterexample to Vaught s Conecture (VC), a scattered first-order theory with ω many countable models. VC has a precise formulation in Section 5. In [5] the following bounding result was established: if T is scattered and satisfies (.5), then T has only countably many countable models; furthermore, every countable model of T has a countable copy in L(β, T ) for some β < σ2 T, the least α such that L(α, T ) is 2 admissible. Hence Vaught s Conecture holds for T if T satisfies (.5). The proofs given in [5] were somewhat sketchy, so missing details needed in later sections of this paper are given in Sections 3 through 5. If Vaught s Conecture is false, then results for scattered theories yield information about models of counterexamples to VC. Theorem 4.9(vii) says if VC fails for T, then T has a model of cardinality ω not elementarily equivalent in the sense of L ω,ω to any countable model (Harnik and Makkai [6]). Theorem 5.3 describes an ω -sequence of atomic and saturated models that every counterexample must possess. Section 5 includes a related absoluteness result implicit in Morley [3]: VC(T ), Vaught s Conecture for T, is a L(ωL(T ),T ) predicate of T, hence, 2. Steel [8], as reported in [], used an assumption stronger than (.5) to prove VC(T ). In Section 2 an arbitrary countable structure A is associated with a theory T A contained in a countable fragment of L ω A ω,ω canonically generated from A. By an argument of Nadel [4], A is a homogeneous model of T A. Steel s assumption ω A is equivalent to for every A a model of T, T A is ω-categorical. Assumption (.5) ω A is equivalent to for every A a model of T, A is an atomic model of T A. Sacks and ω A Young [2] produced a structure A such that A is an atomic model of T A, but T A ω A ω A is not ω-categorical. (In addition, ω A = ωck and T A is a ω A subset of L(ω CK).) Sections 7 through 9 are devoted to bounding for weakly scattered theories.

4 8 Gerald E. Sacks 2 Scott Analysis and Rank This section revisits [5] as promised in Section. Scott [7] showed that an arbitrary countable structure A with underlying first-order language L can be characterized up to isomorphism by a single sentence of L ω,ω. In essence there is a countable fragment L A of L ω,ω such that A is the atomic model of T A, the complete theory of A in L A. Nadel [4] pointed the way to a canonical choice for L A. The admissible set L(ω A, A) is Gödel s L relativized to A as an element2 and chopped off at ω A, the least γ such that L(γ, A) is admissible. Let Nadel [4] showed that L A ω A,ω = L ω,ω L(ω A, A). (2.) A is a homogeneous model of its complete theory T A ω A,ω in LA ω A,ω. (2.2) It follows that A is the atomic model of its complete theory in L ω,ω L(ω A +, A), (2.3) since the types over T A ω A,ω realized in A are first-order definable over L(ωA, A) and so become atoms of the complete theory of A contained in (2.3). A recursion defines a canonical choice for L A and yields the definition of Scott rank for A.. L A = L. 2. L A λ = {LA δ δ < λ} for limit λ. 3. Tδ A = complete theory of A in L A δ. 4. L A δ+ = least fragment L+ of L ω,ω such that L + L A δ, and for each n >, if p( x ) is a nonprincipal n-type of Tδ A realized in A, then the conunction {F ( x ) F ( x ) p( x )} is a member of L +. Note that if A is isomorphic to B, then L A δ = L B δ and Tδ A = Tδ B for all δ. For some δ < ω, all the n-types of Tδ A realized in A are principal. To see this, fix γ and suppose some nonprincipal type p γ + of Tγ A + is realized in A. Let p γ be the restriction of p γ + to Tγ A. Since p γ + is nonprincipal, there is a formula G( x ) of such that both L A γ + x [p γ ( x ) G( x )] and x [p γ ( x ) G( x )] belong to T A γ +. Then there are n-tuples b and c of A such that A [p γ ( b ) G( b )], and A [p γ ( c ) G( c )]. Thus a distinction between b and c is made by a formula of L A γ + but not by any formula of L A γ. Since A is countable, only countably many distinctions can be made. Let d A be the the least δ < ω such that every distinction ever made is made by a formula of L A δ. Then A is the atomic model of T A d A +. (2.4)

5 The Scott Rank of A is defined by Bounds on Weak Scattering 9 sr(a) = least α [A is the atomic model of Tδ A ]. (2.5) If A is isomorphic to B, then sr(a) = sr(b). Nadel s proof of (2.2), p. 273 of [4], sketched below, also shows Hence d A ω A, and so sr(a) A is a homogeneous model of T A ω A. (2.6) ω A +. (2.7) Note that L A δ and Tδ A, as functions of δ < ωa, are L(ωA,A) ; that is, their graphs are definable subsets of L(ω A, A). Since the formulas of LA and T A are ω A ω A enumerated in increasing order of complexity, L A ω A and T A ω A are L(ωA,A). (2.8) To prove (2.6), let p ( x ) be an n-type, and q( x, y) an (n + )-type, of T A, and ω A a, b n-tuples of A. Suppose p( x ) q( x, y) and A [p( a ) p( b ) yq( a, y)]. (2.9) For homogeneity, a d A is required so that A q( b, d). Suppose no such d exists. Let q δ (x, y) be the restriction of q(x, y) to L A δ. Then {q δ (x, y) δ < ω A } is L(ωA,A). (2.) For each d A, there is a δ < ω A such that A q δ( b, d). Since δ can be defined as a L(ωA,A) function of d, the admissibility of L(ω A, A) implies there is a δ < ω A such that A y q δ ( b, y). But then A y q( a, y). (2.) A typical use of Scott rank in conunction with Barwise Compactness and Grilliot type omitting is as follows. Proposition 2. Suppose L(α, T ) is countable and admissible. If for each β < α, T has a model of Scott rank β, then T has a countable model such that sr(a) ω T,A = α. (2.2) Note that the A of (2.2) must have Scott rank either α or α + by (2.7). Forcing the outcome to be α + is a problem addressed in this paper but far from resolved. 3 Small ZF Sets The following is one of many variations (e.g., Makkai []) on a theme initiated by Barwise [2], an extension of a recursion theoretic fact needed for the enumeration of models of both scattered and weakly scattered theories. The variation below was mentioned and used in [5]. The recursion theoretic fact is as follows: if a set S of reals is and has cardinality less that 2ω, then there exists a hyperarithmetic real H such that every member of S is Turing reducible to H; in addition, an index for H can be computed uniformly from an index for S. The latter uniformity is

6 Gerald E. Sacks key to establishing the character of the enumeration of models in Sections 4 and 8. Recall that a ZF formula is a formula in the language of set theory with only bounded quantifiers ( xɛy) and ( uɛv). Let D(x, y) be a ZF lightface formula and A a countable admissible set. Suppose p, b A. Define S p,b = {x xɛv x b D(x, p)}. (3.) Theorem 3. If S p,b / A, then the cardinality of S p,b is 2 ω. Proof Let the language L consist of, bounded quantifiers x y and x y, an individual constant e for each e A, and a special individual constant c different from all the es. Let Z be the following A set of sentences of L.. The atomic diagram of A: d e d e; d / e d / e for d, e A. 2. c b, D(c, p), and c = e for all e A. Suppose Z is not consistent in the sense of L ω,ω. Then there is a z A such that z Z and z is not consistent. And z consists of some A A such that A is a subset of the atomic diagram of A, and c b, D(c, p), and {c = e e f } (3.2) for some f A. Since z is inconsistent, there is a deduction E A of [c b D(c, p)] c f (3.3) from A. But then S p,b f and so S p,b A. Suppose Z is consistent. Then a Henkin-style construction in ω many stages yields a model of Z, hence, an actual c (S p,b A). At stage, a sentence σ of L is considered, and σ is either σ or σ so long as Z {σ i i } is consistent. If σ is an infinite disunction (e.g., σ begins with x e ), then some component of σ is added immediately. The construction can be varied so 2 ω many cs are produced. Let t be a one-one map of ω onto {g g b}. After σ is chosen, and before σ + is chosen, create a split as follows. Choose an n so that (t(n) c) and (t(n) / c) are each consistent with Z {σ i i }. Then the construction takes 2 ω different paths, and different paths produce different cs. Such splits always exist. Otherwise there is a such that Z {σ i i } is consistent and for each n there is a deduction D n A from Z {σ i i } of either (t(n) c) or (t(n) / c). The admissibility of A puts all the D n s in some D A. D decides which elements of b belong to c. Hence there is an e A such that (c = e) is deducible from Z {σ i i }, a contradiction. Corollary 3.2 S p,b is countable S p,b A. Theorem 3.3 There exists a lightface ZF formula F (u, v, w) such that for any countable admissible set A and any p, b, s A, S p,b is countable A wf (p, b, w); (3.4) ( s A){[A F (p, b, s)] s = S p,b }. (3.5)

7 Bounds on Weak Scattering Proof The existence of F is implicit in the proof of Theorem 3.. Thus Z is inconsistent if and only if S p,b is countable if and only if S p,b A. The statement A F (p, b, s) (3.6) says (i) there exist A A and E such that A atomic diagram of A, and E is a deduction of (3.3) from A ; and (ii) s = {x x f x b D(x, p)}. (3.7) 4 Enumeration of Models for Scattered Theories Let L be a countable fragment of L ω,ω for some countable first-order language L and T L a theory with a model. Throughout this section T is scattered as defined in Section. For convenience assume T mentions all formulas of L ; thus L and L are recoverable from T. 4. Review of ω-completeness and finitary consistency for fragments Let L be a countable fragment of L ω,ω and T L a set of sentences. T is ω-complete in L if and only if () and (2) hold.. For every sentence F L, either F T or ( F ) T. 2. For any sentence ( i F i ) T, there is an i such that F i T. Say T is finitarily consistent if and only if no contradiction can be derived from T using only the finitary rules of L ω,ω. The infinitary step being avoided is deriving an infinite conunction by deriving each of its components. Say T is ω-consistent if and only if for any sentence ( i F i ) L, if T { i F i } is finitarily consistent, then there is an i such that T {F i } is finitarily consistent. Proposition 4. If T is finitarily consistent and ω-complete, then T has a model. Proof Note that T is ω-consistent. The model is constructed by extending T to a finitarily consistent and ω-complete set of sentences that includes Henkin axioms. At each stage of the construction, the set of sentences up to that point is ω-consistent. Proposition 4.2 Suppose for all β γ < λ, T β is finitarily consistent and ω- complete in the fragment L β, T β T γ, and L β L γ. Then {T β β < λ} is finitarily consistent and ω-complete in the fragment {L β β < λ}. Morley [3] showed that the scatteredness of T implies the countable models of T can be arranged in a hierarchy of height at most ω based on Scott rank with at most countably many models on each level. The current section revisits [5] and presents a enumeration of the countable models of T with a recursion-theoretic eye on some constructive details. The enumeration is a continuous tree TR(T ) with at most ω levels and at most countably many nodes on each level. Each node is a theory T finitarily consistent and ω-complete in a fragment L T with T T and L L T. Each T has an atomic model, and the class of all such models is the class of all countable models of T. The enumeration of TR(T ) is as follows.

8 2 Gerald E. Sacks Level Call T a node if and only if T is a finitarily consistent and ω-complete extension of T in the fragment L (= L T ). Level λ (limit) Call T a node if and only if there is a sequence T β (β < λ) such that T β is on level β, T β T γ (β < γ < λ), and T = {T β β < λ}. L T = {L Tβ β < λ}. Level δ + Suppose S is a node on level δ, that is, a finitarily consistent theory ω-complete in its fragment L S. If S is ω-categorical, then S has no successors on level δ +. Otherwise, S has a nonprincipal n-type p( x ). Let L S be the least fragment of L ω,ω extending L S and containing the conunction {F ( x ) F ( x ) p( x )} (4.) for every nonprincipal n-type p( x ) of S for all n >. Say T is a successor of S on level δ + if T is a finitarily consistent and ω-complete extension of S in the fragment L S (= L T ). Proposition 4.3 β. If β < ω, then TR(T ) has only countably many nodes on level Proof By induction on β. Level is countable by clause (b) of the definition of scattered. Suppose S is on level δ. Assume L S is countable. The set of all nonprincipal n-types of S is countable by clause (a) of the definition of scattered; hence, L S is countable. The set of all successors of S on level δ + is countable by clause (b) of the definition of scattered. Let T be any node on the countable limit level λ. Let L λ be the least fragment extending all the L S s for all theories S on all levels below λ. By induction, L λ is countable. Let T be any finitarily consistent and ω-complete extension of T in L λ. The set of all T s is countable, so the set of all T s is countable. Let TR(T ) β be the restriction of TR(T ) to the levels below β. Proposition 4.4 (i) If β < α < ω and L(α, T ) is admissible, then (TR(T ) β) L(α, T ). (ii) There exists a lightface ZF formula G(u, v, w) such that for all scattered T, all countable admissible L(α, T ), and all b L(α, T ), (TR(T ) β) = b L(α, T ) G(T, β, b). Proof By a L(α,T ) recursion that relies on Theorem 3.3. Suppose (TR(T ) (δ + )) L(α, T ), (4.2) and theory S is on level δ. The set of nonprincipal types of S is the unique s L(α, T ) that satisfies the F of Theorem 3.3 with p and b both equal to S. The statement q is a nonprincipal type of S is lightface ZF and corresponds to the formula D(x, y) of (3.). The fragment L S was defined ust before Equation (4.). The set of successors of S on level δ + is obtained from Theorem 3.3 with parameters p, b equal to S, L S.

9 Bounds on Weak Scattering 3 Let A be a countable model of T (a scattered theory as above). The Scott analysis of A differs little from its tree analysis: T (, A) = theory of A in L, and L T (,A) = L. T (λ, A) = {T (β, A) β < λ}. L T (λ,a) = {L T (β,a) β < λ}. L T (δ+,a) = L T (δ,a) (defined similarly to L S on level δ + of TR(T ) above). T (δ +, A) = theory of A in L T (δ+,a). Recall from Section 2 the definition of d A, the distinction rank of A, and the argument that the Scott rank of A is either d A or d A +. Clearly, there is a δ < ω such that for all n, any distinction made between n-tuples of A by a formula of L T (ω,a) is made by a formula of L T (δ,a). The tree rank of A is defined by Proposition 4.5 tr(a) = least δ [A is the atomic model of T (δ, A)]. (4.3) tr(a) sr(a). Proof Lδ A was defined ust after Equation 2.3. By induction on δ, LA δ L T (δ,a). Thus Tsr(A) A T (sr(a), A). But A is an atomic, hence homogeneous, model of, and so A is an atomic model of T (sr(a), A). T A sr(a) Proposition 4.6 Suppose A T and L(α, < T, A >) is admissible. Then tr(a) < α sr(a) < α. Proof Suppose not. Then D, the set of all distinctions between n-tuples (all n > ) of A made by formulas of L T (tr(a),a), belongs to L(α, T, A ) by Proposition 4.4. And there is an unbounded L(α, T,A ) map of D into α, a violation of the admissibility of L(α, T, A ). The map carries each distinction d D to the least δ such that d is made by some formula of L A δ. A theory T can be scattered up to a point. The tree TR(T ) is said to be scattered below β if the notion of scattered enumeration succeeds for T on all levels below β. To be more precise, TR(T ) has only countably many nodes (perhaps none) on each level below β. Proposition 4.7 Suppose α < ω, L(α, T ) is admissible, T is scattered below (α + ), and T has a model of Scott rank β for each β < α. Then there exists a theory T α on level α of TR(T ) such that T α is L(α,T ). Proof By Proposition 4.6, TR(T ) has nodes on all levels below α if an A can be found that satisfies the hypotheses of Proposition 4.6 and also sr(a) α. To find A through Barwise Compactness, consider the following set Z of sentences. (Z) Introduce a constant e to name each e L(α, T ). Add the atomic diagram (in the sense of L ω,ω) of L(α, T ) to Z. For each β < α, x[x β {x = γ γ < β}] (4.4) is a typical member of (Z). Any model of (Z) is an end extension of L(α, T ). (Z2) Introduce a new constant d and add sentences saying d is an ordinal greater than β for each β < α.

10 4 Gerald E. Sacks (Z3) Add A T and sr(a) > β for each β < α. (Z4) Add the axioms for admissibility. Let M be a model of Z that omits α but extends L(α, T ) as in [5] or [7]. L(α, T, A ) is admissible; otherwise α M. (Z3) insures sr(a) α. Let T denote an arbitrary node below level α. Call T unbounded if T has extensions to theories on arbitrarily high levels below α. Then T can be regarded as an unbounded node. Suppose T is an unbounded node below level β for some β < α; then T has an unbounded extension on level β. Otherwise, the admissibility of L(α, T ) implies T is bounded. There exists a β < α and an unbounded node T β on level β such that for all β (β, α), T β has a unique unbounded extension on level β. Otherwise, a tree U of unbounded nodes can be constructed such that U is isomorphic to the binary branching tree 2 <ω, and the branches of U define a continuum of nodes on some level α α of TR(T ) (α + ). The set S ub of unbounded nodes above T β form an expanding sequence whose union is the desired T α. To see S ub is L(α,T ), let N γ be the set of all nodes on level γ extending T β for each γ (β, α). Then N γ, as a function of γ, is L(α,T ) by Proposition 4.4, and (N γ S ub ) L(α, T ) since N γ S ub has ust one element. There is a L(α,T ) function that takes each node e (N γ S ub ) to a bound on the levels occupied by extensions of e. But then there is a strict upper bound b < α on the levels occupied by extensions of members of (N γ S ub ). Any such b singles out the unique member of N γ S ub. Proposition 4.8 Suppose α ω, L(α, T ) is 2 admissible, T is scattered below α, and T has models of arbitrarily high Scott rank less than α. Then there exists a theory T α on level α of TR(T ) such that T α is L(α,T ). Proof The proof is similar to that of Proposition 4.7. The only difference is in the handling of U. Then and now U can be defined by a L(α,T ) 2 recursion of length ω, since the set of unbounded nodes is L(α,T ). But now the 2 admissibility of L(α, T ) implies U L(α, T ), and so the branches of U define a continuum of nodes on some level α < α of TR(T ). Two L-structures are said to be L ω,ω-equivalent if they satisfy the same sentences of L ω,ω. (Recall that if A is countable and L ω,ω-equivalent to B, then A is L,ω - equivalent to B.) Theorem 4.9 Suppose Vaught s Conecture fails for T. Then there exist T β, A β, and L β (β ω ) such that (i) if β < ω, then T β is an ω-complete theory in the countable fragment L β ; (ii) if β γ ω, then T β T γ, A β A γ, and L β L γ ; (iii) if λ(limit) ω, then T λ = {T β β < λ} and A λ = {A β β < λ}; (iv) T ω is L(ω,T ) definable; (v) if β ω, then A β is an atomic model of T β ; (vi) if β < ω, then A β+ realizes a nonprincipal type of T β ; (vii) (Harnik and Makkai [6]) The cardinality of A ω is ω, and A ω is not L ω,ωequivalent to any countable model. 3

11 Bounds on Weak Scattering 5 Proof A uncountable model A ω of T is constructed so that it is not L ω,ωequivalent to any countable model. By Proposition 4.8, there is a theory T ω on level ω of TR(ω ) such that T ω is L(ω,T ). Thus T ω = {T γ γ < ω }, and (γ δ) (T γ T δ ). p, the parameter used in the L(α,T ) definition of T ω, belongs to L(α, T ) for some α < ω. Define K = {β α < β < ω L(β, T ) L(ω, T )}. (Recall that X Y means X is a ZF substructure of Y.) Let {γ δ δ < ω } be an increasing enumeration of K. Then L(γ δ, T ) is admissible, and so T γδ = T ω L(γ δ, T ) by Proposition 4.4(i). Also T γδ is L(γ δ,t ) definable via the same definition that works for T ω, since p L(γ δ, T ) L(ω, T ). Structures A δ (δ ω ) and inclusion maps i β,δ : A β A δ (β < δ) are defined by recursion on δ. The map i β,δ will be elementary with respect to the language L γβ ; that is, any sentence of L γβ with parameters in A β and true in A β will also be true in A δ. Stage Structure A is the countable atomic model of T γ. Stage δ + Assume A δ is the countable atomic model of T γδ. Extend A δ to A δ+, the countable atomic model of T γδ+, so that the inclusion map i δ,δ+ is L γδ - elementary. Stage λ (limit ω ) Let A λ = {A δ δ < λ}. For all δ < δ < λ, assume the inclusion map i δ,δ is L γδ -elementary. Then for each δ < λ, A λ is an L γδ -elementary extension of A δ and so is a model of T γδ. Thus A λ is a model of T γ λ. To see that A λ is an atomic model of T γ λ, let a be an n-tuple of A λ. For some δ < λ, a is an n-tuple of A δ and realizes some atom F ( x ) of T γδ. Then F ( x ) is an atom of T λ, because L(γ δ, T ) L(λ, T ). Hence, a realizes F ( x ) in A λ, since i δ,λ is L δ -elementary. If A ω were L ω,ω-equivalent to some countable model, then it would be an atomic model of T γδ for some δ < ω. But A δ+, hence A ω, realizes a nonprincipal type of T γδ. 5 Absoluteness of Vaught s Conecture Let VC(T ) be the predicate Vaught s conecture holds for T. Morley s work [3] implies that VC(T ) is absolute. The enumeration tree, TR(T ), of Section 4 is applied below to make the statement of VC(T ) more precise and to see in some detail how T can satisfy Vaught s Conecture. Suppose an attempt is made to develop TR(T ) and the attempt fails to produce a tree with only countably many nodes on each level and ω many nonempty levels. Then there must be a countable β such that one of the following holds: () β = and T has uncountably many finitarily consistent, ω-complete extensions in L ;

12 6 Gerald E. Sacks (2) β = δ +, some theory S is on level δ, and for some n, the set of n-types of S is uncountable; (3) β = δ +, some theory S is on level δ, for all n the set of n-types of S is countable, and the set of all finitarily consistent, ω-complete extensions of S in L S is uncountable. L S is defined ust before (4.). (4) β = λ and the set of nodes on level λ is uncountable. (5) level β is empty. Define the Vaught Rank of T, vr(t ) to be the least countable β that satisfies one of () (5) above. (If there is no such β, let vr(t ) be ω.) Define the predicate VC(T ) by vr(t ) < ω. Suppose vr(t ) = β < ω. If β =, then T has 2 ω finitarily consistent, ω-complete extensions in L by Theorem 3., hence, 2 ω many countable models. The same holds in cases (3) and (4). If (5) holds, then T has only countably many countable models, and each one is the atomic model of a theory on some level of TR(T ) below level β. Suppose case (2) holds. Then for some n, there are 2 ω n- types of S by Theorem 3., hence, 2 ω many countable models of T. Recall that Proposition 5. ω L(T ) = least γ [L(T ) (γ is uncountable )]. (5.) The predicate, Vaught s Conecture holds for T, is L(ωL(T ),T ), hence 2. Proof By Proposition 4.4, TR(T ) L(ω, T ) and is L(ω,T ). The statement VC(T ) says at some level γ < ω, either (a) TR(T ) ends or (b) blows up; that is, a perfect kernel of theories or types is manifest. Let α be the least α > γ such that L(α, T ) is admissible. Suppose (a) holds. Then Levy-Shoenfield Absoluteness implies α < ω L(T ), and there is an L ω,ω sentence K L(α, T ) that expresses the fact that every model of T is an atomic model of some theory on some level at or below γ of TR(T ). Suppose (b) holds. Theorem 3. implies the existence of a perfect kernel of theories or types. A coding of some such perfect kernel by a real is constructible from any counting of α. The proof of Theorem 3. relies on the consistency of a certain set Z of axioms. Z is L(α,T ), and the consistency of Z is L(α,T ). Hence Levy- Shoenfield Absoluteness implies α < ω L(T ), and so a code for the perfect kernel belongs to L(ω L(T ), T ). Proposition 5.2 Suppose T is a counterexample to Vaught s Conecture. Then there is a theory T ω on level ω of TR(T ) such that T ω is L(ω,T ). For all countable β, T β, the restriction of T ω to level β, has an atomic model whose Scott rank is β. Proof By Proposition 4.8. Suppose L(α, T ) is admissible, A is a countable model of T, and ω A = α. According to (2.6), A is a homogenous model of Tα A ; A is said to be α-saturated if every n-type (n ) of Tα A is realized in A. Theorem 5.3 Suppose T is a counterexample to Vaught s Conecture. Then there is a L(ω,T ) theory T ω on level ω of TR(T ) and a closed unbounded set C ω such that α C: T α, the restriction of T ω to level α, has an atomic model A α of

13 Bounds on Weak Scattering 7 Scott rank α and an α-saturated model B α of Scott rank α +. The atomic models form an expanding chain and each inclusion A β A γ (β < γ ) is elementary with respect to the language of T β. Proof Proposition 4.8 provides T ω. Let p L(ω, T ) be the parameter needed for the L(ω,T ) definition of T ω. For any α, let α + be the least β > α such that L(β, T ) is admissible. For x L(T ), let H (x) be the hull of x in L(T ). Recall that x H (x) L(T ) and that x and H (x) have the same cardinality in L(T ). An expanding sequence of countable hulls, H δ (δ < ω ), is defined by recursion on δ. H is H ({tc(p), ω, tc(t )}). (tc is transitive closure.) Note that ω +, ω H ; if d < e < ω and e H, then d H. Let c be the lub of the countable ordinals in H. Let L(β, T ) be the transitive collapse of H. Then c = ω L(β,T ) and L(c +, T ) L(β, T ). (5.2) Stage δ + Assume H δ is countable in V. Then H δ ω is a proper initial segment of ω. Let c δ be the least countable ordinal not in H δ. H δ+ is H (H δ {c δ }). Stage λ (limit) Let H λ be {H δ δ < λ}. Then C = {c δ δ < ω } is a closed unbounded set. Let L(β δ, T ) be the transitive collapse of H δ. Then c δ = ω L(β δ,t ) and L(c + δ, T ) L(β δ, T ). (5.3) Let T cδ be the restriction of T ω to level c δ of TR(T ). Then T cδ is L(c δ,t ) via parameter p, and N, the set of nonprincipal types of T cδ, is nonempty and countable in V. Then T cδ L(c δ +, T ), and so N L(c+ δ, T ) by Theorem 3.. Hence the structure L[c δ, T ; T cδ, N] (i.e., L(c δ, T ) with x T cδ and x N as additional atomic predicates) is admissible because no subset of c δ in L(β δ, T ) can define a counting of ω L(β δ,t ). Now the construction of M in the proof of Theorem 6. can be imitated to produce a model B of T cδ such that B realizes all the types in N and ω B = c δ. The atomic A β s are supplied by Theorem Bounds on Scattered Theories Once again L is a countable first-order language, L is a countable fragment of L ω,ω, and T L has a model. L and L are effectively recoverable from T. T is scattered below β as was defined ust before Proposition 4.7. Theorem 6. Suppose α < ω, L(α, T ) is 2 admissible, T is scattered below α, and for each β < α, T has a model of Scott rank β. Then T has a model A such that ω A = α and sr(a) = α +. Proof By Proposition 4.8, TR(A) has a theory T α on level α such that T α is α and T α is {T β β < α}, where T β is a node on level β. Let Z be the following set of sentences. (Z) (Z2) The atomic diagram of L(α, T ) in the sense of L ω,ω. Add (d > β) for all β < α. d is a constant not occurring in (Z).

14 8 Gerald E. Sacks (Z3) Let T d be a theory on level d of TR(T ). Add A is the countable atomic model of T d and F T d for each sentence F T α. (Z4) Add (b( x ) is an atom of T d ) for each b( x ) that is an atom of T α ; that is, b( x ) generates a principal type of T α. (Z5) Add the axioms of admissibility. The set Z is L(α,T ) 2, since the set of atoms of T α is L(α,T ). Suppose β < α, L(β, T ) is admissible, and Z β is Z L(β, T ). A fundamental fact of forcing in the setting of set theory is the Levy collapse of a cardinal to V preserves replacement; furthermore, the preservation of n replacement needs only n replacement in V. To check the consistency of Z β, augment L(α, T ) by adding a generic counting of L(β, T ) to L(α, T ) that preserves the 2 admissibility of L(α, T ). The set Z β can be modeled by the augmented L(α, T ). By Proposition 4.4, T β L(β, T ). Interpret d as β. Interpret A as the atomic model of T β. Such an A belongs to the augmented L(α, T ) because there T β is countable. If b( x ) is an atom of T α and belongs to L(β, T ), then b( x ) is an atom of T β. The set Z has a model M that is a proper end extension of L(α, T ) but omits α. ω A α; otherwise, α is recursive in A, and then α M. By design A T β for all β < α; hence, sr(a) α by Proposition 4.5, and so ω A = α by (2.6). Suppose sr(a) = α. Then α M as follows. By supposition A is the atomic model of T α. The rank of an atom b( x ) of T α is the least β < α such that b( x ) is an atom of T β. Let f be the function that carries each a A to the rank of an atom of T α that generates the principal type realized by a in A. Thanks to (Z4) f is definable from T d, and so f M. Then lub(range f ) = α M. Corollary 6.2 ([5]) Suppose for every countable model A of T, the Scott rank of A is less than or equal to ω A. Then Vaught s Conecture holds for T. Proof Suppose VC(T ) fails. Then T is scattered below ω, and TR(T ) has nodes on every countable level. Choose an α < ω such that L(α, T ) is 2 admissible. Then T has a countable model A such that ω A = α and sr(a) = α +. A more effective version of Corollary 6.2 is as follows. Define σ T 2 = least α [L(α, T ) is 2 admissible]. (6.) vr(t ), the Vaught rank of T, was defined at the beginning of Section 6. Corollary 6.3 Then vr(t ) < σ T 2. Suppose T does not have a countable model A such that ω A = σ T 2 and sr(a) = σ T 2 +. (6.2) Proof If vr(t ) σ2 T, then T is scattered below σ 2 T level below σ2 T. and TR(T ) has nodes on every As a warmup to the main bounding results of the paper (Section 8), the above is recast as an effective bounding theorem. Corollary 6.4 Suppose T is scattered and sr(a) ω A for every countable A T. (6.3)

15 Then β < σ T 2 such that sr(a) < β for every A T. (6.4) Bounds on Weak Scattering 9 Let SA(T ) say for every countable model A of T, the theory T A is ω-categorical. ω A Steel [8], as reported in Makkai [], showed that VC(T ) follows from SA(T ). Theorem 6.5 is an effective version of Steel s result. The set L(α, T ) is said to be recursively Mahlo if L(α, T ) is admissible and every L(α,T ) closed unbounded subset of α has a member β such that L(β, T ) is admissible. Define Note that rm(t ) < σ T 2. Theorem 6.5 rm(t ) = least γ [L(γ, T ) is recursively Mahlo]. (6.5) Suppose T is scattered and T A ω A Then β < rm(t ) such that is ω-categorical for every countable A T. (6.6) sr(a) < β for every countable A T. (6.7) Proof Suppose there is no such β. Let α be rm(t ). Then Proposition 4.7 supplies a L(α,T ) theory T α on level α of TR(T ). Then T α = {T β β < α}, and T β, as a function of β, is L(α,T ). There is a L(α,T ) function f such that T β L( f (β), T ) for all β < α. Iteration of f leads to a L(α,T ) closed unbounded set C = {γ T γ L(γ, T )}. (6.8) A similar argument produces a L(α,T ) closed unbounded set C such that [ γ C (Tα L(γ, T )) is L(γ,T ) ]. (6.9) Then there is a L(α,T ) closed unbounded set K such that γ K [ T γ L(γ, T ) and T γ is L(γ,T ) ]. (6.) Hence, for some γ K, L(γ, T ) is admissible. Consequently T γ has a model B such that ω B = γ. But then T B, hence T ω B γ, is ω-categorical and so has no extension to a node on level α. 7 Iterated Classical Bounding In this section classical bounding (reviewed in Section ) is translated into the language of admissible sets and revised to allow for iterated use in recursive definitions in Section 8. Let B(x) be a ZF formula with parameter p. The formula B(x) is β-bounded if and only if c[b(c) L[β, p ; c] B(c)]. (7.) The set L[β, p ; c] is the result of iterating first-order definability with y c as an additional atomic predicate through the ordinals less than β starting with the transitive closure (tc) of {p }. Assume B(x) is β-bounded. Define c β = c L[β, p ; c]. (7.2)

16 2 Gerald E. Sacks Then B(c) B(c β ). For all z let A z be the least admissible set with z as a member; thus A z = L(ω z, tc({z})). (7.3) Let F (u, v) be a ZF formula with parameter p, and let p be {p, p }. Suppose for all c, if B(c), then there exists a unique δ A {p,β,cβ } such that designate δ by δ p,β,c. Theorem 7. (i) There exists a δ p,β A {p,β} such that for all c, A {p,β,cβ } F (c β, δ); (7.4) B(c) δ p,β,c δ p,β. (7.5) (ii) δ p,β can be construed as a partial function of p and β whose restriction to any admissible A has a A definition uniformly in A; that is, one formula works for all A. Proof Let Z be the following A {p,β} set of sentences. Let α = ω {p,β}. (Z) Introduce constants c and c β, and put c β = c L[β, p ; c] and B(c β ) in Z. (Z2) Add constants that name the elements of L(α, tc({p, β, c β })) (7.6) and sentences of L ω,ω that define each element in terms of elements of lower definability rank. (Z3) Let F (u, v) be wg(u, v, w) for some ZF formula G(u, v, w). Add G(c β, δ, r) for all δ < α and every r that names an element of (7.6). (Z4) Add axioms for admissibility. Suppose Z is consistent. Assume for a moment that Z is countable. (7.7) As in the proof of Proposition 4.7, Z has a model M that is a proper end extension of (7.6) but omits α. Then (7.6) is admissible, and so A {p,β,cβ } = L(α, tc({p, β, c β })). (7.8) But then A {p,β,cβ } F (c β, δ) for all δ < α, a contradiction since δ p,β,cβ A {p,β,cβ }. Thus Z is inconsistent. To remove assumption (7.7), generically extend the universe V to V so that Z is countable in V. Then Z is inconsistent in V, hence in V, by the absoluteness of provability in the sense of L,ω. Since Z is A {p,β}, there must be an inconsistent W Z such that W A {p,β}. The set W consists of (W ), (W 2), and (W 3): (W) (W2) (Z) and (Z4) above. Some A A {p,β} such that A set of sentences of (Z2).

17 Bounds on Weak Scattering 2 (W3) For some δ < α, the sentence G(c β, δ, r) for all δ < δ and every r of (Z2) that names an element of L(δ, tc({p, β, c β })). Then there is a deduction D A {p,β} from (W ) and (W 2) of {F (c β, δ) δ < δ }. (7.9) Let ρ be the least ρ such that there is such a D L(ρ, tc({p, β})); let δ {p,β} be the least δ associated with any such D L(ρ, tc({p, β})). Then δ p,β,c δ p,β (7.) for any c such that B(c) holds. The ZF formula H that defines δ p,β as a partial function of p, β uniformly owes its existence to the effective nature of deducibility in L ω,ω. The formula H singles out a deduction in A {p,β} that establishes the value of δ p,β and can be formulated to succeed in every admissible A, because p, β A implies A {p,β} is a A definable (uniformly) subclass of A. 8 Enumeration of Models under Weak Scattering Let L be a countable fragment of L ω,ω for some countable first-order language L and T L a theory with a model. Assume T is weakly scattered as defined in Section. For convenience assume T mentions all formulas of L ; thus L and L are recoverable from T. Since T need not be scattered, there is no hope of enumerating theories in L(ω, T ) whose atomic models are exactly the countable models of T. But some useful vestiges of the constructive features of scattering carry over to weak scattering, and L(ω, T ) manages to say a great deal about the countable models of T. First consider RH(T ), the raw hierarchy for the countable models of T. On level of RH(T ), put every T such that T T and T is a finitarily consistent, ω-complete theory of L. (If needed, see the beginning of Section 4 for a review.) Suppose T δ is on level δ of RH(T ). Define δ if δ is a successor, δ = (8.) δ if δ is not a successor. Let L (T ) be L. Assume T δ extends a unique T δ on level δ and L δ (T δ ) is countable. If all n-types (n ) of T δ are principal, then L δ+ (T δ ) is undefined and T δ has no extensions on level δ +. Otherwise, let L δ+ (T δ ) be the least fragment of L ω,ω extending L δ (T δ ) and having as a member the conunction {F ( x ) F ( x ) p( x )} (8.2) for every nonprincipal n-type p( x ) of T δ (n ). Since T is weakly scattered, the set L δ+ (T δ ) is countable. On level δ + of RH(T ) put every T δ+ that extends T δ and is a finitarily consistent, ω-complete theory of L δ+ (T δ ). Put T λ on level λ if there is a sequence T δ (δ < λ) such that (a) T δ is on level δ; (b) T β T γ if β γ ; and (c) T λ = {T δ δ < λ}.

18 22 Gerald E. Sacks Define L λ (T λ ) to be {L δ (T δ ) δ < λ}. It is straightforward to verify that A is a countable model of T if and only if A is the atomic model of T δ for some countable δ. Define the raw tree rank of A by rtr(a) = (least δ)[a is the atomic model of some T δ ]. (8.3) Propositions 4.5 and 4.6 hold when tr is rtr. Thus and if L(α, T, A ) is admissible, then rtr(a) sr(a), (8.4) rtr(a) < α sr(a) < α. (8.5) What matters more is what can be expressed inside L(α, T ) when α ω and L(α, T ) is admissible. Let A δ be the set of all T δ s on level δ of RH(T ). The set A δ will be defined by a β-bounded ZF formula (7.), and its definition as such, denoted by A δ, will belong to L(α, T ) when δ < α. The fragment L δ (T δ ) will be constructible from T δ via an ordinal ρ δ < α for all T δ A δ. The pair A δ and ρ δ will be defined by a simultaneous L(α,T ) recursion uniformly in α, that is, the same formula will work for all α ω such that L(α, T ) is admissible. Consider an arbitrary T δ on level δ of RH(T ). There exists a natural recovery process that can be applied to T δ to recover the unique sequence T γ (γ < δ) such that T γ is on level γ, γ γ 2 T γ T γ2, and (8.6) T λ = {T γ γ < λ} for all limit λ δ. The recovery proceeds as follows. It begins with T is T δ L. If γ is a successor, then T γ = T δ L γ (T γ ). (8.7) If γ is a limit, then T γ = {T β β < λ}. The recovery process can be used to decide whether or not an arbitrary set c is a theory on level δ of RH(T ). The answer is yes if and only if c passes the following tests at all levels γ δ. Level Let c be c L ; c is an extension of T and a finitarily consistent, ω-complete theory of L. Level γ + δ Let L γ + (c γ ) be the least fragment extending L γ (c γ ) and having as a member the conunction {F ( x ) F ( x ) p( x )} (8.8) for every nonprincipal n-type p( x ) of c γ. Let c γ + be c L γ + (c γ ). c γ + extends c γ and is a finitarily consistent, ω-complete theory of L γ + (c γ ). Level λ (limit) δ Let c λ be {c γ γ < λ}; let L λ (c λ ) be {L γ (c γ ) γ < λ}. In short, c is a theory on level δ of RH(T ) if and only if c satisfies the recovery process on all levels γ δ and c = c δ. It will follow below that A δ is β-bounded ZF definable (7.), where β is large enough to define the recovery process. An effective version of the recovery process is woven into the L(α,T ) recursive definitions of ρ δ and A δ for < δ < α. The set L δ (T δ ) is constructible from

19 Bounds on Weak Scattering 23 T δ via the ordinal ρ δ for all T δ A δ, and A δ is a β-bounded ZF definition of A δ. The definition A δ specifies the value of β and the ZF formula. Stage L (T ) is L ; A is the set of all finitarily consistent, ω-complete theories of L extending T. Since L is recoverable from T, the set A is β-bounded ZF definable with β = and parameter T. Stage δ + Assume the recursion has produced sequences {ρ γ γ δ}, { A γ γ δ} L(α, T ) (8.9) such that A γ is a β-bounded ZF definition of A γ, and L γ (T γ )(γ δ) is firstorder definable over L[ρ γ, L ; T γ ]. (8.) (The definition of (8.) follows (7.).) Consider an arbitrary T δ A δ (δ > ). Use the recovery process to construct the unique T δ A δ such that T δ T δ L δ (T δ ). (8.) The recovery is effective thanks to the sequence ρ γ (γ δ). Now L δ+ (T δ ) can be defined as above (8.2) but with an effective twist. Let ST δ be the set of all n-types (n ) of T δ. Since T is weakly scattered, Corollary 3.2 implies the least admissible set with T δ as a member. Let ST δ L(ω T δ, T δ), (8.2) γ Tδ = (least γ)[st δ L(γ, T δ )]. (8.3) By Theorem 3.3, the ordinal γ Tδ, as a function of T δ, is uniformly ; the same ZF formula singles out γ Tδ in L(ω T δ, T δ) for every T δ A δ and for all δ. By Theorem 7.(i), there is a γ δ such that ( T δ A δ )[γ Tδ γ δ < α]. (8.4) Hence ST δ L(γ δ, T δ ) for all T δ A δ. Theorem 7.(ii) implies that γ δ, as a function of δ, has a uniform definition utilizing the parameters occurring in A δ and the uniform definition of γ Tδ. Any n-type p( x ) ST δ for any T δ A δ is constructible from T δ via some ordinal less than γ δ. A set P δ of first-order definitions can be assembled at level γ δ of L(α, T ) as follows. Let {p T δ J δ } (8.5) be the set of all first-order definitions over L(γ, T ) for all γ < γ δ with parameter T δ. For each T δ A δ, the obect p (T δ ) is the set defined by p (T δ ) when the parameter T δ is assigned the value T δ. The set (8.5) has a natural well-ordering W δ definable at level γ δ, since each p T δ is specified by its level γ < γ δ and its Gödel number e < ω as a formula of ZF. The type d δ (T δ ), the default type for T δ, is defined by its action on T δ A δ : (T δ ) = (least in sense of W δ )[p (T δ ) is an n-type of T δ ]; (8.6) d δ (T δ ) = p (Tδ )(T δ ). (8.7)

20 24 Gerald E. Sacks The formula p T δ is a slight variant of p (T δ ) and is defined by its action on T δ A δ. p (T δ ) if p (T δ ) is an n-type of T δ, p T δ = (8.8) d δ (T δ ) the default type, otherwise. Let P δ = {p T δ J δ }. Then. for all T δ A δ and p( x ) ST δ, there is a J δ such that p T δ p( x ) at level γ δ of L(α, T ), and 2. p T δ ST δ for all T δ A δ and all J δ. defines It can happen for some T δ A δ and, k J δ that = k but p T δ = p T δ k. Such repetitions are the price paid to have P δ L(γ δ +, T ). The ordinal ρ δ+ < α is chosen ust large enough to develop the sequence ρ γ (γ δ) needed for the recovery of T δ from T δ (δ > ) and the ordinal γ δ needed to assemble P δ. The set L δ+ (T δ ) is first-order definable over L[ρ δ+, L ; T δ ]; its definition begins with L δ (T δ ), adds the conunction of all formulas in p T δ for each p T δ P δ, and closes under the finitary operations that generate a fragment of L ω,ω. To complete stage δ +, construe A δ+ to be the set of all x such that the effective version of the recovery process applied to x reports that x is a theory on level δ + of RH(T ). The effective version uses the sequence ρ γ ( < γ δ + ) to define L γ (T γ ) from T γ for all T γ A γ. Thus A δ+ is β-bounded ZF definable with β equal to ρ δ+, and A δ+ L(α, T ). The parameter specified by A δ+ is T. Stage λ (limit) Assume for < γ < λ that L γ (T γ ) is constructible from T γ via ρ γ for all T γ A γ. Use the effective version of the recovery process to define A λ as a β-bounded ZF class. For T γ A λ, effectively recover the unique sequence T γ (γ < λ) such that T λ is {T γ γ < λ}, and then define L λ (T λ ) to be {L γ (T γ ) < γ < λ}. Makkai [] showed that if T is a counterexample to Vaught s Conecture, then T has a model of cardinality ω that is L,ω equivalent to a countable model. The following are variants of his results. Suppose A is a countable admissible set and T A. Assume T L, L is a countable fragment of L ω,ω, and L is a countable first-order language. Also assume every symbol of L is mentioned in T so that L is recoverable from T. Let L denote an arbitrary fragment of L ω,ω that extends L, and T an arbitrary finitarily consistent, ω-complete theory contained in L and extending T. Call T weakly scattered in A if and only if ST A for all T A. According to Theorem 3.3, we have the following. Theorem 8. Suppose A is a countable model of T. Assume T is weakly scattered in L(ω T,A, T, A ), and sr(a) ω T,A. Then A is L,ω equivalent to a model of T of cardinality ω. Proof Let α =ω T,A. Thus ω A = α, since ωa + sr(a). Let T β A (β sr(a)) be the Scott analysis of A as defined in Section 2. By Theorem 3.3, STβ A L(α, T, A )

21 Bounds on Weak Scattering 25 (and so Tβ A has a countable atomic model) for all β such that β + < sr(a). The set Z is L(α, T,A ) and consists of the following sentences: (Z) the atomic diagram (in the sense of L ω,ω) of L(α, T, A ); (Z2) d is a countable ordinal and d δ (all δ < ω T,A ); (Z3) y[y < d T A y has a countable atomic model]; (Z4) axioms of admissibility. The set Z is consistent since it can be modeled by V (the real world). Every model of Z is an end extension of L(α, T, A ). Let M be a model of Z that omits α. Thus M has nonstandard ordinals greater than every ordinal less than α. Hence sr(a) α in V and α / M, so sr(a) γ for some nonstandard γ M. Now work inside M. Let Tδ A (δ γ ) be the Scott analysis of A up to level γ. Choose a nonstandard β < γ. Then Tβ A has a countable atomic model A β. There is a map i βγ : A β A (8.9) that is elementary with respect to all formulas of L A β (defined in Section 2). Note that i bγ is not onto, since A β is not isomorphic to A in M. But A β is isomorphic to A in V. Now ω A β α since α / M; also sr(a β ) δ for all δ < α. Hence sr(a β ) α, and so ω A β α. Thus both A β and A are homogeneous models of Tα A by (2.6). To see they realize the same types of T α A, choose p α STα A and first suppose A β p α (b). In M, note that A β p β (b) for some type p β of Tβ A and that A p γ (i βγ (b)) for some type p γ of Tγ A. Then p α p β p γ, (8.2) since i βγ is L A β elementary. Hence A p α(i βγ (b)). It follows that i βγ is L ω,ω elementary, (8.2) since the types of Tα A realized in A β are atoms of L ω,ω. Now suppose A p α (a). In M, the tuple a realizes p γ in A, a type of Tγ A. Choose a nonstandard δ < β. Let p β be the restriction of p γ to Lβ A, and let p δ be the restriction to L A δ. Then p α p δ p β p γ. So A x p δ (x). (8.22) But then x p δ (x) T δ+ T β, so p δ, hence p α, is realized in A β. Thanks to the above there exist structures B and B, both isomorphic to A, such that B B and the inclusion map i is L ω,ω elementary. A strictly expanding L ω,ω elementary chain B δ (δ ω ) is defined by iterating i. For δ < ω, assume B δ is isomorphic to A. Then enlarge B δ to B δ+, another copy of A. For limit λ ω, let B λ be the union of the B δ s (δ < λ). B ω is an L ω,ω elementary extension of B, hence L ω,ω-equivalent to A, consequently L,ω -equivalent to A. Corollary 8.2 Suppose T is weakly scattered. For each β < ω T, assume T has a model of Scott rank β. Then T has a countable model A such that sr(a) ω T,A = ω T,