Annals of Pure and Applied Logic 42 (1989) 1-19 North-Holland

ark H Purdue Uniuersiry, West Lafayeae, IN 47907, USA

Communicated by A. Nerode Received 8 January 198’7

A tree is said to have unique limits if 110 two distinct nodes on the same limit level have the same predecessors. If a is a regular cardinal, then a tree of ight K is narrow if it is <K branching, has unique linnits, and does not embed 2’“. A set of nodes is stationary if the set of heights is stationary.

the folIowing extension of Fodor’s theorem: If T is narrow and S is a stationary set of nodes, then S has a stationary subset

along one branch. If 91Tz is a countable model, then say that !DZ is isolated at Q if the a-theory of 911 has only

countably many countable models. For # E Lala, let Mod* be the colltxtion of isomorphism types of countable models of #.

If #J is a counterexample to Vaught’s conjecture and for each .lDI E Mod+, Am is a thin set of &mtable ordinals at which sit is isolated, then beM

“;R Aw is thin.

We use this theorem to prove several new consequences of e existence of a counter- example to Vaught’s conjecture, md to prove extensions of some of the known consequences. For example we show:

eorem. If 4~ is a counterexample, then there is some (Y < o1 such that every nontrivial a-theory extending Q is a counterexample. m. If + is a counterexample, then there is an !SRE Mod, such that for every

!R E Mod, if o(%) = o(SR), then the o(S)-theory of ‘3 is a counterexample.

In 1960, after proving that a countable theory cannot have exactly two countable models up to isomorphism, Vaught innocently asked ‘phether the number of countable models of a countable theory can be strictly between & and 2% This conjecture has received a great deal of attention and has emerged as one of the most notorious open questions in

njecture has been shown ld for many types of theories (see arrington-Makkai-Shelah [lo]), but all of these arguments depend

quite heavily on the special properties of these classes of models. Vaught’s conjecture has also generated a considerable amount of activity in Descriptive Set Theory, but there are very few general model-theoretic results about the question.

*This paper is a revision of part of the author’s Ph thesis written at WC Berkeley under the

direction of Jack Silver. I would like to thank him for his advice and encouragement.

3.50 @ 1989, Elsevier Science Publishers

be the collection of isomorphism types of countable

the least admissible above !IR) for every jecture. It turns out that we can obtain

eoremasan queflce of Fodor’s theorem on 2). In fact, this argument

If to “If $ is a counterex le to Vaught’s conjecture, then

counterexample II,

~~~~~~v~ent to any countable model. In this paper we show that an extension of the Steel-Makkai-Sacks theorem

the results of I-Iarnik-Makkai are immediate consequences of Schlipfs and an extension of Fodor’s theorem. The extension of Fodor’s theorem

if K is reguku and uncountable and T is a <K branching tree of height K

has unique limits and does not embed 2’*, then any stationary set of nodes of T has a stationary subset -along one branch. We also use this to give a simple

f a ~s~~~~) weaker form of the main theorem of Sacks 1241 and to prove new coizequences of the existence of a counterexample to Vaught’s

conjecture.

The subject of this paper is an application of Fodor’s theorem to the model theory of countable fragments of L,,,. Ostenstbly I assume familiarity with Barwise [4] and Keisler [12], but most of the arguments require little beyond

ory. L is a fixed finite language throughout this paper, and C is countable collection of constants disjoint from L. If F is a countable

and c is a finite sequence of constants from C, then F(cj is the ent of L&c) generated by F. F(C) = !JeECCm F(c). Note that F(C) is a

t of L,,,(C). If X is a collection of countable L-structures, ction of F(c)-theories of expansions of models in X to L(c),

Throughout this paper F-type means an eie,ment of routinely identify a type with its n. If GE SFcc,, then

moCels of @. I will ),_, then 58 =_F !R means that

Vaught’s conlectw 3

%! and ‘8 satisfy the same sentences in F. X c o1 is thin if o1 - X contains a club set.

The starting point of this paper is [ 181, in which orley proves that if @ E L,,,

and IMod,J >K,, then (Mod,( =2 % His proof was in two parts and actually showed considerably more than this. I will give a quick indication of the argument, as the result is central to this paper and it provides a convenient framework for introducing many of the definitions I

Let X be a collection of countable L-structures and let F be a nt of L,,,. X is F-scattered if there are only countably many

F-types realized in structures in X. X is scutterea’ if X is F-scattered for every countable fragment F.

The first part of Morley’s argument was the following lemma:

.2. If IMod, < ZK”, then Mod, is scattered.

Morley’s proof of this lemma was (mildly) topological, but there is a simple model-theoretic argument which is perhaps more suggestive. To motivate it let me formally introduce some well-known notions:

.3. If B is an L(C)-structure, then 8 is a C-structure if every element of A4 is denoted by some constant in C.

rc F(C) is an F-Henkin theory if r is the F(C)-theory of a C-structure. r is formally consistent if there is no @ E F(C) such that {#, 19) c r.

F-Her&in theories have a simple syntactic characterization. This fact is due to Karp and is referred to as the weak model existence theorem in [4]. All that we need to know about this characterization is that there is a countable Z” of subsets of F(C) such that r c F(C) is a Henkin theory iff r is formally consistent and rna#I?Ffor every a~&

Let C be a collection of subsets of E (C). G c F(C) is C-generic if G is formally’ consistent and G t7 (7 # 8 for every cr E Z: The set of Z-generic subsets of F(C) will be denoted by 5!&. For A c F(C) let @= = {G n A: G E %$I.

If G is &-generic, then G is a enkin theory. He }, then @ is satisfiable there is a (.ZF il [

% is a consistency property ) such that for every s E % a

f ‘g is a set of fo ry 0 E P: there is

is A-per&t if for every s E V there are

ll-s of any two b~ncsaeS differ on A. This proves:

rley’s theorem, and armed with this Lemma 1.4. First notice that if SF{0 is

r some CE CC”. Furthermore note that theory extending @) = S$si91, so by 1.8, II of Qi realizes at most countably many

.

IX+ the second part of Morley’s theorem can be n of ?&x&t’s theorem on the existence of Soott

theorem is the case X = (Sn).)

. pf @)E(W,,,, then G is a Swtt fkory (sentence) for 92 if

u cowtabie fragment and let X be scattered. There is a such that if 92 E X and 9il is F-homogeneous, then the

them-y for D!.

Let Fg be the least fragment e,:tending F and containing the sentence

for each c e Cc”. Let % be countable, %I E X, 9R F-homogeneous, and XR zF; 92. on of F$, every F-type realized in ,Jz is realized in X. If p(c) is an

in X, then the sentence Ekp(.rr) is in F& so %11 and 12 realize the pie back and forth argument shows that 32 is F-

(&)* 3~ q@, y)) for au F-types p(c) E q(c, d) real- rices are ail in Fi, so 92 is F-homogeneous. Another

argument shows that any two countable F-homogeneous same F-types are isomorphic, so %I = %. IJ

If X is scattered, then there is a comtikwus sequence of countable

SR E X there is an (Y < ml such that the

Vaught’s conjecture 5

Define &= I,,,, if II is a limit ordinal and F’,, = F,+. then !8! is F,-homogeneous for

Co is countable theie is a countable QT < o such that + {%a, a) =F’&, . Ia particuku

1, and an easy ent now shows homogeneity mentioned in 1.11. 0

wts de in 1.11 is called the $ is scattered, then will write F$ (or just F,) for F, .

. rJ’X in scattered, then 1x1 s HI.

Let (F,: cw< q} be the rley hierarchy for X. y 1.11 each %k a Scott theory in U4<o)1 SE, and each SE is countable. Cl

y 1.2 and 1.13 we can conclude:

rley’s Theorem). If (Mod,1 < 2%, then IMod+ G X1.

). + is a ~ounterewnple to Vaught’s conjecture iff h?x& is scattered and uncountable.

. Mod@ is scattered iff Mod, is countable.

Let VC and SVC denote Vaught’s conjecture and the Strong Vaught Conjecture respectively. The proof of orley’s theorem shows that ,“JFCkSVC+VC, and by l.lS,ZFC+l

In this section I use Fodor’s lemma to give some new characterizations of scatteredness. For c E Cc” d$ be the set of expansions of models in

ode is scattered iff od$ is scattered. Let Mod:= let L,, be the sei: of rmulas of &,, of quantifier rank

d$ let T’hz(m) be the F:(c)-theory of ?l%! snd ‘I%,(n) be the L&c)-theory of m.

. od, is scattered, fgl E oJ$ and PCV denotes “the least a sue

) = pafIli,(2R) is a Scott theory for %R),

h*(m) = W(

closely related, as is indicated by the

&a, h(sz)scu e h,(sn)s the set of such tt contains a dub

due TV NadeP and can be found in [4],

then every formula in Fg,, has particular, if @ is

%. Note that the set of

is scattered, then for any LY < aI, {%I E h&de: r(%R) =S ar) is

2.2(iv), ( od*:?(~)~CU}={!BE )<a, thent -theory of %?I is a Scott

is scattered there are only countably many complete ere are only countably many such B’.

t indication that the ationships between the ordinals of 2.1 had ecture was probably the following res

Vaught’s conjecture 7

Sacks’ proof of 2.4 was recursion t eoretic, but it can also be obtained from a theorem. Let 0

Mode}. Schlipf prov every admissible above hence o

such that {o(H) EO(

wing slight extensiorr of Schlipfs theorem we can actually charaztetize scatteredness (2.7).

Let A’ be a red. d,) ik uncountable, then it contains a club set.

s a$ is uncountable, then it contains a c!ub

(i) Let (!k@; A, E) be a countable admissible set co #, x) and let f-l o&. Let T be Kp + “m c= #” + (‘“rLp(IDz, x) F a}. T is a

&-theory of LA which is A-finitely satisfiable, hence by the Bat-wise compactness theorem and results on pinning down ordinals [4, Theorem III, ‘7.51, T has a model with wellfounded part B of length 3. B is admissible above (932, x), so o(!!DiZ, x) s cy. cy s o(%& x) by the definition of T, so o((r132, x) = a. a was an arbitrary admissible containing (@I, x), so the set of such CT contains a club set.

(ii) Apply the argument in (i) to T + L,(B) b

We can glean the following theorem of Ressayre from the proof of 2.5:

2.6. If A is countable admissible alrd # is a 2$,-definable subset of LA which k A-finitely sawable, then @ has a model D such that o(m) = A n OR.

then the following are equivalent:

(ii) {m E Mod+: r(m) s cu) is countable for every LY < ol. ) s a) is countable fw every LY < q.

(iv) {o(S& x): !IZ E od,, r(m) < s(rrrZ, x)} is thin for every real x. (v) {0(2x, x): !?R E od,, o(n) < o(%& x)} is thin for every real x.

(i) =$ (ii): 2.3. 3 (iii): 2.2.

(ii) 3 (iv): If (o(!III, x): 9X E dor’s theorem t e is an 0 such

then by

stationary, hence uncountable.

8 M. Howard

(iii) 3 (v): See (i) 3 (iv). (iv) * (v): 2.2. (v) + (i): &sume that is not scattere orley’s theorem there is a

le fragment F(c) such that is a perfect tree of satisfiable F(c) es. Let A be a countable set which contains the tree, and let a!

of A. By standard arguments there is a perfect subtree T with -generic. Each branch is contained in an admissible set with the

als as A, so by 2.6 each branch has a model %I2 such that o(XR) = cy. which codes T. For any real x there is a branch !DS of T such that x the pair (ax, a), hence x is hyperarithmetic in (!!l72, z) for any

particular every real is hyperaritbmetie in @R, z) for some 92 such = cu, so (o(m, n): o(n) = cu} is uncountable. By 2.5 this set is

stationary. Cl

Of course by 2.2, 2.7 shows that Mod,+ is scattered iff for any cy < ol, t(9Q G cu} is countable, is any one of the ordinals 3. In Section 5 we will se is scattered iff {o(XR, x): 9R E

o(m) s o(m, x)} is thin for any real x. Note also that since the countable union’of thin sets is thin we can replace Mod+ by Mod; in (ii)-(v).

. K always denotes a regular uncoul.table cardinal. A c K is Turin if A is not stationary. Let (T, X) be a tree. For d E T !zt ht(t) be the order type {x:x<t}. If AcT, then ht(A)=sup(ht(t)+ktEpl}. A branch of T is a maximal linealy ordered subset of T. If b is a branch and (x < ht(b), then b(m) is

ue t E b such that ht(t) = cy. b is bounded if ht(b) < ht(T). If S s T, then (t): t E S}. Call S stationary if [S] is.

If .T = (T, <) is a tree of height K, then T is narrow if: t E T has fewer than K immediate successors. )=ht(i) is a limit ordinal and {x:x<s)={x:x<t}, thens=t.

(iii) T does not embed 2?

ee Lemma). If T = (T, 0 is narrow and S s T is stationary, nary for some branch b of T.

Assume that S n b is thin for every branch b of T. For t E T let s E S: t XS}. Let S, be stationary. We will sho-w that t has incompatible

at S, and S’ are stationary. If not, then {x > t: Sx is

Vaught’s conjecture 9

node below x which is on &. For Q! E [R], let X, E R be such that ht(x,) = cu, and let g(x(y) be the maximal element of b below x,. Let f(cu) = ht(g(x,)). f is regressive on a stat; onary set, hence constant on a stationary set. Let /9 be such that f-‘(@) is stationary, and let X =g-l(p). Note that X G U {SX: x is an immediate successor of b(j3) which does not lie on b). Hence the set of heights of nodes in X is the union of fewer than K thin sets, hence is thin, which is a contradiction. Thk shows that {t: St is stationary} embeds 2’“, contradicting (iii). Cl

In l-articular, if 3 has height K, then 9 is stationary, hence 3 has a stationary brar,&. This proves:

3.3* Ijr T k roarrow, then 9 has a cofinal branch.

The next two corollaries are also immediate.

colro . Let K s 2% I(f 3 has unique limits, is <K branching, has height sK

and 3’ is the tree obtainer from 3 by extending every branch to one of length K,

then 3 has a per;fect subtree iff 3’ has a stationary antichain.

thin. 3.5. If 9 is narrow, then the set of lengths of bounded branches of 9 is

Let 9 be a tree of height K. Call a node isolated if it lies on fewer than K branches of 3. Let (3) be the set of branches of 9. If f: (2’) --) K and t E (T), then let 9 be the set of isolated nodes of t of height <f(t).

3.7. If 3 is narrow of height K, and for each branch t of 3, A, is a thin set of isolated nodes of t, then Uic(T) A, is thin.

Otherwise UtE ( s) A, has a stationary subset S along one branch 6. Let x be the least node of S n b. If y E S f? b, then y E A, for some t. x lies below y so x lies on t. This shows that S n b G U,,, A,. x E S so x is isolated, so this is the union of CK sets of size CK, hence thin. q

The rest of this section is independent of the rest of the paper. These results are intended to show that the Tree Lemma is not as peculiar as it may at ‘first seem. The next example shows that it is actually an extension of Fodor’s theorem.

e (Fodor’s Theorem). Let f: K --9 K be regressive on a stationary set. r the following tree 3. F has a main branch b of len ol, and for each

node b(e) there is a branch b, of length o1 which splits obviously narrow. For a! c o1 let S, = (6: f (/3) = a}, ;iad iet IV,,

10 hf. Howiud

union of the IV, is stationary, hence has a stationary subset along some branch bs. But then the preimage of j3 is stationary, so f is constant on a stationary set.

Let wr = (T,, -Q: r E X] be a collection of trees, and let or be the least node If (X, gx) is a tree, then we can form a new tree by attaching each 9’ to X

at r. lh;ormally:

. CreX T, = (T, <) where:

(i) T = {(r, t): t E Sr}. (ii) Ifr+q,then(r,t)<(q,s) e t=u,andt<q.

(ii) (r, t) qr., s) @ e<,s.

If (X -4 is narrow of height K and each Z& is narrow of height SK, then CreX T, is uarrow of height K. (Let P be a perfect subset of crsx T,. Since X is narrow, (q t) E P for some t, t Zo,. But then {(r, s): t < s} contains a perfect subset of P, contradicting the narrowness of T=.)

CkU 9’ a simple ex&nrion of T if 9 E 9’, ht(T) = ht(T’) and every branch of Sris extended by exactly one branch of 5’. A simple extension of a narrow tree is narrow.

If S has height K, let Z(s) = (ht(6): b is a Lounded branch of 9). If f: K+ K

is non-decreasing, then by extending branches, any 9 has a simple extension 9’ such that %‘( 9’) =f[ %‘( .T)].

. A c K is thin a A = Z(s) for some narrow 9 of height K.

(3): Obvious for IAl < K. Let C = {cd: (Y < K} be a club subset of K - A. Let Ta be narrow such that BY&) = {/3: (cy C j3 E A) A (a + /? < c,+~)}. Clearly ~(C,,, Z) = A.

(e): Let 9’ be a simple extension of 9 such that R(T) = {cu + 1: LY E A). (b(ht(b) - 1): b is a bounded branch of 9’) is a stationary antichain, contradict- ing 3.5. cl

For this section let Mod+ be scattered and let {F,: LY< ol} be the Morley hierarchy for od,. For 8: E C’” let T’,= {@: @ is a complete F,(c)-type extelPtding $ for some & c 0,). T$ is a tree under c. This tree has the

property that every branch is consistent (4.2), hence it is narrow is a complete F,(c)-theory, then call 11 triuial if 7~ is implied by its

me /!I < ty. Note that q is isolared in the sense of Section 3

Vaught’s conjecture 11

T$, is the tree of complete F,(c)-types extending # for QI < o,, ordered by inclusion. T# = T$. Call an F&)-type ~k&zZ if it is implied by its restriction to Q(c) for some /3 < a.

4.2. If @ is a branch of T”,, then @ has a model.

f. Let 4p, = @fl F,(c). If ly E F,(c), then call V# @je-complete if ly is &(c)-complete with respect to @. There are two cases:

(i) h(@)C al: Let an be increasing such that U,.=, a;, = ht(@). By the omitting types theorem @mm has an Fax(c)-atomic model !2Rn. !ll2,, is F’“(c) prime, so w.l.0.g. 92, <Fem(cj ZR,,,. Clearly U,,, ZIRn k @.

(ii) ht(@) = ol: Let A = {a: if q? E F, is @W-complete, then ly i, @-complete}. Assume for the moment that A is club and let {yo: (Y < wl} be an increasing enumeration of A. Define !l& for (Y< w; by induction on LY. Let llJ&, be a F,,(c)-prime model of GyO and let !lR,+, be a F,,=+,(cj-prime model of @,,a+, such

that na -=$.J~) ~I&+P If 12 is a limit ordinal, then let !Il& = UOsA me. I+Iote that if a E M,‘w, then a E Mz” for some cy c A. By induction, rrJz, is FJc)-prime, so a realizes some @,,P-complete formula ly E &(c). By the definition of A, I/J is @complete, hence @,,,-complete. In other words snl, is F&)-prime, so we can carry the induction through limit stages. U,,,, !Ili& is the desired model of 0. To see that A is indeed club, for 3 E F,, let mV = 0 if rj~ ie @-complete and let a,,, be the least cy such that V/.J is not @,-complete otherwise. A = {cu: g[a] E a-}, where

g(a) = u*ci’, aiuY but {ar:f[a]=a} is a club set for any f: ml+ ml. Cl

eoreun 4.3. Zf Mod,+ ir scattered, then T; is narrow.

f. Let JZ be a perfect subtree and let (Y be thz sup of the heights of nodes in IE: LY < q. Every branch of n has an F&)-type extending it, hence Mod, is not F,-scattered. Cl

If LY and j3 are ordinals, then [cu, #l] denotes the set of ordinals acy and G/3. In particular, if #l < a; then [a; @] is the empty set. [cw, /3) = [ ca; /I] n 8.

The next theorem is the main theorem of this paper.

is thin. Zf Agn G [o,(!lR), w,) is thin for each %! E od$ then L&IM,.w; APR

f. Since the countable union of thin sets is thin it sufbces to show that 2R E lkIod$ is thin for each c E Cc”, but this follows immediately from 3.7 and the fact that T$ is narrow.

The fact that X in the next corollary is nonempty is new. Call an I,,,-type a counterexample if it is realized in uncountably many countable models of 9.

12 hf. Howard

S. There & Q club set Xc o1 such that for LY E X, every nontrivial &_-type extending # k a cou.Wezmmple.

q(c) is a nontrivial L--type extending # which is not a counter- d os h(?IR). s not a

y 4.6, UlR EMoa; [o,@Z), t(!DZ)] is thin. Let X be the complement of this set. ‘Note that r&iR) G r(a) so if r,,(%i!) E X, then r,&Ui!) s t(!lR) < w&?lR). By 3.60, h&B) s t&D2) + 1. Since m&N) s h,@R), if r,&D2) E A’, then

r$!R) = r(Sn) < q,(rtrc) = h,(Sn)

= r&N) + 1 d o(‘82) s h(9R) G t(2R) + m + w. cl

The follotig is a slight extension of a theorem of Hamik and Makkai. (Notice that any such v is a minimal counterexample in the sense of Narnik-Makkai [S]. Conversely the existence of the club set below is easily derivable from their theorem that any counterexample can be extended to a minimal counterexample).

For every c E C’” there is a counters v~:.FJF $.+> extending 9 and t X ; o1 such that for every u E X, ~($1 c p 3 rmique nontrivial extension

Otherwise if r@(c) is any :*~.~a; +*mfgE= extending $, then there is a t S such that for every (Y E S, I@ has at least two nontrivial extensions

y Corolhtry 4.5 there is au IY E S such that every nontrivial F,(c)-theory is a counterexample. In other words every F’,,(c)-counterexample extending # has two incompatible F,,(c)-counterexamples extending it, but then T$, embeds 2 ‘“, contradicting narrowness. 0

is as i en cali 11 a minima; counterexample.

. Let ly be a minimal counterexample and let X be the corresponding club set. If tv E X, then 9 together with the set of negations of Scott sentences in F,(e) is F,(c)-complete.

g. Any complete extension is nontrivial, hence it 5. I as exactly one completion to F,(c). 0

Let #_J be a minimal counterexample. There is a club set X such that if a E X, then every nontrivial Fp-type extending ly is omitted in only countably many countable models of qgO

Let X be the intersection of the club sets from 4.7 and 4.8. Let t(c) be a nontrivial F=-type for some IY E X. t is realized in uncountably many models of q, hence 3x r(x) has nontrivial extensions to every F;,. If VZCT(X) A ly has uncountably many models, then it has nontrivial extensions to every F,, contradicting the fact that r# has a unique nontrivial extension to %;, for a club set of LY. So t is omitted in only countably many models of $9. 0

5.

In Section 2 we saw a connection between admissibility and Vaught’s conjecture. Steel (assuming analytic determinacy), Makkai, and Sacks have independently improved 2.4 to

5.1. If Mod, is scattered and uncountable, then o(m) < o(2J2) for some ID1 E Mod,. In fact, the set oJf suc.h o(B) contains a club set.

Theorem 5.1 is an immediate corollary of 4.4 and Schlipf’s theorem. In fact, somewhat more is true:

If Mod, is scattered and uncountable, then there is an a E o(Mod,) such that if 2R E Mod: and o(m) = (Y, then o(!lJZ) c w(n). In fact, the set of such a contains a club set.

of. Y = UrnGMod$ [o(B), o(m)] is thin, so by Schlipfs theorem o(mod@) - Y contains a club set. Cl

Of course, Corollary 5.2 can be relativized to any real, and this provides another characterization of scatteredness:

3. BAod@ is scattered ifl for every real x, {o(B, x): ?N E Mod;, o(!lQ) < o@R, x)} is thin.

M. Howard

uncountable. For any real x there is a if o(%t, x) e X, then o(!#, x) = t&iR) =

s x), then d s o(m, x), hence this follows from 4.7 and

re~xthereisaclubsetxsuchthatif a!EX, 9!Jk , x) (i.e. (Y is the ordertype of the ordinals q some

an isomorphic copy of il!R), then (Y is the tih , .r) or (Y = o(%R, x).

Let S be station such that for each cy E S there is some !I?2 E Mod, such that cuisthe@h e above (%R, x) and 1 < p c cy. By Fodor’s theorem we can stabilize the least such /3 on a stationary set, w.1.o.g. each ordinal in S is the /3th admissible ordinal above (%, x) for some %I E Mod,. For 2R E Mod, let f (!#) be the lpth admissible ordinal above (92, x). o(9tz) s o(%!, x) + o <f(rat) for

hence S c {f @2): 92 E +} c UPM,M+ [o(m), f(m)] which is

If T is a theory cxeending then o&I!LR, x) is the least ordinal of the ordinals of a transitive model of T containing x and

e scattered. If T extendis KP +&-separation, then

then h(M) <oT(SR) for any countable d,} c uSnEMod, [m(m), o#)I which is

tell us something about pairs of models: th of which can be found in [4].

need two

de1 [21]). If A is countable admissible and 1111, !J! are in A, :hen %.

issible and %R E A, then for any %, ZR sLA Yl iff Di! =cy %

Vaught’s conjecture 15

Let 11 be a minimal counterexample. There is a club set X such that i%, if o(2DZ) = o(S) = o(SR, 8) E then =%.

be the intersection of the club set in 4.8 wi he club set in 5.6. Let 8, then by 5.1 !@k 8, hence .2 9!F! fobs) 9, hence

!@2&,, 8. If o(m) E X, then there is only one nonprincipal F,,+heory extending hence w.1.o.g. the FO(mJ -theory of !82 is principal and hence a L&ott sentence. t then h(W2) <o(?I!R) contradicting 5.6. q

Let 11 be a minimal counterexample. There is a club set X such that for !D2, YI E I&M&,, i.f o(SR, ‘8) E X, then o(m) = o(rXn, ‘32) or o(S) = o(ZR, 92).

Let X be stationary such that for CVE X there are rXn, % such that o(m) CO@!& 8) and o(Z) co@%, 8) = cy. For G! E X let (ma, 8,) be such a pair, and let f(a) be the greater of o(%&) and o(&). f is regressive, so f stabilizes on a stationary set. But that means that there i {o($R, %): o(%R) c 16, o(.%) < /I} is uncountable, hence {m E uncountable, contradicting 2.6. 0

ac eolre

In this section we show how the tree lemma can be used to prove a variant of the main theorem of Sacks [24]. Let Mod,,, be scattered. We will need the following rather surprising connection between the Strong Vaught Conjecture and a strong downward Lowenheim-Skolem theorem for scattered theories in L Q)lO (the connection is that they are equivalent). (i) + (iv) (t

ction) is due independently to both Narrington and Harnik kai [15]). An argument similar to (ii) _i (iv) (without using stationary sets)

appears in PIarnik-

. The folio wing QTC cquiwalent (recall that Mod, is scattered):

0 i ode is uncountable.

0 ii There is a complete nontrivial &,-theory extending @I. . . .

( ) 111 $I has a model which is not L,,, -equivalent to any countable model.

0 V $B has a model with no countable L,,,-elementary submodel.

(i) 3 (ii): For Q! I, ol, let @a be any complete &theory with uncoun- tably many countable models. { Qa: a < w,} is a stationary set of n hence has a stationary subset along some branch @. In particular, uncountably many countable models for every cy c wl. Note tha has a Scott Sentence in F,, extendi Sentence. ence for every CY C wl,

is

is an a! C o1 such that

nding Qi, then there is

Since Qp is nontrivial, there is a @?= nonprinci type p(c) for every is countable, there is a stationary set and some c such that

ar< o1 there is a complete @+-nonprincipal p(c). By the tree a there is a stationary subset along some branch Y(C) has arbitrarily

rincipal initial segments, so is nonprincipal. .

3. Let @ be a complete nontrivial F,,-theo extending @. lye F,,(c) is call& extendible if J/J is contained in a @-nonprincipal F,,(c)-type.

There is a club set X such that for every QY E X, every complete er extendibk or @complete.

Let S be stationary such that for each LYE S there is a complete Fe-nonprincipal type qua t va is not extendible. Since C’” is countable,

A0.g. every ty, is a c- ce the set of t/~~ is a stationary set of nodes of ce has a statio ubset along one branc ut this branch has

tial segments, hence is a onprincipal &,-type, e of this branch is extendible, which is contradiction. This

shows that there is a club set Y such that if cy E Y, @-nonprincipal is extendible. By the proof of 4.2 there is a cl such that every cipal is @complete, sc let X be. the intersection of Y with this

Let q(c) be extendible. There is an extendible q(c) extending q(c) such tht for each cy E X, I/J has exactly one complete extendibk

f is identical to that of 4.8. Cl

of the club sets from 6.4 and

. Vaught’s conjecture 17

(3): Let /3 c cy be such that j3 E X. 7: is in the only extendible &type and every other type is an atom, hence every extension of it is an atom.

(C_): If every extension of ~z is an atom, then qt is not extendible, hence t is extendible. 0

Let {A,: QI < wl} be a closed unbounded sequence of countable admissible set: such that F, c A,. Let A = UaCU, A,. If Qp is a sub say that @ is E,,-definable on a club set if there is a club set X such then @ n A, is a &-definable subset of A,.

CO . There is a complete nontrivial LA-theory @ extending # which is &-definable on a club set.

Let 9~ be a minimal counterexample and let X be the co t @ be the unique nontrivial extension of q and let *a =

if Q! EX, then q together with the negations of the Scott Sentences in LAll is LA,-complete. Nadel [21] shows that there is a club set 2 such that for cy E 2, every Scott Sentence in L A, implies a canonical Scott Sentence in LA,, and the set of canonical Scott Sentences is a &-definable subset of LA,, hence the set of Scott Sentences is &-definable. The set of negations of Scott Sentences is then &-definable, hence q together with the set of negations of Scott Sentences is &-definable. By the extended Barwise completeness theorem the set of conse- quences of this theory is &-definable, ence @& is &-definable for or E Y n 2. (Note that we could have shown that was &-definable on a club set without resorting to Nadel’s theorem by observing that the set of L&-complete sentences of LA, is a &-definable set, and replacing Scott Sentences in the argument by L&-complete sentences.) 0

Let @ be a complete nontrivial LA-theory which is &-definable on a club set. Tien there is a nonprincipal type extending @ which is C,+I-definabfe on a club set.

e, z(c) is not an atom w.r.t. iff there is a O(c) such and 3c (z(c) A l@(c)) E @. ) is not an atom s an atom w.r.t. W’ is X be as in 6.6. By

6.6, Y is z&+1- definable on the set of limit points of X. 0

our attention to Sacks’ theore e main interest of this it is not true for Friedman’s C classes. There are some t

argument which we circumvent by weakening the statement of the theore

18 . Howard

ened form also distinguishes between EC and PC counterexamples, and is relatively easy to prove using the above machinery. The proof below was greatly infhie by ideas of Silver, and the last part is from Sacks [24].

let s ) (the sucks rank of be the least Q such

re k a set X such that for each (Y E X there are SO,

(ii) s(YQ = @I,) = (Y + 1.

and let A, be a continuous sequence of countable admissible

0) F&A,, 3 UC,, A, is a model of Zp;‘. 6.g and 6.9 we can find a nonprincipal LA-theory 9, a @-nonprincipal

and a s(c)-nonprincipal type t(c, d) which are &detinabIe on a e that cy is the set of ordinals of A, for a club set of a: and that

ere is a club set of a! such that A, is an elementary submodel of A, hence a sub set of 4y such that (A,, T) is admissible for any definable T c A. By (the proof

re is a club set of 4y such that sL,, =A, and =Fp agree on models in X be the intersection of Y with these sets and the club set of 5.5. Let

letp=snL,, andq=fn&_. By (the proof of 2.2, k&c) -p(c) fl ++q(c, d) fl F,(c, d), so we will not distinguish between them. the F,-atomic model of @ n F,, and IIR, be the F,(c)-atomic

find an Sm such that t(m) = (Y and the F,+,-theory of 2JI is nt from that of %i, and 9&. Note that (A,, q(c, d)) is

admissible, so by 2.6 q(c, d) has a model (!DZ, m, n) such that o(%!) c cy. If <LY contradicting the fact that @ is nonprincipal, so

=r(%). By 5.4 we can assume that r(32) =o(%R), hence e characterization of homogeneity in 1.10, 9IIkV.x (p(s)+

is atomic over some realization of p(s), and q(x, y) is on, so this sentence is not true in %&, hence

I = a, %Z = the F,+,-atomic model of the F,,,- +,-atomic model of the F,+,-theory of 2Jl (in fact,

therr 913=%). The Fa+, -theories of ‘%I and SR, ensure that they are not , hence s(%~) = s(%,) = (Y + 1. Cl

of ordinals of countable models s set in {e’: cy is admissible} U of the ~~tiQ~a~s}.

Vuughr’s conjecture 19

set of nonstandard ordinals in the Friedman example is also a PC class; just add the sentence which says there exists a set X which has no minimal element. 6_12(ii) is not true of the Friedmcln example, neither 6.12 (i) nor (ii) is true of the modified example.

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