Countable Theories Categorical in Uncountable Power
COUNTABLE THEORIES CATEGORICAL IN UNCOUNTABLE POWER
JOHN T. BALDWIN B.A., Michigan State University, 1966 -x,w i3.S. University of California at Berkeley, 1967
A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY In the Department
of Mathematics
@ JOHN T. BALDWIN, 1970 SIMON FRASER UNIVERSITY DECEMBER, 1970 APPROVAL
Name : John T. Baldwin Degree: Doctor of Philosophy Title of Thesis: Countable Theories Categorical in Uncountable Power
Examining Committee:
Y - I 1 (M. Morley) External Examiner , Professor Cornell Universitv- Ithaca- N.Y.
- /.+' (B. Alspach) Examininn Committee
(J .L. ~er&rdJn) Examining Committee
A " - 1.
Examining Committee -
---TT---. - . (S .K. Thomason) Examining Committee
Date Approved : January 14, 1971
(ii) ACKNOWLEDGEMENT
The author wishes to express his gratitude to Professor A. H. Lachlan for hio never-failing help
and guidance during the preparation of this thesis. He wishes also to thank the National Research Council
of Canada since most of the research for the thesis 'F 'F was supported by National Research Council scholarships.
Particular thanks are due to Miss i4arian Birch for typing the thesis in an extremely short time. TABLE OF CONTENTS
Page
Section Introduction and Terminology 1
Section Algebraic Closure and Strongly Minimal Sets
Section Strongly Minimal Theories
Section N1 Categorical Theories
Section Almost Strongly Minimal Theories
-Sq$t ion Proof of Vaught's Conjecture
Section Proof of Morley's Conjecture
Section A Note on Definability
Section Related Results
Bibliography 50 Introduction and Terminology
Most of this thesis is concerned with countable complete theories having only infinite models. A theory T is categorical in power if all models of T which have power X are ismorphic. has focused attention on Nl-categor- ical theories with his conjecture [51 that a countable theory was #l-categorical if and only if it was categorical in every uncountable power. Vaught suggested another property of
.bk l-categorical theories. In [211, he conjectured that an -. %l-&ttegorical theory was either iO-categorical or had exactly Hg isomorphism types of countable models. Morley [71 proved the bos conjecture. In the course of the proof he attached an ordinal aT to each countable complete theory T enb ccnJectured that if ': were #l-categorical then a would T be finite. In his thesis [6] Marsh introduced the notions of
11 algebraic closure" and "strongly minimal set." This thesis investigates the properties of % -categori- cal theories. The principal tools of this investigation are the notion of strongly minimal set, Vaught's two cardinal theorem, and two of Morleyls theorems. The theorems of
Morley state that a theory categorical in any uncountable power is totally transcendental [7; 3.81 and that if X is a subset of the universe of a model of a totally transcendental theory there is a model of T prime over X.
,Sectton 1 of this thesis exhibits the basic properties of algebraic closure and strongly minimal sets. The simplest sorts of sl-categorical theories, strongly minimal theories, are studied in Section 2. In Section 3 the notion of strongly minimal is combined with the idea of Vaught's two cardinal theorem [g] to reprove the hs conjecture. A slightly more complicated sort of H1-categorical theories, those which are almost strongly minimal, is considered in Section 4. vsughtls conjecture is proved in Section 5. Morley's conjecture is proved in section 6. Section 7 deals with a problem in definability. Section 8 summarizes some related results.
Sections 1, 3 and 5 are from [I]. To the best of the author's knowledge those results in s,ections 2, 4, 6 and 7 not expiicitiy credited eisewnere are new. The notation used here combine8,that of Morley [7] and Shoenfield [20]. We deal with countable first order languages. For convenience, we will assume that each such language L con- tains only relation symbols and constants. L has vaf:~~bles vO, vl, .... Following [20; p. 171 we let Ay o, ... W n
(so, sn ! denote the formuls obtained by replscing each occurrence of the variable by the term ai for 0 5 i 5 n. w i Whenever such an expression occurs we assume no variable has become bound by the substitution. We omit the subscripted variables wo, ... w n if they are clear from context. For each k natural number k we admit quantifiers 31 v and 3'kv which mean intuitively "there exist exactly k elements v" and "there exist at most k elements v" respectively. If c is a constant in L anda is an L-structure then the value of c in R is
denoted as in [20] by (c) r'. R We may extend the language L in several ways. 1f@ is
an L-structure there is a natural extension ~(d)of L
obtained by adjoining to L a constant a for each a E 1 c7 I
(the universe of 4). For each sentence ~(a an) & L(A) 1' ..., we say (1 satisfies A(al, an) and write QF*(al, ..., ..., an ) if in Shoenfieldls notation a(~(a,, a,))= T [20; p. 191. - ..., -- If c7 is an L-structure and X is a subset of 101 then L(X) is the language obtained by addoining to L a name x for each x E X? ( Q ,X) is the natural expansion of d to an L(x)-struc-
ture. A structure B is an inessential expansion [20; p, 1411 of an L-structure a if 8 = (0 ,XI for some X 2 I a 1. We also extend a language L by adding additional relation symbols.
If L' extends L and Q1Ws an L1-structure the r?duct, denoted a' IL, -of 01 -to L is the L-structure obtained from 4'. by omitting those relations.and constants which occur in L1 but
not L. Shoenfield calls this concept "restriction" [20;
Sn(~)denotes the set of formulas of L with free
variables among vo v , . . . , n-1 ' If A is a formula such that u u in the natural order are the free variables in A, .... n then A(@) is the set of n-tuples bl, ..., bn such that
(bl, ..a b,)- If p is a unary predicate n eynbol we abbreviate pvo(@) by p(g).If 8 an L-structure Y = 1 I and X 5 I Q lk then X is said to be definable (Q ,Y) if there is a formula A in s~(L(Y))such that X = A(Q). X is said to be definable -in Q if X is definable in ( R , 141 ).
A consistent set of L-sentences is a theory in L., If T and T' are theories in L then T1 extends T if T -c T1. If T is a theory in a language L then T' is an inessential extension of T if there is a model a of T and a subset X of I C7 1 such that T' = Th( C7,X) (i.e. the set of all sentences in L(X) true of (0,X)). T1 is a principal extension of T if T' is aninessential extension of T by a finite number of conswts and a set of nonlogical axioms for TI' can be obtained by addoining a finite set of sentences to a set of nonlogical axioms for T.
CI 5 8 and Q E 8 abbreviate "Q is an elementary sub- structure of B", "a is elemeatarily equivalent tc 8" respectively [20; pp. 72-74]. Suppose Q and 8 are L-struc- tures, X 5 la] and f is a 1 - 1 map of X into 181. Let 8; be the L(x)-structure obtained by setting (x)~,= f(x) for each x E X, Then f is an elementary monomor~hism (elementary embedding) of X into 181 if (0 ,X) o 8'. The cardinality of a set X is denoted by K(x); we abbreviate K( ( Q 1) by K(Q).
Let r be a subset of Sk(~). Then r is a -k-type in T if there is sbme model Q of T and elements a1~ a*+ ak 101 such that iz b(al, . . . ak) if and only if A t r. If Q is a model of T and X -c 1 0 1 then a k-type --r is realized X if there exist xl, x E X such thata LA(~~,xk) for ... k ... each A € r. A k-type r is a principal k-type in T if there is a formula A & Sk(L( 0))such that for each formula B in 0 i--Vvo, r . . . vvk-l (A + B). Since T is complete there is- Qn-e -0-types , truth- -. Following Morley [?I we assume that each T = C*for some
C and thus that each n-ary formula 4 is equivalent in T to an n-ary relation A. N(T) is set of all substructures of models of T. The following summarizes with slight changes in .-. notacion the second paragraph of 12 in [7]. If 67 is an
L-structure &( a) is the set of all open sentences in L(Q ) which are true in (a,fAl). If C7 e N(T), T(Q ) =B(Q )UT is a complete theory in ~(a).Let sk( C7) denote the Boolean algebra whose elements are the equivalence classes into which s~(L(~))is partitioned by the relation of equivalence in
T( q ) , and whose operations of intersection, union, and complementation are those induced by conjunction, disjunction and negation respectively. The Stone space of S1 ( a), the set of dual prime ideals of Sl(t7), is a topological space denoted s(Q). A dual prime ideal of Sk(Q) is a k-type of
T(a ). This is a special case of the definition of k-type in the preceding paragraph. Note that if p & s(Q) and at is an inessential expansion af 0 p is naturally a member of s(QrI). 51. Algebraic Closure and Strongly Minimal Sets
This section contains the definitions used, and the
basic theorems proved, by Marsh [6]. For completeness, the rseults are reproved in a manner similar to that used by
Marsh.
Let a be an L-structure and X a subset of 1 Q 1, The algebraic closure of X, denoted by cl(X), is the union of all finite subsets of I G' definable in (C? ,x). This notion was
expl~,ed by Park as "obligation" in [ll]. X spans Y if
Y -c cl(~).X is independent if for each x E X, x E cl(~-{XI). X is a basis for Y if X is an independent subset of Y which
spans Y. If every basis for Y has the same cardinality p, we define the dimension of Y to be p and write dim(^) = &I. Let G'be an L-structure. A subset X of 101 is minimal
in Q if X is in-f-inite,definable in a, and for any subset Y of la1 which is definable in either Y fl X or X - Y is finite.
If D E s~(L(0)) and X = D( G') then X is strongly minimal --in 0 if for any elementary extension 8 of@ D(B) is minimal in 8. Let go and Ul be models of a complete theory . Since up to isomorphism any two models of T have a common elementary extension, D(UO) is strongly minimal inaoif and
only if D( G' ) is strongly minimal in gl. Thus, without 1 ambiguity we define a formula D E ~~(1)to be strongly minimal --in T if there is a model C? of T such that D(Q) is strongly minimal in R.
We now exhibit some of Marshvs results. It is trivial that X -c cl(~)and that cl(~)5 cl(~)if X 2 Y.
Lemma 1. cl(cl(~))= cl(~)
Proof, Let Q be an L-structure and X 2 1 G' 1.
Suppose X & cl(cl(~)).Then there exist elements al, ...,
a in cl(~)and a formula A E Sn+l(L) such that for some n k k Q t~(x,a,, . . . , an) A 31 vo A(v0, al, . , a. But .-., then 'f6r each i, 1 9 i I n, there exists a formula Bi in
S1(~(x)) and an integer ki such that c? pBi(a i) ; . ki A 31 v B.(v ). Let A1 be the formula 3vl, 3vn 010 ..., n k (%=I Bi v - A, A. A. 31 vO A). Then A1 r S1(~(x)),
Al( a) is finite, and@ A1(x) so x E cl(~).
The next lemma asserts that within strongly minimal
sets the "exchange principle" holds.
Lemma 2. Let D be in S1(~)and C7 b& an L-structure. Suppose D(Q) = X is strongly minimal in R . If Y g X, a
and b are elements of X, a € cl(~),b s cl(~),and
Proof. We suppose that Y is empty since if that case
is proved the general case may be deduced by adjoining
names for the members of Y to L. Since b E cl({a)) there k exists an A E S2(~)such that a 1- ~(a,b), 31 vl ~(a,v~) for some positive integer k. If (A(v0,b) ,. D)(4) is finite the lemma is proved. If not, since D( Gf) is.strongly minimal, for some positive integer m (- ~(v~,b)A D)( (I) has
cardinality m. Write c(vl) for D(V~)A 3lmvo(-A(v A 0' v1 ) D(v0)). ~henai-'~(b) and b $ el($), so c(@) is infinite. k Urite B(V~)for D(v0) , 31 vl~(vo,vl). Since Q..B(a) and a E cl(@),B(# ) is infinite. Let bo, . . . , bk be chosen from nf 4 \ fA#\ (~(v (u ) w 1. Since 0 ) , - ~(v~,b~))has cardinality m for each i 5 k, and B( R ) is infinite and contained in D( d ) , for soae a' E B( 4)(I A( sl,bi) for i = 0, . . . , k. But this 'F contradicts the definition of B so the lemma is proved.
The proof we have juet given differ8 from Marsh's in that we did not invoke an elementary extension of a. The following is proved from the exchange principle just as the corresponding result is proved in the theory of vector spaces.
Lemma 3. Let X be strongly minimal in 0 and let Y & X.
If Z is an independent subset of Y then Z can be extended to a basis for Y. Moreover, any two base8 for Y have the same cardinality.
The relationship between strongly minimal sets and elementary aonomorphisms is expressed in the following lemma.
Lemma 4. Let D be strongly minimal in a complete theory
T and let C? and 8 be models of T. Let f be a 1 - 1 map from X -c I)(@) into D(B) such that X, f(~)are independent in t?, 8 respectively. Then f is an elementary monomorphism.
Proof. By using the device of adjoining names it suffices to treat the case in which X is a singleton, say x. Let A be any formula in S1(~). Since D is strongly minimal, just one of (D A A) (4), (D , - A) (d) is infinite.
Without loss of generality we may suppose (D A A) (Q) is infinite. Then (D ,A) (8) is infinite since T is complete.
Since {x} is independent in Qand x& I)(@) we have x € A(@).
Similarly f(x) E ~(8);since A is arbitrary f is an elementary
mn".n."Allb~ r... Auvuvuvr yAAL~uI.
One may easily obtain the following slight variant on
Pr~position4 of [6]. 'F Lemma 5. Let @ and 8 be L-structures, X c Id 1 and f mapping X into 8 an elementary monomorphism. Then f can be extended to an elementary monomorphism of cl(X) into 8. The image of this extension is cl(f(~)). 52 Strongly Minimal Theories
A theory T is said to be a strongly minimal theory if the formula vo = v 0 is strongly minimal in T, that is if the universe of each model of T is strongly minimal. The following result is due to Marsh [6].
Theorem 1. If T is a countable strongly minimal theory then T is categorical in every uncountable power.
Proof. Let a and 8 be models of T each with power -, Let X be a basis for 1 0 1 and Y a basis for 181.
Then K(X)= K(Y)= X since for each X a subeet of a model of
T K(c~(x))5 K(X)+ 8 f 0' Then there exists a 1 - 1 map from X onto Y. Then by Lemma 4 f is an elementary mono- morphism. By Lemma 5 f extends to an isomorphism of C? and B.
If we drop for the moment the assumption that L is countable we could still prove in the same way Lemmas 1 through 5. Then, similarly to the proof of Theorem 1, we obtain
Theorem 1'. If T is a strongly minimal first order theory in a language L for each X > K(L), T is categorical in power A.
We return to countable languages. C7 is a prime model of T if for each model 8 of T there is an elementary embedding of a into 8. Theorem 2. If T is a strongly minimal theory and C? is
a prime model of T then T is L*O-categorical if and only if for every finite X -c 1 a ( cl(~)is finite,
Proof. Suppose for some finite X = 10 ( cl(X) is infinite. Let T1 = Th(d,~). Then T1 is iO-categorical if
and only if T is ~O-categorical. Let l' be the set of k formulas of the form - (A , 31 vo A) as k ranges through the
generated r then both ~(0)and -B(Q) would be infinite *tc which is impossible since la( is strongly minimal. r is therefore not principal; thus by Ryll-Nardjewski's theorem
[20, p. 911 T1 and hence T is not KO-categorical. Assuming that for each finite X -c 141, cl(~)is fPaite, we pDave by contradiction that for each n T has only finitely
many n-types. The theorem then follows by Ryll-Nardjewskils
theorem. There are only two 0-types since T is complete.
Let n + 1 be the least natural number such that there are
infinitely many n + 1 types. Then some n-type I' has
infinitely many extensions. Since T has only finitely many
n-types r must be principal and hence be realized in C? by
C E sl(l({al, a 1)) and both 3~~6~B) and ..., n . - 3vo(c , - B) are consistent with T', then either (C , - B) (a)or (- C ,, - B)( 61) is a finite nonempty set since 1 a) is strongly minimal. But then B( G' ) = cl(al, . . . , an). Hence there are only K(B(~))+1 1-types in Tt so r has only finitely many extensions. So for each n T has only finitely many n-types and T is * o-categoricsl. Although to the author's knowledge this result has never been published, it has probably been known for some years. A related result, also unpublished and due to Vaught, as -.far as we know, states that there are no finitely axiom- atizakLe, ~o-categorical strongly minimal theories. In 171 Morley makes the following definition. For each ordinal a and each a E N(T), subspaces Sa( ) and ~r~(a ) of
S(R) are defined inductively by
(2) p E ~r~(a) if (i) p 8 ~"(d7)and (ii) for
every map (f*: ~(8)-+ S( 0))where 8 E N(T) and f is a monomorphism from a into 8, f*-l(p) fl SO(%) is a set of isolated points in ~'(8). (See [7; p. 5191 for the definition of f*.) If i~~ is an elementary embedding of a into 8 then i* QB maps ~(8)onto S( a). Note that q E i:il *(p) is equivalent to q n Sl(l(@ )) = p. 0 An element p of ~(a) is algebraic if p E Tr (a); p is a transcendental in rank a if p € Tr (a1.
Morley defines T to be totaUy transcendental if there is an ordinal $ such that if a 2 $ andC7 E N(T) sa(a ) = $. The least such 6 is called aT. The notion of total trans-
cendence will be discussed in detail in the next section.
However, we wish to note here the value of aT for strongly minimal theories.
Theorem 3. If T is a strongly minimal theory, aT = 2.
in a, The uniqueness is guaranteed by the strong miq$mality of 1 d? 1 . \F An example to indicate that the converse of this theorem is false and a partial converse will both be
exhibited in 54. The prototype strongly minimal theory is the theory
of algebraically closed fields of characteristic 0. Other
examples include the theory of infinite, divisible, torsion-
free, abelian groups and the theory of the integers with the successor relation. Each of these theories has No
isomorphism types of countable models.
An example of an ~o-cate60ricalstrongly minimal theory is obtained by adjoining nonlogical axioms to the pure theory of equality asserting that the universe is infinite. A less trivial example was noticed by hs IS].
Let T be theory of infinite abelian groups such that each P element has order p, Then for each prime p T is an P M oategoric~let.rang1.y minimal theory. 03 8 c6tegorical Theories 1
A theory T is totally transcendental if for each modela
of T and each countable substructure 8 of S(8) is countable
Morley proved in 171 that this definition is equivalent to his
origins1 definition mentioned in 52.
! If C7 is a model of T, 8 E N(T), and 8 & I dl then 67 is prime over 8 if every elementary embedding of 8 into a model
C of T can be extended to an embedding of L7 into C. @ is a
prime model of T if a is prime over $. -. we-'wish to show that if T is totally transcendental then
for each X € N(T) there is a model of T which is prime
over X. Morley's original proof in 171 depended on his notion
of transcendence rank. To avoid this notion we must prove
from our definition of totally transcendental that for each
nodela of T and each X c (c? 1 the principal 1-types are dense
in s(x). That is we must show for each A E S ~L(x))such that 1 (€? ,X)', 3v0 A, there is a principal 1-type I' E S(X) with A E r. Suppose for contradiction T, d , X and A constitute a counter-
example. Construct a sequence of formulas in S (L(x)) as 1 follows. Let A1 be A. If An has been chosen with n t 1 choose
B E s~(L(x))such that 0,%,(A A B) and ,gl-3vo(~A - B). Let
= A A A2n B and A2n+l = A - B. For any real number a, 1 5 a 2 2, there is a 1-type I' in S(X) containing A [a],A[20~* .... Let X* consist of all x in X such that x,the name . Ofx, occurs in some A . Then X* is countable and K(s(x*)) = n 1 2 No which contradicts T being totally transcendental. Now we may apply 4.3 of [7] to obtain:
Lemma 6. If T is totally transcendental and X r N(T) then there is a model d of T prime over X.
In his proof of the two cardinal theorem Vaught [21] showed that if T ia a complete theory, D a formula in S1(~)and C7, 8 models of T such that 0 4 8, D(a) = D(B) but 1~712181then there exists a model C of T with K(C) =K1 and K(D(C))= Ho. If a theory T has such models and 8 we say T- satisfies -the hypo- thesis of the two cardinal theorem. No theory T which satisfies the-hy9othesis of the two cardinal theorem can be fil categorical.
For in addition to C by the ~6wenheimSkolem theorem T has a
K(Q) K(D(@)) 3 An obvious consequence is model a with = = 1'
Ledma 7. If D is strongly minimal in a complete K1-cate- gorical theory T and if a is a proper elementary substructure of a model 8 of T then D(Q) is a proper subset of ~(8).
Lemma 8, tet T be a complete totally transcendental theory then either some principal extension of T has a strongly minimal farmula or some inessential extension of T satisfies the hypothesis of the two cardinal theorem.
Proof. Let T be a complete totally transcendental theory and (7 a prime model of T. There is a formula D r S~(L(~)) such that ~(0)is minimal in Q. Consider all sets X c- 1 Q { definable in (7. The sets X form a subalgebra •’3 of the Boolean algebra formed by all subsets of 1 C7 1. It is clear that the Stone space of 8 is homeomorphic to the space S(0)
described in 51. But then there exists X in 8 such that X is
infinite and for no Y in 8 are X fl Y and X - Y both infinite. Otherwise there would be 2 8o dual prime ideals of 8 whence
K(S( a )) = 2 . But K( a)= do so K(S( )) = Ho since T is
totally transcendental. Choose D E S1(~( )) such that X = D( 6;then D( ic" j is minimai in ~2' . Form L1 by adjoining to L the constants which occur in D.
Let Q*'.be an inessential expansion of to sn L1-structure. 'r If IC = D( = D( 0 ' ) is strongly minimal in 0 then D is
strongly minimal in TI. The theorem IS then proved. For, T1 is a principal extension of T since any n-tuple from I realizes a principal type.
If X is not strongly minimal in then there exists an
inessential expansion 8" of an elementary extension 8' of 0.'
such that for some C E S1(8") both ~(8")fl ~(8")and ~(8")- ~(8') are infinite. Now C can be written as ct(v0, cl, ...
where C1 E Sm+l (L') and cl, ... c m are constants adjoined in expanding 8' to 8". Let I" = L U {C l... cm}. Consider all possible expansions 0''of U' to an L"-structure. Since X is minimal for each such 0" one of D(Q") n C(CZ") or D(C7 ") -
C(CF") is finite. If as " varies neither D(Q") fl C(a3") nor D(0") - ~(d')can have arbitrarily large finite cardinality then I is valid in and hence in 8. This contradicts the fact that (D,c)(B') and (D,-C)(B') are both infinite. Without loss suppose that as d" i varies D(Q "1 n c(G"') can have arbitrarily large finite cardinality Let L* = L' U where p_ is a unary relation symbol. Let i? be the set of nonlogical axioms of T. Let A be a set of sentences which are true in an L*-structure C* just if there is an elementary substructure C of C*]L" such that = 1 lcll p(C*) and such that (D A c)(C*) = (D , C.)(C1). Let A" be a sentence which is true in ff" if and only if (D , c)(Q") has cardinality greater than n. Let T* be a theory with nonlogical axioms r U A U {A~In < w 1. To show that T" is consistent it suffices to show for arbitrary k that the set of nonlogical axiomkr U A U {A" : n 5 k} yields a consistent theory Tk*. To obtain a model of T * choose such that (I) c)( (7") has k a'' ,. finite cardinality greater than k. Let 6" be a proper elementary extension of a". Expand 8" to B* by letting p(8*j = ia"j. Tnen 8* is a nodel of Tk*. Hence T* is con- sistent. Let 8% now be a model of T* then ~h(S*lL") is an inessential extension of T which satisfies the hypothesis of the two cardinal theorem. Our original proof of Lemma 8 used Keislerls result [ 3 ; 4.261 that an kt1 categorical theory cannot have the finite cover property. By Theorem 3.8 of [ 7 ] if T is K1 categorical T is totally transcendental. Thus by Lemma 8 if T is g1 categor- ical a principal extension of T has a strongly minimal formula. We now derive some properties of strongly minimal formulas in i! -categorical theories to make use of this fact. 1 Lemma 9. Let D be strongly minimal in a complete theory T. If P is an infinite cardinal there is a model C7 of T with K( 0) = dim(D( 0))= p. If Q is a model of T and n is finite with dim D(Q ) = n then there is a model 6 of T with dim(~(B))= n + 1. Proof. Suppose that D, T satisfy the hypothesis, If p is infinite by the completeness theorem there is certainly a model C of T with K(C) = rlim_[r)((l)) = y; Le+; C be a model of T with dirn(D(C7 )) finite. Let c be a new constant symbol and Tf the theory obtained by adjoining to -.. 'F- Th(Q , ~(a))the nonlogical axioms D(c) and {c # b ibr~(C?)}. Let 0 I be a model of Tf; by renaming the elements of la' 1 , if necessary we may assume that the interpretation of b in a' is b for b&~(a). Applying Lemma 4 to Th( a, D(a )) we see that Tt is complete. Again applying Lemma 4 the 1-type I' in L' = L U {c} of a point d E D(@") - cl(~(c7)U {(cb,)) is indepenaent of the choice of d and 0'. Suppose r were principal and generated by A E S1(Lt) then ~(ff')would be contained in D(C?') infinite and disjoint from D( a). This would contradict the strong minimality of D in T. Thus r is not principal and by Ehrenfreuht's theorem there is a model 8' of T' omitting r. Let 8 = ~'ILthen dim ~(6)= 1 + dim D( C?) since if X is a basis for D( 4),X U {(c)~}is clearly a basis for ~(6). Let a s, o; then 6 is a prime extension of 0 if I a1 -c 15) and every proper elementary embedding f of a into a modal C extends to an elementary embedding of 6 into C. Let a < u; then 8 is a minimal extension of if la /#IBI and C( 8 implies C =a or C = 3. Lemma 10. Let D be strongly minimal in a theory T which does not satisfy the hypothesis of the two cardinal tneorem. Let 0 be a model of T and X a basis for D(Q ) then Q is prime over X. The isomorphism type of Q is uniquely determined by the cardinality of X. Moreover a has a minimal prime extension 8 and there exists y c ~(d)- ~(d) such tkat X U {y) is a basis for ~(6). Proof. Assuming the hypothesis let a' be an elementary submodel of c7 prime over X then D(Q I) = D(Q ) = cl(~). Hence d1 = Q or T would satisfy the hypothesis of the two cardinal theorem. Suppose C is a model of T, that Z is a basis for D(Z) and that K(Z) = K(x). Let f be a bijection from X to Z then f is an elementary monomorphism by Lemma 4. Since 0 = a' is prime over X, f can be extended to an elementary monomorphism g of a into C. By Lemma 5 g(~(c7)) = D(C) so g is an isomorphism of Q and C; otherwise T would satisfy the hypothesis of the two cardinal theorem. To show that a has a minimal prime extension let C now be a proper elementary extension of a . Since T does not satisfy the hypothesis of the two cardinal theorem there exists y E D(C) D(Q ). Let C be an elementary submodel of - 1 C prime over X U ty}. Then C1 has an elementary submodel C 2 prime over X and hence isomorphic to 0. Since y # cl(~),C1 is a proper extension of a . Further if C2 8 Cl and Ic2J+llil then D(C1) = D(d) by the exchange principle e em ma 2) and so C1 = 6 since T does not satisfy the hypothesis of the two cardinal theorem. Thus C1 is a minimal extension of C 2 ' Let f be an elementary embedding of C2 properly into a model b of T. By Lemma 7 there exists y1 € (D(B) - ~(f(c~))). Let g extend r with domain g = jc2j u iyj and gi~j= YI. Since D is strongly minimal in Th(C IC2 ) by Lemma 4 g is an 2 ' elementary monomorphism. Since C1 contains a model prime over ',.- lc21 U Iy} and C1 is a minimal extension of C2, C1 must be prime over Jc2J U {y). Therefore g extends to an elementary embedding of C1 in B. Hence C1 is a prime extension of C 2 ' Now if h is the isomorphism from C, onto 2 then h can be C viewed as an embedding of C 2 properly into C. Let hl be the extension of h which embeds C1 into C. Then hl(C1) is the required minimal prime extension of a. We now combine Lemmas 8 and 10 to obtain Theorem 4. Let T be an K1-categorical complete theory; T is p categorical for every uncountable cardinal p. there are at most&0 isomorphism types of countable models of T each model of T has a minimal prime extension every uncountable model of T is saturated if a countable model of T contains a strictly increasing elementary chain then it is saturated. (vi) an elementary extension of a saturated model of T is saturated. Theorem 4 (i) and (iv) are due to Morley [7]. blorley proved Theorem 4 (ii), (iii), and (v) in [8]. Marsh proved Theorem 4 (vi) in [6]. Proof. If T is an 81-categorical complete theory then no inessential extension of T satisfies the hypothesis of the tu~cardinal theorem. By 3.8 of [7] T is totally transcendental. Hence by Lemma 8 there is a principal extension T' of T which has a strongly minimal formula 9. For any model a' of T' the isomorphism type of is determined by dim(^( at)). Thus T' is categorical in every uncountable power. A fortiori, so is T. Parts (ii) and (iii) are proved similarly. To prove (iv) consider a model a of T with K( d) = IJ > NO* Let X be any subset of I with K(X)< IJ. We must show for any p E S(X) that p is realized in a. Let 8 be an elementary submodel of Q prime over X. Then ~(6)= K(X) + No < IJ. By the Lowenheim Skolem theorem any point p & ~(8)and hence any point p E S(X) is realized in an elementary extension 8' of 8 with ~(8')= ~(6). By (iii) there exists a strictly increasing elementary chain Y+l 6 d = U idfi ( B < a) if a is a limit ordinal, and 8* = 6'. Y' a Again by (iii) using induction on y, for each y 5 6, since ~(6) < Q) is 8 CI Y K( there an elementary embedding of Y into which preserves inclusions and is the identity on 6. Thus p is realized in 4 proving (iv), Since K(s(~))= K(Q) for any model d of T, there 0 exists an elementary chain 0 < al 4 . . . of countable models of T such that an+,realizes every point of S( an) for each 21. < Q; !yher? 0 = n {Q ! rr < ! is C1eerl:rCatl?r=te5. T~ n prove (v) let 6, C be two countable models of T each containing a strictly increasing elementary chain; it suffices to show 8 'F and C are isomorphic. Expand 6 and C to models 8' and C' of Tt by interpreting the new constants in the first member of the chain in 8, C respectively. The strictly increasing 8' % C' chains force dim(~(6'))= dim(~(C'))= H o. Hence and therefore 73 2 C. Note that (vi) is now immediate. We wish to produce a proof of Morley's theorem that a countabl first order theory is categorical in power K if and only it is 1 categorical in every uncountable power which does not depend on the notion of transcendence rank. In this endeavor we use two results from Morley. In Lemma 6 we showed that 4.3 of [71 whose proof does not depend on the notion of transcendence rank could be applied without reference to the notion of transcendence rank. We also use 3.8 of [71 but Morley's original proof did not employ the notion of transcendence rank. Lemma 11. Let CP be a model of a totally transcendental theory T. Let X -c 1 a 1 have cardinality I M1. There exists an elementary extension 8 of a and a subset Y of 161 which properly contains X such that each n-type in T which is realized in Y is realized in X. Proof: By essentially the same argument that was used in the first paragraph of the proof of Lemma 8 there exists XI -c X with K(x') = M1 such that for each Z -c 1 dl definable in some inessential expansion of a just one of X' fl Z and XI - Z has power H1. (1nsteaJ. of considering all subsets of I G71 aB' in Lemma 8 consider subsets of X and replace "infinite" by I1 of cardinality HI1'.) We now construct a theory T' such that the required 6 and Y can be obtained from a model 6' of TI, Let Li be the lan- guage obtained by adjoining a new constant c to L(Q). Let T' have nonlogical axioms A where: r To show A is consistent consider fixed finite Z c I a1 and 1 - A E S~(L(Z=))such that ~(c)E A. By choice of X' there are at most Ho members a of X' such that ( 4,~). - A Since there are only KO formulas A E S1(L(z)) we have for all but Ro members a of X' and for each A E S1(~(z))such that ~(c)E A ( a,~). A(a). This shows A is consistent. Let 6' be a model of T' and 6 = 3'1~. Without loss, since 6' . Th( Gi, ( a]),we may suppose i3 is an elementary extension of a. Let Y be (c)B, and Y = X U iy}. Then 6 and y satisfy the conditions of the theorem. For, let be an n-type realized in Y by xl, x Then for each At 6 S1(~(xl,...... n-1 ' Y. n-1 ) > where At = A(v~,X1, ..., Xn-l ) either At(c) or -'A'(c) is in A. Moreover A(vl ... vn) E I' is equivalent to ~"(c)E A which is in turn, equivalent to ~({aE Xt I ( 0, {xl, . . . x n-1 1) !Z nt(a)}) = N,. But then since there are only countably many nt E S1(L(x X jj some x E Xi must be such that 1, .. . n < xl, . . . x~-~,x > realizes f . .,Careful examination of the proof of 4.3 of C71 yields the .c following: let Q be an L-structure with subset X such that a is prime over X and constructed as in 4.3 [7] then for eacn n, every n-type in ~h( ,x) realized in ( C? ,x) is principal. With this observation we can reformulate the last lemma to read: if ~h(0)is totally transcendental and K(0') 2 1 then @ has a proper elementary extension C such that for each n each n-type in ~h(6')realized in C is already realized in C7 . To see this first apply Lemma 10 with X = 1 a 1 to obtain 8 and Y. Then let C be prime over 1 a I u {y}. Theorem 5. Let T be \I-categorical for some \I > KO then Proof. Assume the hypothesis and for contradiction that T is not H1-categorical. By 3.8 of [TI T is totally transcendental. Then by Lemma 8 there is an inessential extension T' of T in a language L' extending L which satisfies the hypothesis of the two cardinal theorem. Hence there exists a model C of T1 and a formula D & S1(~') such that ~(c)= u, and K(D(C))= . Let C' = (C , ~(c)). Applying Lemma 11 and Tarski's lemma we obtain an elementary extension 8' of C1 and an X with K(X)= u, and (CtI 2 X -c 18' I such that every n-type from T~(CI) realized in X is realized in C', Lemma 11 can be applied because D(C') is countable. Now we use the idea in the reformulation of Lemma 11. Let G" 4 6' be prime over X. Suppose for cont'radiction there exists d in ~(4I) - D(C) then there exists A E S1(~'(~))generating the 1-type in L'(D(C)) realized by d. Bow A has the form ~(v~,a1 * . . . a m ) where a a E X and B E S1(~'(~(C))).Choose (c cm) in 1' ... m 1' ... lclm realizing the sane T~(c') type in C1 as (a,, .. . a ) & D does in 13'. Then B(V~'cl, ... cm)(cl) n (D(c') - D(c)) f. $ which is absurd. Hence D(&' ') = ~(c).But tnen a' IL is a model T of power in some inessential expansion of which is definable a set of cardinality # 0. dut using the completeness theorem and Tarski's lemma it is easy to construct a model of 5' of power p such that any set definable in an inessential expansion of it has power p, Hence T is not U-categorical contrary to hypothesis. This completes our development of the main results known concerning theorems categorical in power. The spirit of our approach is captured in the following observation which is obvious from the proof of Theorem 4: a theory is categorical ! in uncountable power if and only if it is totally transcend- ental and no inessential extension of it satisfies the hypothesis of the two cardinal theorem. 24 Almost Strongly Minimal Theazies A theory T la almost stronglyl minimal if there is a principal extension T' of T with strongly minimal formula Dl such that for each model Q' of T1 la'l = cl(~(c7~)). In this section~weprove a theorem characterizing such theories. We also make use of this concept to prove the prqmised partial converse to Theorem 3. If T is almost strongly minimal then T is % categorical. For, let ,.- a and 8 be models of T with K(Q ) = c(B) = HI. Let a' and 81 be inessential expansions of G7 and 8 which are models of T1. Then by Theorem 1 and Lemma 5 dl' is isomorphic to 8'. Hence is isomorphic to 8. Let T be HI-categorical and 0 a model of T. We prove some technical lemmas about the structure of S(O ). sn(d ) and ~r"( 0.) are defined in Section 2, just before the statement of Theorem 3. If T is an H1-categorical theory we can associate with any model of T a strictly increasing elementary chain ~=aoGI$ . .. by letting an+1 be a minimal prime extension of an for each n. Rote that by Lemma 10 this chain is unique up to isomorphism. If p is an element of S( 0)we say p is first realized in on if p is realized in an but p is not realized in a for each m < n. For:each n we take {dl, ... dn} to be a subset of D( an) - D( 0)which is algebraically independent in L( (I). Thus by Lemma 10 an is prime over 1 ~7 1 U {dl, . . . dnl . Lemma 12. Let T be an :tl-categorical theory, with strongly minimal formula D, 0 a model of T and p E: S( Q ) . There exists a natural number n such that (i) p is first realized in Qfi* (ii) There exists a formula A c Sn+l(L( a))such that 3v,, ... 3v,A is in p and if 8 is an elementary extension of * a* a and yl, ew., yn, b are elements of 181 such that 8 ' ~(b,yI, ... , yn) and b realizes p then yl, . . . , yn are algebraigally independent in L( 0) but {yl, , , . , yn} 5 cl(l &I U {b}). Proof. (i) Let 8 be an elementary extension of Q such that p is realized .in 8 by b. By Lemma 10 B is prime over I U Y where Y is a subset of D(B) - ~(0)which is algebraically independent in L( a). Then for some n there is a formula B E S,+l(~( 0))and elements yl, ... yn in Y such that B(V~,yl, ... yn) generates the principal type in ~h(8,1~71 U Y) realized by b. Let n be chosen as small as possible. Then an is isomorphic to the model prime over I 01 u {b}. Hence p is realized in On but not in amfor m < n. (ii) Let p be first realized in On by c and suppose dl, . . . dn are a basis for D( CIn) in L( a). Let ~(c,vl, v ) generate the principal n-type in ... n ~h((In, la, I U {c}) realized by dl, . . . , dn . Clearly 3vl ... 3vnA is in p. For i = 1, ... n there is a formula Ai E S2(~(Q')) and an integer N such that For let Ai( t,vl) generate the principal type in ~h(On, la I U {c3).realized by di and suppoae A~(C.V~)(Qn) is infinite. Then since D is strongly minimal and Ai(c,vl) (a .?5 P( cn)there is an element cv E D(~) such that C7n13vl(~i(c,vl) A v1 = c'). But then for each element e of A,(C,V&Q ,) an 1 e = ct. This is absurd so each ,.- A~(c,v~)(an) is finite and we may assume each of these sets has less than N elements. Note that in ~h(a,, (01 u {GI) Suppose 8 is an elementary extension of 0 and Y1 ' e . . . yd, b are elements of 181 such that B /2 ~(b,yl, ..., yn) and b realizes p. Since b realizes p Th(8, 101 U {bj) I= Th(0 n, la I U {c}). Then since A(C, vI, ... vn ) generates the principal t~~erealized by dl, ..,, dn which are algebraically independent in L( (7). Yp *ao Yn are algebraically independent in L(Q ). Similarly, since each di E cl( la I U {c3) each yi Lemma 13. Let T be an i(l-categoricU theory with strongly minimal formula D and0 a model of T. If p is an element of S(Q which is not realized in 0 for k < m then Proof. The proof is by induction on m. If m = 0, the result is evident. Suppose the lemma holds for eacb model of T and eachm < s. Let 0 be amodel of T and p € s(Q) such that p io not reilized in am for m < s. Then by Lemma 12 (i) p is first realized in for some n 2 s. Choose A o Sn,,(l(a 1) satisfying Lwma 12 iii). Let . .& 8 be an elementary extension of 0 such that D(B) - D(O ) is infinite... Let Bn-l 8 and suppose yl, . . . yn-l are a basis in"'L(8) for D(B,-~). Let bly b2 ... enumerate D(8) - D(Q ), Let f be an isomorphism of an into Bm determined by 3 - f (d ) yi for 15L 5 n 1 and fj(dn) = b Let 3 i - 3 ' a = ). Let q be the element of s(B) realized by a 3 3 3' Then q is not realized in Bk for any k c n-1.For suppose 3 8' 2 8 and at o Bt realizes q Then if 83 is prime over 3 3 ' of B~ into 8' But by 181 U {a.J 1 there is an isomorphism . Lemma 12 (ii) y ... y1c cl('Q1 U {a3}) so {ylY ... Y,-~} &D(d). Thus ~(d)and hence ~(8')contain at least n - 1 elements algebraically independent in ~(8).So is not realized in any Bk with k < n 1 and in particular q3 - with k C s 1 so q E ssdl(B) by induction. Since each a - 3 3 realizes p each q E iff Moreover there are infinitely 3 a B -l(PL many distinct q For if not there exsits an elementary 3' extension C of 8 and c E ICI such that c realizes infinitely many of the q C i: ) b 3 . Then B,(C yb J for infinitely many 3 in ~(8)- D( 0). But since c realizes p Bn(c,vl)(C) is finite. Thus i* -1 (p) fl sS-l(8) is infinite so p e ~'(8)which ~78 was to be proved. Lemma 14. Let T be an K l-categorical theory with strongly minimal formula D and prime model 0' such that for each model a of T I = cl(~(0) U I at1 ). For each model 0 of T if p E S( is first realized in 4 E a) m then p ~r~(C7). Proof. The proof is by induction on m. If m = 0 the -* lemma is 'cclearly true. Suppose the lemma holds for m < n and let C? be a model of T and p c; S(C7 ) such that p is first realized in C? by c. Let B e Sn+l(~(a ) and suppose ~(y~,dl, d ) generates .. , n the principal type in ~h(g, 10 I U dl, . . . dn) ) realized by . Then by hypothesis B(vO, dl, . . . dn)(a n) is finite, say with cardinality k. Let A be chosen as in 12 (ii). Let C = A ,B. Clearly 3vl, ..., 3vn C is in p. Let B be an elementa- ry extension ofa and suppose yl, ..., yn is a basis in ~(8)for D(BJ. Then c(v0, yl, ..., yn)(8,) has cardinality k so there at. most k types in S(8) which are realized in c(v0, yl, . . . , yn) (Bn). Let q e i* -L(p). Let 8' L B and suppose q is 08 realized in 8' by b. Since b realizes p there exist yi, ... y; in ~(8')such that 8' k~(b,pi, . . . , y') and by Lemma 12 (ii) is ow*, yI. are algebraically independent in L(Q ). Moreover b E cl(l0 I U {yi, ..., y;}). Thus if yi, ..., y; are not algebraically independent in ~(8)q is realized in gk for some 3 *. k < n and q $ sn(a ). If y, . . . y: are algebraically independent in L(B), let f map yi to yi. Then f extends to an isomorphism taking a model prime over y'1' ... y: into Bn. Then f(b) is in C(vO, yl, . .. , ynnso q is one of the at most k types in S(B) which are realized by a member of I c(vO, Y~,. , ynnThus for every 82 0 i* tl dB ~"(8)is finite so p E T~"(Q). Lemma 15. Let T be N i-categorical with strongly minimal formula D. Let 0 be a model of T and p E S(d ) such that p is first realized in Q m. Let C be the formula in Sm+,(L(C?)) 1( associated with p in the proof of Lemma 14, Suppose m+l c(v0, dl, ,.. dm)(Qm)is infinite. Then p E S (a 1. Proof. The proof is by induction on m. Let m = 1. Let 8 (1 and suppose D(B) - D( 5') is infinite. Let bl, b2, . . . enumerate ~(8)- ~(a).Since c(v0, dl)(C is infinite, for each j < w C(vo, bj)(B) is infinite. Thus for each j there is a qj 2 S' ( B 1 w i t h C (v*, bj.)~gj,Aa in Lemma 13 infinitely -1 2 many are distinct. Each q E i* (p) SO p E s (a 1. 3 3 Q B Suppose the lemma holds for each m < n and Q is a model of T with p E ~(6') such that p is first realized in an and c(v0, dl, . . . dn)(c7 ,) is infinite. Let 82 C1 such that ~(8)- D( 0) is infinite. Let yl, ... yn-l be a basis in L(B) Bn-l and let bl, b2 . . . enumerate ~(8)- D( for each j which is infinite consider C(vo, yl, .... Y,-~, b 3 (8) since C(vo, dl, d )( an) is. Consider the types in ... n Th(Bn, IBI U {yl, **. Yn-l 1 ) of elements of c (vO,yl. .Y,-~, .. b~ ) (8n-1 ) If some such type q is realized infinitely often in C let q i where BI; = B U { , a y Then by n-1 J q-l . . . induction q E sn: 8). If each type in S( 8:) is realized at J most finitely often in c(vOa yl, . . . , y n ) (8 .) then there are infinitely many such types in ~(8;) so there exists and c a such that c realizes Then if C1 is prime 3 fCI 3 qt, over 181 U ic i the dimension in ~iBjof ~(2:)is 8 for soae 3 S L n. Bet q be the type in ~(8)realized by c Construct 3 3' C & Si+l(~(8)) from q3 Ju8t as C is constructed from p. Then 5 as in Lemma 12 (ii) any 8' P 8 which realizes q, must contain a set of s elements in ~(8')which is algebraically independent in ~(23). So q3 is first realized in 8.. Since 2 n by Lemma 13 q r sn(8). As in Lemma 13 infinitely many 'I n+l of the q are distinct. Hence p E s (0). 'I Lenuna 16. If T is Hi categorical and for some model of T, p r S(Q) is first realized in am implies p E ~r~(dl ) for every m, then T is almost strongly minimal. Proof. Since T is H1 categorical, by Lemma 8 there is a principal extension T1 of T in a language Lt with a formula D E Sl(Lt) which is strongly minimal in T1. For each model of T let at be the natural expansion of (2 tto an L'-structure. Let 0 be as in the hypothesis. We may assume for contradiction that there is an elementary extension 8' of@' and an element b E cl(~(8')~Ia1 1. Let p be the element of ~(a)realized by b. By Lemma 12 (i) p is first realized in 0, for some n. Choose C as in the proof of Lemma 14. If C(vo, dl, . . . dn)(C7 1;) is finite,mapping ~7;l into 8' yields a contradiction. Hence p satisfies the hypothesis of Lemma 15. n+l Thus p E S ()But p is realized in CIA so by hypothesis p e ~r~(c7I). From this contradiction we conclude that for each 8' P Q and each b e I B1 I there are natural numbers n and k, a formula Ab I)) and elements dl, dn & D(8') %+l(~(a ..., But then bx the compactness theorem ~h(0I, lC7 11) models for some natural numbers N and k. But then if 8 is a prime model of T, 8 must model that sentence with the constants from 1 a 1 replaced by names of members of 181. If the set of these constants is X, Th(8,~)is the required principal extension of Let T be an l-categorical theory, 0 a model of T and p an element of ~(0). Let T1 be a principal extension of T with strongly minimal formula D. For each model B of T let B1 be the natural expansion of 8 to a model of T. Then p is naturally a member of s(G I). Moreover p is first realized in amif and only if as $member of S(a ') it is first realized in 0 ',. Furthermore for each o sa( a ) = sa( @ ) . With this observation we can collect the preceding results Theorem 6. Let T be an H1-categorical theory. The following are equivalent: (i) T is almost strongly minimal (ii) For some model d of T and every natural number m, if p is first realized in ffm then p E ~r~(a ). (iii) For every model of T and every natural number m p E ~r~(d j if. and only if p is first realized in Q m , By Lemma 12 and Theorem 6 (iii) if T is almost strongly minimal'*a'r is finite. From Theorem 6 we can further deduce the following partial converse to Theorem 3. Corollary. If T is B1-categorical and aT = 2 then T is almost strongly minimal. Proof. aT = 2 implies that for each model Q of T each p c S( a) is in ~r*((l)or ~r'(C7). Thus for each natural number n, if p c ~r~(O) then p is first realized in dn. Hence by Theorem 6, T is almost strongly minimal. The following construction shows that for each positive integer n there is an almost strongly minimal theory T with aT = n. Let T be a totally transcendental theory in a first order language L which has only relation symbols and constants. For each positive integer b we construct a theory T'~). The language L (k) is formed by adjoining to L a unary relation symbol p and Dinary relation symbols fl, ..., fk. Let C7 be a prime model of T. We will construct a prime model 8 for T (k). Let 1 BI = I@l u lalk. Without loss we may assume la1 n lalk - 0. Following Shoenfield 120; p. 18[ if p is a symbol in L and 4 is an L-structure (q)a denotes the interpretation of q in Q. Define (P)B to be lal. Each (fi)* is the graph of the ith coordinate function mapping 1C7lk onto 1~11. If q is an n-ary relation symbol in L then (Q)~= (q), U - If c is a constant in L (c)~= (c)~. Let T'~)be Th(B). Then B is a prime model of T(~)and it is easy to verify .c Lemma 17. If k is a positive integer and T is totally transcendental (i) T(~)is a totally transcendental theory- (ii) IfT is categorical so is T (k) (iii) If T is a strongly minimal theory then for each k T(~)is an almost strongly minimal but not a strongly minimal theory. Moreover a = k + 1. Applying Lemma 17 (iii) with k = 1 shows that the corol- lary to Theorem 6 could ng,t be strongthened to: if T is H1 categorical and aT = 2 then T is strongly minimal. In each of the particular # t -categorical theories T that we have considered so far there has been a strongly minimal formula. This is not always the case. There is no strongly minimal formula in the theory of algebraically closed proJective planes of characteristic zero formulated in a language containing unary relations picking out points and lines and the incidence relation. But if s principal extension is formed naming a line and two points not on the line, not only is this set strongly minimal but its closure is the entire model. H # We now exhibit an K1-but not 0 -categorical theory T which is not almost strongly minimal. For coavsnience in presenting the next set of examples we will consider languages which have function symbols. We first define a structure for the language L whose only nonlogical symbol is a ternary function symbol f: let Q denote the set of rationaLs and let Id I consist of all pairs (q,l-q), q E Q. Define F = (f)a by : where addition in I QI is point-wise. We show that every n-ary relation definable Q is & Boolean combination of relations ----R of the form: (1) R(E1, ..., 6,) * q1 El + ... + qn Ea = 0 , where ql, qn ere all in Q and El =O. We first . . . , ~j ~nqj show that if A E Sn(l) is an arbitrary atomic formula s = t then the n-ary relation defined by A in d has the form (1). This is easily accomplished by showing that for any term s containing at most the variables el, ..., zn there exist s 1, ... s & QwithC = 1, such tt-k if i is the n 15 j s n 'j 3 name of a for 1 I j 5 n then For the rest suppose that the relation defined in 0 by B is a 38 I Boolean combination of relations of the form (1) for all B in L it with length less than that of A. It is sufficient to show that A defines the same kind of relation. The only case which is not obvious is that in which A has the form 3 z B. Without loss assume that B E S~+~(L)and that z is z From the induction n+l ' hypothesis, using disjunctive normal form, we may suppose that F 2efines an (n+l) -any relation S of the form: where for each pair (r,s) we have C Q = 1 5 j 5 n+l r,s,J = 0. We have to show that the n-ary relation '1 5 j 5 n+lQ r,s,j' T on 1 4 defined by : is a Boolean combination of relations of the form (1). Since 3 and v commute we need only examine (2) when m = 1, i.e. we may suppose that (3) sK1, , 5 * : If Qs ,n+l = 0 for each s there is nothing to prove, because T(E~,. . . , cn) is clearly equivalent to the r.h.s. of (3) with all the conjuncts in which q:,n+l # 0 deleted. Otherwise we may suppose that q1 ,n+l # 0 in which case: Since C1 5 J 5 n(ql,j'ql,n+l ) = -1, in each conjunct the sum of the coefficients is zero so the induction is complete. From (2) we now see that if B E s (L) then for some n+l positive integer k the formula is valid in a. In fact we can take k = max {m,mi, ..., m;}. This proves that I Ql is strongly minimal in 0,and hence by Theorem 1 that ~h(b ) is '& l-categorical. Since I 0 I has dimension two ~h(0) is not No-categorical. It is essential to the construction of T' below that C7 have the following properties: for any @ , Y c la1 AS F (@ ,Y,&) is an automorphism of a mapping @ into Y, and if AS F ($ , Y, 5), A6 F (@ loY', 5) agree at one point they are identical. These properties may be easily verified from the definition of F given above, Their necessity below is implicit rather than explicit, # Let L', the language of T , have as its nonlogical symbols a binary predicate symbol q and a ternary function symbol f. The axioms of T' are to be such that if 8 is any model of T # then the following conditions are satisfied: i Let C = {b Ib~lBl& B(3voqbvo) = TI and = {c IcrIBI & 8(q b c) = TI then C # $ and Cb Cb (d just Also, cn u fc,la~c}= @; cb n cb, = 0 b,bl c C. (ii) There is a substructure C of 8 such that Ic~= C and CIL q , and for each b E C there is a substructure Cb of 8 such that 1 Cbl = cb and Cb ]L a . (iii) If b € C and c E Cb then hxf8(b, c, X)~Cis an iaomorpbism of CIL and Cb/~taking b into c. Further if c' E Cb then (4) fB(b, c x = fg(c, ct, fs(b, c, x), x € C. (iv) For all x, y, z E 181 (x, y, Z) & implies fB(x, y, z) = z. Since (i)- (iv) are all elementary properties, to show that the axioms of TW can be chosen it's sufficient to exhibit 8 satisfying (i) - (iv). Let 181 = 101 U (1 0 1 x 10 1) and define q+, Y) ++ : x e la1 & Y E t x /al. Further, fg is to be the unique function satisfying (iv) and for all 4, @I, Y, x E Id 1. The reader will easily verify that (i) - (iv) are satisfied. # To show that T is 8 l-categorical consider two models 1 # go, 8 of T both of power ?A whose universes arc disjoint. We carry over all the notation developed for 8 adding apprcpriate superscripts, ,,,,,C:-nc, ,,,#A- j = 0, 1 j andU'since IC(Cj ) = K(Cb) for each b E c', ve have K(C3 ) = H 1. 0 1 Hence C (L, C IL are isomorphic, by G say, since both are models of T~(Q). For j = 0, 1 and each b E C' choose Cb E c;. We extend G to all of (~'1 as follows. Recall from (iii) that for b r 2 j,~$=Axf (b,cb,r)j~'iaan Bj 0 0 f-isomorphi~m of cJ and For any c E IB I - C let b be the unique member of CO such that c E CO define b' 0 1 0 ~(c)= G&~)~(0~)- (c). Clearly G(c~)= c;(~) for each b E co, whence G is a 9-isomorphism of 8' and gl. To show that G is an f-isomorphism we must show that This is immediate from the choice of G if x, y, z E C" or if 0 for some b E CO we have x, y, z E Cb. Suppose that Y 0 0 x = b E C , y E Cb, and z L C0, then using our last remark and (4) we have: -. xnere only remains the case in whIch ix, y, 2) satisfies (ivj with rrspect to BO, in which case f O(x, y, z) = z. But since 8 respect to 8, whence f (~(x),~(y), ~(z)) = G(z). This Ua1 completes the proof of (5) and shows that T' is indeed an -categorical theory. 1 The particular model 8 of T' constructed above is prime, because it has no proper elementary submodels. This follows inmediately when one observes that the strongly minimal subset la I of 8 has dimension two, and that for any a in 10 1, a = a. By naming a point in la I we obtain a structure (8, {a)) which has two strongly minimal sets I C7t and C of different dimensions. a In [lo] Morley conjectured that if T, C characterize X < Ho then T has infinitely many algebraic types, (i.e. types realized by only a finite number of points). T, C characterize A if C is a set of 1-ary formulas consistent with T and there is a model of T in each cardinal less than X which omits C but every model of T with power X realizes C. Several 1 false in general. However we may note the following special b. s case. If T is H1-categorical and T, E characterize Hl then T has a principal extension T" with infinitely many algebraic types. For, let Tt be a principal extension of T with strongly minimal formula D and let Qt be a prime model of TI, Since T, 6 characterize K1 T and hence TI is not H0 categorical. 61 Hence D(Q') has finite dimension, Let T" be ~h(d,X)where )( is a basis for D( Qt). Lachlan has pointed out that the weakening of the conclusion to allow a principal extension is Y necessary with the following example, Let be the structure we constructed above whose language contained a single unary function symbol f and whose universe was {(q, 1-q) I q E Q). Define a structure 3 by letting / = j x / , and interpret two nonlogical symbols over this set: f again a ternary function symbol, and p a binary predicate symbol. We define % by $((aOs al), ((a1, F(al, ai, a2)) otherwise. Intuitively 4 is obtained by replacing each point in 0 by a copy of 0. T~(.B)is K1-categorical. Let B be obtained from 6 by addoining unary relations naming two equivalence classes of p. B' is H1-categorical. There are infinitely many 1-types in ~h(.4')so T~(B')is not 5 5 Proof of Vaught's Conjecture In this section we prove Vaught's conjecture that modulo isomorphism an H1-categorical theory has either just one or exactly R0 models of power Ho. It was shown in Morley [8] that the number of isomorphism types of countable models is always 5 No for HI-categorical theories. Thus we have oniy to demonstrate the impossibility of an it -but not 1 H 0 -categorical theory having only a finite number of isoaorphism types of countable models. In this endeavor we r rely on the results of Sections 1 and 3 and Lemma 17 from Section 4. Let T be an H1-but not Ho-categorical theory in a countable first order language L. Let L' be an extension of 1 m L by constants c , ..., c and suppose TI is a principal extension of T in L' which has a strongly minimal formula D'. Then Dl has the form D(V~,cl, . . . , cm) where D E sm+l(~)* 1f go1is a prime model of T' then dim(lI1 ( go7)) is finite for otherwise by Lemma 10 T' and hence T is No-cate- gorical. By forming a principal extension of TI if necessary we may assume dim(^'( ')) = 0. For each a 0 C? positive integer n let n ' be a minimal prime extension of R ' and let = n-1 Qn AIL. By Lemma 10 for each n dim(^'(^ A)) = n. An unsaturated model 8 of T can be expanded to an unsaturated model 8' of TI. Clearly 8' r n for some n whence B iZ :. Conversely each an is an unsaturated model of T . We now assume that T is a counter example to Vaught's conjecture. Thus we may choose N such thrt every unsaturated model of T is isomorphic to one of CI09 , . a . . Clearly for each n there exists an elementary embedding fn of a into aN. For each n let Xn be a basis for D1(Q')n and let arbitrary positive integer. When we write "cl" we shall mean closure with respect to L rather than L'. We show that f;(xM) cl(Z) where Z = XN U YN U fM(YM)a Suppose for contradiction that x E fM(xM)- cl(Z). Notice that aN is prime over Z because some elementary submodel C of ON is prime over Z and if l~lfw~lthenT' satisfies the hypothesis of the two cardinal theorem. Hence the type p in S(Z~ i realized by x in aNis a principal type. If cM is the name i i of f,((ci)Q ,) in GIN, i.e. if fM((c ) ') = (C ) t ":-7. .i M OM d~ then clearly D(v0, cMs ' . . . , cm)M is in p. Let X be the set of all =embers of laNl which realize p. Then X is infinite since 1 x 4 cl(z) and X -c D(V~, cM9 , , , c)( . Recall that by 1 m the choice of T' cl(yy) 2D1(Q';). Hence D(vos cM, ..., CM) (a,)- X, being a superset of fy(D1(Q 6)). is infinite. Consider the expansion a f; of a obtained by letting i (cilai = (~~1~forlTi5m. Clearly, ON" is a model cf N T1, X is definable in a{, and both D1(O{) fl X and ~'(af)- X are infinite. This contradicts the strong minimality of D' in T' ; hence fbI(XM) 5 cl(Z). Now fM (x~) is an independent subset of D'(a i) and the members of fM(yM)are named in a i. Hence, now with respect to L', cl(XN U YN) contains an independent subset of ~'(ff;.) of power M. Note that the cardinality of XN U Yg is m + N and does not depend on M. Let k = m + N and apply the construction of Lemma 17 (k) be T" to T9. Let T' . Let 8 be a model of T'! such that p(8) is isomorphic to af;by an isomorphism g. Let b be theyelement of - p(B) whose components are the elements in 1( p(B) which are identified with the elements of XN U YN by g. Then in 8, cl({b}) 2 U Y )) contains an independent N subset of ~"(8)of power M where D'' is the relativization of D' to p. Thus we have established Proposition. If Vaught's conjecture is false there exists an l-but not Yo-categorical theory TI' and a formula D" strongly minimal in T" such that for every M there exists,.a model B of T" and b E 181 with cl({bl) containing an independent subset of ~"(8)with M elements while b 4 ~(8). We now show no theory T" can satisfy the conclusion of this proposition. We again exploit Vaughtts two cardinal theor'&, Let T be a complete theory in a first order language L and p a unary predicate symbol in L such that if a is a model of T p( a) is infinite. Let T be extended to T' in a language L' by adjunction of a new constant c but no new axioms. Introduce special enki kin) constants one for each closed instantiation exactly as in [20], Chapter 4. Let A, B, B' be the least subsets of the set of Henkin constants satisfying the following conditions where C denotes A U B U B': (i) If 3vOA is closed and contains only symbols from L' and C bad if & ha8 the form -'VO .* .A. 3vo(pYo A B) ' B then the constant for 3vOA is in A. dii) If 3vOA is closed and contains at most symbols from L and AUB, and if A has the form then the constant for 3v0A is in B. (iii) If 3vOA is ciosed and contains at most symbols from L' and C, and if A has the form - pVo .A. 3v0 (- pVo A B) + 8 then the constant for 3vOA is in B' if it is not in B. It is easy to check that A, B, B' are indeed well defined by redefining them in terms of the "levels" where the constants are introduced. Let be the set of nonlogical axioms of T, Let A be the special axioms for all the special constants in C. Let O be the set of formulas I' U il U {-PC) U {-c = rlr e B). Suppose that T( O), (the set of logical consequences of O) is consistent and C* is a model of T(@ ). It is easily shown that there are substructures 8, 8' of C such that It is further easily shoun that 8 4 8' < C, that (c)~E 18'1 - 181, and that p(8) = p(Bt) = {(rlC I r E A}. Thus if T(8) is consistent T satisfies the hypothesis of the two cardinal theorem. The converse is almost immediate so we have shown Len~s18. Let T be a comglete theory in a language L and p a unary predicate symbol in L such that p( 67) is infiqite for each model G' of T. Then T satisfies the C hypothesis op the two cardinal theorem with p for D if and only if..T(0) is consistent. Below we shall apply lemma 18 to an '%l-categorical theory T in which p is strongly mi~imal. Before we can do this wenneed the following technical lemma. Lemma 19. Let T be a complete theory in a first order language L. Suppose D is in S1(~) and D is strongly minimal in T. Let CI be a model of T, al, a E 101 . . . , n , and X 5 D( a) such that a > realizes a principal Proof. By adjoining nemee it suffices to consider the case where X is empty. By induction we may further suppose X! - X is a singleton, say {a). There are two cases depending on whether or not a s cl(0). Suppose a cl.(@) and let A E s,(L) generate the type in T realized by adjoining to L(a) the constants al, ... a n and extending T by the nonlogical axiom ~(a~...., an ) is incomplete. Thus there exists c formula B E Sn+l(~)such that B(a, al, a ) is neither provable nor refutable in TI. ..., n WitlPouf loss of generality assume B(V~,al, .*.,an) A D( 51) is infinite. D(O ) fl cl(0) is a disjoint union of sets Ai(a ), each of which is finite and minimal in the sense that there is no formula C E S (L) such that 1 (C ,A. )( a)is a proper nonempty subset of A. ( CI), For 1 1 each formula Ai either Since (~(v~,a1' ..., an) A D)(Q) ie Infinite and D(Q) is strongly minimal there can be only finitely many Ai such that (ii) holds. Thus for each k k k 3'v0 Vvl, Vvn (#(vl, vn) +BA D). T' ... ,.., i Since D is strongly minimal, for some m t Since a elk) by assumption A(al9 ... an) ,. B (a,al, ) A D(a) is refutable in T' , whence - . . . , an Now suppose that a e cl(@)and choose C E S (L) such n+l that C(vO, al, ..., an)(Q ) = C and is the smallest definable subset of 16' I containing a. Clearly C is finite .....A .....A C is contained in JJ\U).,. t A\ Extend L to Li by adjoining constants a, al, a to L. Form T' by adding the . . . , n nonlogical axiom ~(a~,..., an) A C(a, al, an) to "- ... ~h(aa It suffices to show T' is complete. To show this let B(a, al, a ) be an arbitrary sentence in L1 ..., n where B E Sn+l(~). Let Eo , El , E2 be the formulas Since A(V~, v ) generates a principal type in T one of ... n is a theorem of T'. Now C is the least subset of Eo, El, E 2 16'1 definable in {al, a whichcontains a. Hence (a , ... n 1) Eo cannot be a theorem of T'. Thus either El or Ep is a theorem of T' so either B(a, al, a ) or its negation . . . , n is a tiw,g,r.emof Tf so T' is complete. This proves the lemma. We are now in a position to apply Lemma 18 to an K l-categoricel theory which has a strongly minimal ' formula. Lemma 20. Let T be a complete .K1-categorical theory in which pvo is strongly minimal. There exists a positive integer N such that if 8 is a model of T, b E 181, and D(%) fl cl(@)is infinite, then there exists c7 < 8 and X cp( 6') such that b E 16'1, K(X) c N and is prime over X. Proof. Assume that T: 8: b are given swtisfying the hypothesis of the lemma. Assume also that b & ~(8) because otherwise the lemma is trivial. By Lemma 18 T(O ) -i * is inconsistent, By the compactness theorem there is a finite subset A. of A such that replacing A by AO in the definition of O would leave T(O ) inconsistent. Choose such a AO with the additional property that if the special constant appears in a member of A then the special r 0 AO. axiom for r is in Let rl, ... r N be the special constants occurring in A so that if ri is in a lower level 0 than r then i < j. We expand 8 to C by interpreting c and J in 181 so that all the sentences in AO are true **** N C. in We first let (c)C be b and then we choose (ri)C for i = 1, N in that order. In choosing (K ) when r E ..., 3 c J A U B' we are concerned only to ensure the truth in C of the special axiom for r However, if r E B we choose 3' J (rj)C not only to satisfy the special axiom for r but also 3 such that the type of (r ) in ~h(8,Z ) is principal where J c J Z = {(ri)C I ri E A U B 15i 5:jI. This is possible 'I ,. because, as we pointed out in the proof of Lemma 6, in any 52 model CI of a totally transcendental theory the principal types are dense in s(Q ). From Lemma 19 it is clear that if we let X = I ri E A ,, 1 5 i 5 N), Y = {(ri)C ( r E B 1 9 i 5 N}, and if bl, bk is an enumeration of i . . . , Y, the#the k-type of bl, .. . , bk in ~h(8,~)is principal. Let C7 8 be chosen prime over X U Y and C7 ' 4 8 be chosen prime.over X. Since principal types are always realized in models there exists bit, . . . , b; in latl such that embedding f of Q into 12' which takes bi to bf 1 5 i C k and is the identity on X. Hence 6' is prime over X since c? ' is. Also, since the special axioms in A 0 are a sufficient subset of A to make T! 0) inconsistent no+, all of +,he axioms .- c = ri, ri E B and1 5 i 'N, are true inc. Thus b E Y and since Y -c la 1 the lemma is proved. We can now prove Vaughtls conjecture. Theorem 7. If T is an ?.$-but not Ho-categorical theory then there are H0 isomorphism types of models of T i). of power 0' Proof. If not, from the proposition proved at the '.I beginning of this section there must be an &\ -but not 1 ~o-categoricaltheory T" such that for every M there exists a model 8 of T" and b E 181 such that cl({b)) contains an independent subset of ~"(8)of power M, where D" is strongly minimal in T". Without loss we can suppose that D" is pv 01 and that in any model 8 of T", p(8) tl cl((3) is infinite. 0 This follows because dim(p(8 )) is finite if 8' is the prime model of T". Now we apply Lemma 20 to TI1. Suppose the value of N for T" is N". Choose M > N" and correspond- ing 8, b so that cl({b)) contains an independent subset of p(8) of power M. Then there exist Q 4. 8 and X -c p(a) such K(X) 5 N", b E and a is prime over X. Since p(8) n cl(@) is infinite any principal 1-type in ~h(8,x) which-"cogtains the formula pv is realized by only a finite 0 number of members of 181. Otherwise pvo would not be strongly minimal in TI1. Thus p( a)c- cl(~)which shows that dim(~(a))5 K(x). But cl({b)) contains an independent subset of p(B) of power M so p(Q) contains an independent subset of power M. Hence M 5 K(x). Since K(-X)5 N" we have contradicted the choice of M. Theorem 8. If C7 is a model of an #l-categorical complete theory T, then a is homogeneous. Proof. By Theorem 4 (iv) and (v) it suffices to consider an unsaturated model of T. Let al, a and ..., k a ai be elements of 1 1 such that al, a 1 ' . . . a . .. , k and a . . . , ak have the same type in T. We shall show there is an automorphism f of a such that f(ai) = a{ for 1 5 i 5 k. Since any model of T of power N1 is saturated it is easy to see that Th( C, {al, . . . 1)is H1-categor- ical. Let T* be principal extension of Th( a , {als . .., ak}) in which D is a strongly minimal formula. Without loss we may assume that T* comes from Th( 4 , {al, *, akl) by adjoining names a a for ak+l, "'9 a k+l ' ..., k+m k+m la I and a nonlogical axiom A(al, a ) which is true in 0. ..., k+m ~etB = ( a , {al, ... ak+mI), then 8 is a model of T*. Since principal types are always realized in models we can choose ai+l, a1 € such that C7 rA(ai, a;+m). . . . , k+m 1 a I . . . , Hence C7 can be expanded to a model 8' of T* by letting (ai) = a' for 15i 5 k+m. Now let X, X' be bases of B' i D(B'7, ~(8')respectively. Let f be a 1-1 map of X into XI and onto if possible. Since B is prime over X by Lemma LO, f can be extended to an elementary embedding g of 8 into 8'. Now if f is onto and B" is the image of 8 under g then ~(8")= D(B), whence !B"! = 181 by Lemma 7, so g is the required automorphism. If g is not onto a is isomorphic to a proper elementary submodel of a. Let < C n cur> be a sequence of models of T such that Co n I is a prime model and Cn is a minimal prime extension of C Let ho be an elementary embedding of Co in g(a ). n-1 ' For each n > 0 such that hn - l(Cn-l ) is a proper elementary submodel of g(a ) we extend hn-l to an elementary embedding 4 hn of Cn in g(Q ). Since g(0) is not saturated by Theorem (v) we have, for some N, hN(CN) = g(a). We now extend hN to an elementary embedding h into a and so on. 1?+1 Of Cg+l Again 1.y Theorem 4 (v) since CI is unsaturated there must exist M > N such that hM(cM)= a. Since CM s CN it follows that C for all j > 0. That is at most M of M+3 c~+j Co, C1, ... are nonisomorphic. But just as we proved that g( @) is isomorphic to some C we can prove each unsaturated model is isomorphic to some C Since modulo isomorphism J . there is only one countable saturated model it nos follows , . from Theorem 7 that T is do-categorical. Hence is saturated contrary to hypothesis so g must be onto. This completes the proof. * ..Notice that the reasoning given is adequate for a * stronger result in that we may replace "homogeneous" by "countably homogeneous1', i .e. the finite sequences (al, ... ak ) and (a;, ... a')k by sequences of length o. Theorem 8 is more or less equivalent to Theorem 7 since each can be deduced from the other purely on the basis of Marsh's results, Morley has observed [6; p. 161 that if some 1-type is N not realized in a modeld,of an 1-categorical theory then 4 cannot contain an infinite set of indiscernibles. We can sharpen this to: a model of an 8 -categorical theory - --- 1 -is saturated if and only if it contains an infinite --set of indiscernible~. The "only if" part is immediate. For the "if" part we continue the line of reasoning begun in the last proof where it was shown that Co, C1, . . . are all distinct and that none of then is isomorphic to a proper submodel of itself. Let X c 1 be an infinite set of indiscernibles. Without loss suppose X is maximal in the ii P set of indiscernibles. Let Xo be any proper subset of X with K(X) = K(x~).Let 8 Q be prime over X and h a 1-1 map from X onto Xo. Since % is prime over X h can be extended to an elementary embedding h' of 6 in itself, Since X is maximal h'(l~l)n (X - X,) = 8. Thus 8 is isomorphic to one of its elementary submodels. Hence 8 is saturated whence, by Theorem 4 (vi) , is saturated, $6 Proof of Morleyts Conjecture In this section we prove Morleyts conjecture that for an H1-categorical theory T, uT is finite. Our first step in this project is to introduce a concept of the rank of a .formula in a model of a theory, We will compare this notion with two other sorts of rank. then in 9.1 we defined A to be minimal in a if A(C~) is infinite and for each formula B E S1(~(G')) B ,A(U) or * -B ,A(a) is finite, We will define a notion of rank of a formula in a model such that minimal formulas have rank one. Well order the class X consisting of (-1) and the direct product of the class of all ordinals with the positive integers by placing -1 first in the order and then following the natural lexiocogr~phicorder. For each L- spa)) structure CT define fQ : X + 2 by induction A s f ( from s~(L(~)) such that the.sats..~i(Ci') partition A( a) there exists an x < Let T be totally transcendental, 0 a model of T, and A c S1(~(C)). Call a formula A rankless if A not in the range of f,. Let A be a rankless formula and let< Bol> list the formulas in SI(L(c7 )) such that Ba( &')and -~~(a)~artition A(c~)where X = r(Q) + >to. There exist an x s X such that if Ba is not rankless Ba & f (y) for y that if there is a rankless formula A in a model of T Thus ifC? is a model of a totally transcendental theopy we may define for each A € S1(~(c7))the rank of .z A(G) (the rank of A in 6') which we denote by %(A). In [6], Morley introduced for a countable first order theory T, X s N(T), and p E S(X) the concept of the transcendental rank of p. In [4] Lachlan interprets this notion in terms of the rank of a formula A in S1(L(d)) as follows We relate r a(A) to Ra (A) in the following theorem. Theorem 9. Let a be a model of a totally transcend- ental theory T and A & S1(~(C?)). 59 (ii) For some 8 an elementary extension ofa and some integer k, R8(A) = (ra(A),k) (iii) For some 8 an elementary extension o.f d To prove this theorem we need the following extension of a lemma in [4]. Lemma 21. Let T be a first order theory, G7 a model of T, A E S1(~(a))and suppose ra(A) = athen for each B < a there'exjst an elementary extension 8 of C7 such that -1 B i*08 (UA) tl Tr (8) is infinite. Proof. If the lemma is false there exists a model of T and a formula A E S,- (L( a))with rg(A) = a and some . -1 (U,) fl TrB(I3) is < a such that for each 8 0 i*Q'ag finite. Suppose q E ~r''l(8). Then for each C 8, i*8C-1(q) tl sB+l(c) is a set of isolated points in sB+l(c). But then if A E q, i*8C-l(q) tl S B (C) is a set of isolated -1 -1 points in S B (C) since i* QC (U,) = i*gc (UA) and -1 B B *"c (UA) tl Tr (C) is finite. Thus q E Tr (6) but q was chosen in TrB+l(g) so this is impossible. Hence -1 i*~~(UA) fl ~r~''(8) is empty and by induction for each -1 Y >6+1,- for each C) C? T~'(c) tl i*Oc (uA) is empty ~6 rb(~)# a. Proof of Theorem 9. (i) The proof proceeds by induction on ra(A). If rQ(A) = - 1 then iT - 3vOA and so the theorem holds. Suppose,as the induction hypothesis, the theorem holds for a formula A if ra(A) = y is less than a. We first prove that for each 8 3 R~(A) < (a + 1, 1). If not, there is some 8 2 L7 with R (A) >, (a + 1, I), Then there 1 exists a sequence of formulas (A~)~," each Ai E sl(~(B1)) such that Ai(Bl) 5 (A~A Aj)(B1) = O if i # 3, and R A = a.Now we show that for each natural number 87 J. i there is a 1-type pi E U fl ~~(6~). A,A " - Case 1) a is a successor ordinal, say a = A + 1. Since R (Ai) = (A + 1, 1) there exists a sequence of B1 formulas (Ai3 ) j A (8,) 5 Ai(8,), (Aij A Air)(B1) if J k, and i3 = @ # = A 1. Then by induction for each j r > 2 A so there exists pij E UA: a fl T~*(B~).Then for each i , LlJ since s(B1) is compact and UA is closed, there exists i an accumulation point of the p such that pi * 13' Case 2) a is a limit ordinal. a has cofinalkty o since a < w Then there exists a sequence of 1 [h]. ordinals (a and a sequence of formulas (A ) 3 ) 3- ij J (Aij A Aik)(B1) = O if j # k, R (AiJ) = (aj, 1) for each B1 j, and the a increase monotonically to a. Then by J induction r (A~~)= a so there exists a type 3 a. Since is closed and is Pij "A fl Tr J(~l). UA ~(81 ) ij i 6 1 for compact there exists pi an accumulation point of the p13 each i. But pi ~r'(8~)for any y < a so pi EU n A, Since UA is closed there exists p, an accumulation point of a+l thepigandp E UAn S (q) since eachpi 'A n ~r~(3).~ut then r.d(~)2 a +1 so i) is proved.' (ii) Now we show that there exists 8 a such that for some k RB(A) = (a,k). By Lemma 22 since ra(~)= a, for each ? < a there exists an elementary extension (2' of (2 such that Y Ll such that RC(~Yk)? (yak) and A AYj)(c) = @ if i f 3. So RC(A) 2 (a,l). Since for each 8 R~(A)< (L"+ 1, 1) by i), for some k R~(A)= (ask) and C is the required model. (iii) It remains to find B > O such that RB(A) = sup{RC(A) I C > Q 1. It suffices to show that the set of k such that for some C > Q RC(A) = (ask) is bounded. If not there exists a sequence of models Bi such that for each k < w there exists i For eachnatural number j2, for each A E Sg+l(L) and to - 1 and each natural number n assign a set of formulas as follows AT. E U U O(n) and some A! E O (n)) r A vl, ..'. v (al, ... at) by A(al, ... aR). Thus A(al, ... aR) R Theorem 10. Let T be a theory of the kind described above, a a model of T, m E {-11 U 61 A E SR+l(L), and C ' a, ...* aR 6 1 1. The following two propositions are equivalent, i) There exists a formula A* E rA(m) such that ~*!a~.... , a&,.\ ii) For some k %(~(v*,a~,..., aR)) = (m,k) if m 5 0. Notice that there is no loss of generality in this theorem because of our assumption that T has a strongly minimal formula D and that for each model 8 of T D(8) n cl(@) is infinite. For, let T be an arbitram H1-categorical theory in a first order language L. Then there is a principal extension Tt of T with a strongly minimal formula Dl. Let d be a prime model of T'. Let X be an infinite subset of ~'(0'). Then ~h(0' ,X) = T" is a theory of the specified kind. Suppose 8 is a model of T", A E Sk+l(L), A* E for some m, and -a, a, s IBI. Then B ,kA*(al, ..., at) if and only if BIL C ~*(a~,. . . , a,). Moreover R (~(v~,a~,, af)) = 81~ . . . RB (b(vo, al?...... , $)). Thus it suffices 'td move the theorem for T". Proof of Theorem. The proof proceeds by induction on m. If m = -1 diA*(al, ..., a,) for some A* E if and only if A(v0,al, ..., a,)(a) = (8 which is equivalent to Ra (~(v~,a~,..., a,)) = -1. We assume the theorem is true for m < n and prove (i) implies (ii) for m = n. Then we prove a lemma. Finally we assume the theorem holds for v m < n arrd prove (ii) implies (i) for m-= n. To prove (i) implies (ii) consider a formula . A E S,+l(~) and a formula A* E rin) such that a ~*(a~,. . . aL) with Ebl, . . . at E 1 a 1. Notice first that it suffices to prove the case in which A* E @!"I. For, suppose the (i) implies (ii) has been shown for each integer (n) g, each A E S,+l(L) and each A* s (") and that A* E OA , Then since QpA*(a1, ... a,), ~(v~,a~,. . . a,) ( (7) = - U (~,(v~,a~,. a (0))for some a,+l, ... ak in (Ql i-1 . and some A,., ..., A . Moreover for each i satisfies S n-L A2 (al, ..., ak) and each At E ,, $n-l) U @Ln) and some i=l Ai i SO for each i there exists ni 5 n and a ki such At E Ai that Ra (~~(v~,a~,... aQ)) = (ni,ki) and for some i there exists k such that Ra (~~(a~,... at)) = (n,k), by induction and the assumption that the theorem holds for each (n) B* E QB . But then RCI (A(al, ... at)) = (n,m) for some integer m. Thus to prove (i)implies (ii) when m = n, let A E Sa+l(L) and suppose0 /I ~*(a~,. .., a&) where A* E min). Lettins A' = A(v0,al, ... , a&) we wish to prove that for some q %(Aq) = (n,q). From the definition of @Ln) we see A* has the form . . * where p is a positive integer, k 5 k < W c is in Sk+2(~) and C* is in r(n-l). Since C ~*(a~,. . . , ak) there exist ak+l, , a k E la I such that for all but p elements b of D(g ) Q Cr(al, a b). Thus for any and 1: 1: . . . k' al> a d E D(Q,) - D(O ) 0, c*(a1, ..., ak' d) since D is strongly minimal. By induction for some s R (C'V~+~(~)) a1 (n-1, s) where C * = C(vO, al, a v ). Then R?(A') . . k' k+l is 5 (n,s). For, if not there exist L-formulas B1, ..., where each Bi has free variables vO,v~+~, ,, v m with the i following properties. There exist constants ak+*, ..*, R (B') 2 (n,l). We will show that this condition implies a i for each elementary extension 0 of 0 , each d E D(Q,) - D(Q ), and eachithatR (Bi ,C; (d)) 5 01 k+l -11. This in turn implies R (c; (d)) 2 (n-1,s) 01 k+l which is a contradiction allowing us to conclude that Suppose R&B~) 2 (n,l) and for some O1 0 and some d E ~(a,)- ~(a1 % (~iA C' (1) < -1BY 1 Vk+l A for some induction there exists a formula (Bi c)* & I- BiAC() i i r < n-1 such that Q (B~,c)* (al, . . . ak, d, ak+2, . . .am). Since D is strongly minimal, there exists p, € w which may - be assumed larger than p such that for all but pl members of i D(Q) dl (Bi A C)* (a1, '0 ak. b, ak+2, *** a:). Consider the fgrmulas i i NOW F* c rF(r+l) and 0 Fx(al, . . . ak, . . . am) so if Ft i i is the formula F(v0 ,al, . . . , ak' ak+2 ' . . . am) by induction there is an integer q such that R &FI 1 = (r+l,q) < (n,l). For each element c E B~(o)there exists an element b in D(c7) such that a 1: Bi(c) , Ct(b,c) A c*(al, . . . ak, b) since and c7 /Z ~*(a~, ak). Let bl, b be ~l(a)&A'(G: ...... 9 an enumeration of the elements b & D(Q) such that I a k. c*(al, ..., ak ,b) A - (B~A c*) t i (al, . . . ak, b, ag+l, . . . We know there are only finitely many such b from above. Then % (B; A C; (b)) 5 %(C$ (b)) = (n-1,u) for some u < w k+l k+l q by induction. ~ut4I Vvo(~ict F' v IJ (Bi ,CV (bJ ) s jE1 k+l So B;(Q) is the union of a finite number of definable sets each with rank less than (n.1) and thus % (B;) < (n,l) contrary to assumption. Thus we conclude as outlined above Ra(A1) - - '(n.8). Since VvO? k+l (c' ), R~(A".)z (n,l) . Therefore there exists an !I, 1 5 I, 5 &Isuch that R~(A') = (n,I,). We have shown (i) implies (ii) when m = n. Lemma 22. Let QbT A ~s&$L) a,, ... ae c lal, A' = A(v~,~~,... a&) and a 5 w. Suppose the Theorem holds " - ,+ for each m < a and that for each B>Q there is some k such that R*(A') = (ask) then there exists r < a and Proof: Adjoin a new unary predicate symbol q to L to form L' and a new constant symbol f to L1 to form L1. Let A be the set of L1 sentences which are true in an L' structure C' just if there is an elementary eubstructure C* of the reduct of C1 to L such that lC*l = q(Ct). Let D~ be the L1 sentence 3'nvO(~ ,- q). Let rl be the set of sentences {elementary diagram ofd) U A U {~~(n< w). If k < w and F c Sk+*(L) consider the following formulas. Let m = I, + k. Let F1 € Sm+*(L) be the formula F(V~,v~+~, v v ) A A. . m ' m+l Let G(F,F~*) = ~V,+~(D(V~+~)A Fl A F~*). Let G* (F ,F~*,p) be (VV,(G(F,F~*) %-+ G(F,F~*))) =P A (VV,(G(F,F~*) + 3 V~+,(D(~~+~)A F1)) Then if F,* is in , (s +1 A G*(F,F~*,d. P) is in rG(F,p* 1 #!* Let r2 be the set of sentences Pa+l, -** bm, E la I} Now we show that r2 is inconsistent by finding for each L" structure C" such that C" )I fl, for each element f E (A' A - q)(C1) formulas F and F1*, an integer p, and constants c c such that R+1' . . . m L-structure. Let C1 be an L-structure prime over U {f}. Then D(C1) - ~(8)# @. For, suppose D(C1) 5 D(8) and let B1 be prime over D(C1). (B1, C1 exist by 4.3 of [71.) Then = 8 for if not B1+ C C1 while D(B~)= D(C~)But then B1 1 and C1 are models of T which satisfy the hypothesis of the two cardinal theorem so, as in section 3, T is not K1 categorical. Thus there exists d E D(C~)A ~(8).Let C E Sk+2(~)and c 1, C 0 Ck, v ) generates the . . . k €1 I such that ~(f,cl, . . . k+l principal 1-type in ~h(c,lal~{f)) realized by d. Then finite. For not, since D ~tr0nglyminimal and contai.ns infinitely many ~lgehrnicpoints . - there exrsts an algebraic point b € Id ) such that C Since algebraic there exists formula B c S1(L) and an integer t such that C et ~(b), 3- vo B. But since C ~(f,cl, ... c , b), 9 C, ) generates a principal type and c, . ck ' v k+l )(C) ~(f,cl, ... c kD vk+l is infinite, B(C) is infinite. So for some q < w Let the following member Let Ci be obtained from C1 by substituting al, ... a& for v1, . . . vg and cl, . .. c k for v&+~,. . . v rn . For any b E D(C) - ~(8)R (c' (b)) = (d)) since any such lv m+l Vm+l b realizes the same 1-type in ~h(C,{a~,... a&, cl, . . . c& as d and C is homogeneous by Theorem 8. Since D(C) - D(8) C is infinite I,:Yv 03-qvm+lC'; thersf ore if RC(CI (d)) r (a,l) then RC(At) 2 (a+1, 1) contrary to v m~1 *. hypothesis, So for some u < a and some k R (C (dl) = c lv . m+l (u) (u,k). Thus by hypothesis, there exists a formula Cf E I- C1 such that Cf(al, at, cl, c d). Let p be the C 1.- . . . , . . . P' *. maximum of q and the cardinality of F C;(al, . . . ae, cl, ... ck)(CV) which is a finite subset of D(C1'). Then so C" does not model I'2 but C" was an arbitrary model of r1 so r2 is inconsistent. By the compactness theorem there S i* for exists k E w F 1 , ..., F in Sk+2(L) and Fl & r %? some t i < a such that .a*, list the constants occurring in some Fi and are cl, ck assumed to occur in each Fi for notational convenience. If (A' , .. B1)(<7') is infinite then there are models of T of arbitrarily large cardinality with (A' n - ~')(8)- (A' n - B') ( a) # $. Thus there is a model C of rl with (A1 , - B')(C) - q(C) 3 @.But this is impossible. So where u = max(ui) < a. We return to the proof of Theorem 10. The induction hypothesis asserts that (i) is equivalent to (ii) if m < n. We have already proved (i) implies (ii) if m = n and now we -- - wish to sdow (ii) implies (i) if m = n. Suppose A & S1+l(L), a, ... at E 1~1,A' = ~(a,, ... a,) and for some k % (A') = (n,k). The definition of BA(n) allows us to assume that k = 1. We will find a formula A* E rln) such that 6' A*(al, . .. at). We first assert that there is an elementary extension of and a formula B1 & S1(~(8))such that ~~(8)~'(8) R8(Bt) = (n,l) = sup {RC(Bt) I C B}. To see this construct the following sequences of formulas and models. Let B0 = 3 and Bo= A'. Given Bm and Bme S1(~(Bm)) choose Bm+lby theorem 9 such that I? (ern) = sup {R~(B~)I C> P m 1. 8m+l Choose Bm+l such that Bm+l (8m+l) 5 B,(B,+~) and R (Bm) = 'm+l .(A,1) > (n,l) for each C RC(Bm+l ) R~(B,) = (A,&). Hence there exists p < A, there exists q such that RC(BmA ) = (u,q). But then since R (B A-B~+~)5 'm+l m (u,q) and(n,l) < (A,&) R (Bm) < (A,&) which is a 'm+l contradiction. Since there is no finite descending sequence in a well ordered set, for some k R B = 1.Let Bk be %+I-- - the formula Bt = B(bl, . . . b,) where bl, . .. bs o !Ski and let Now Bi and 8 satisfy the hypothesis or Lemma 22 so there exists (k+l) B* E rB for some k < n such that 8 1:. 13*(k1, . . . b,). If k < n-- 1 by the induction hypothesis Rs(Bt) e (n,l) so k = n - 1. t 8 B*(bl, . . . bs) ,Vv0(~(bl, . . . bs) + At) and 8 is an 0 elementary extension of so for some cl, . . . c s & 1 1, 0k B*(cl, ... cs) ,Vv0(B(cl, ... c s ) + At). Since B* E rin), and ill ' we have proved (i) implies (ii) for m = n, for some R % (B(cl, . . . cs)) = (n, Q. L nust equal 1 since B(cl, . . . cs)@ ) -c ~'(0)and Ft,dAt) = (n.1). If C' = C(vo, al, ... all, by induction there exists C* E q1 rhJ) such that J=-1 a!.:c*(a,, . . . a&, cl, . ., cS). Hence letting A B* A C*) A* is in rln) and 0 ~*(a~, a ) proving the theorem. \ ... 11 Recall from Section 2 that a, is defined to be the least 'I ordinal such that for all BE N(T) and f3 > a, Sa'( a = sB( 0 ). I M 'O+ In [TI Morley proved aT exists and is less than (2 ) for every complete theory. In [h] Lachlan shows that aT < wl for each complete theory. We apply Theorem 10 to prove the following condecture of Morley. Theorem 11. If T is HI categorical then aT is finite. Proof. If for some 4' and some 6 I w there exists B p E S (01,then since T is totally transcendental for some Y b 6, p E ~r'( a)and by lemma 21 there exists 8 a,qE ~r~(8)" -1 ih (p)so there is a formula A' = A(V~,al, . . . at) in Qs S1(~(B))with r8(A') = w. By Theorem 9, there exists C F 8 and an integer k such that for every elementary extension C1 of C iiC(&)= (u,k). Now by lemma 22 with a = w, there exists an n < w and a formula A* E l"Ln+l) such that C I ~*(a~,. . . a&). By theorem 10 for some k RC(~')= (n+1, k). This is a contradiction so there is no 0 and no B 2 w and no p with B 3 E S (d). Hence 3 T < w. 97 A Note on Definability Let L be a first order language containing a unary predicate p, a an L structure, and B = p(Q ). Recall that denotes the set of k-tuples bl, ... bk such that 6' the form A(al. ..., an)(c7) ll B an A-relation. We prove undex certain conditions on ~h(0) each A-relation is definable + by naming a finite number of constants from B. There is no assumption in this section that L is countable. For simplicity in notation we assume n = k = 1. Each, . . A relation on an infinite set B can be represented by a pair (h,q) where h is a bijective map of K(B) into B and TI E 2K(B'. Thus A(i) is in the A relation if and only if ~(i)= 1 where i < K(B). There is a natural equivalence relation on such representations produced by calling two representations equivalent if they represent the same relation on B. Ambiguously we denote both the equivalence class containing (X,q) and the relation (A,TI) represents by (A,. We also consider pairs (k,h) where !& E B~ and h E 2n for some n < o. (ash) is a partial representation of To simplify notation in the following formulas we assume that in addition to the sequence of variables vo, vl ... L contains an infinite set of variables Z where a E 2•‹ and 4n a I n is the initial segment of a of length n. We order such finite sequences by length and among those of the same length lexiocographically. If q is the last sequence of length n under this order the prefix 3z0 32 will indicate that ... P each variable zalm, m c n is being existentially quantified. If A is a formula and a E 2@ then A a(i) - uiij = i and A aw= -A if a(ij = 6. Let Cn be the formula 3z0, , 32 . . . 9 .. . n-1 ( A (avo( A A(vo, zali )a(i)) where q is the last s equence as2" i=o of length n in the order described above. Then if Cn is true in T for each model of of T there is a complete tree of length n of distinct A-relations. k Let f, be an element of B~,X = range and h c 2 for some positive integer k. Define H" E S*~(L(X))to be 0,h) where q is the last sequence of length n. Define [(A,n) 1 to be class 1 if 0 HI(a,h) for some partial representation (8,h) of [(h,~)]. For n > 0, define [(A,Tl)l to be class n if [ (A,I?)] is not in class n-1 and for some partial representation (L,h) of [(X,q)]. Lemma 23, If [(A,n)] is in class n for n c w then [(h,n)] is definable in (a, X) for some finite X 5 B. Proof. Let (Jl,h) be the partial representation of [ (A ,n)] such that 0 k -H" Suppose that !?, E B~ and (Jl,h)* k h € 2 . Let X = range L. If n = 1 let G be the formula k-1 A ,,, A(vO, k(i) h(i). If' n > 1 let G be the formula where q' is the last sequence of length n. Since [(A,q)] is not in class n-lif [(A q)] = ~(a)(o) fl B clearly 01. ~(a).Thus, if we can show there is but one A-relation [ (A' ,n' ) 1 such that [ (1' ,173 ] = ~(a')(d) ll B and a $ ~(8')we can define [(A,n)] by 3v0 G(v0) - A(vO,vl) - P(vl)* Suppose there exist A-relations [(AL,t')l)] and [(X2,q2)] and elements al, a2 E 10 1 such that [(A1. ql)l = ~(a,)(a)n B, 1,1# [(A2,n2)l, and@ Idal) ~(a~). Then there is some element b E B such that Qb A(al,b) , 1 -A(a2,b). Then &'bH(Q,h) which is a contradiction if n = 1. If n > 1 let 1, = Q U XI-1 n-1 0 1- H(k ). Since a k G(a2), a 1:. H(& , . But then 1' 1 2 2 This contradicts our hypothesis and the lemma at=%,h) In [19] Shelah defines a complete theory T to be stable if for each n + 1-ary predicate R the following set of formulas t is inconsistent. (a) denotes the length of Cj , a sequenze of zeroes and ones. (Y~S-*- Y i is an n-tuple of variables indexed by a 1 i. In our context we can conclude that if T is stable r A is inconsistent where Theorem 12. Let a be a model of a stable theory in a language L and B be a subset of I I definable in L. Any relation on B which is definable in (a ,la I ) is also definable in ( C7 ,B). Proof. Since T is stable r A is inconsistent. By the compactness theorem for some nQlr. Cn. But then for every ( R,h) a/:, Hence each [(A ,rl)l has Class 5 n and the theorem follows from Lemma 23. We can invoke the compactness theorem once more to produce the following uniformization of Theorem 12. Corollary. If T is a stable theory in a language L, p is a unary predicate in L, and A a binary predicate in L such that 3 vO V vl(~(vO,vl) + p(vl) there exist integers N, M and formulas C1, CM in S (L) such that . . . N+1 58 Related Results In this section we list some recent results on problems related to the subject of this thesis. A natural generalization of the &&s conjecture is: if T is a complete theory having infinite model in a language L such + that there are K sentences in L then T is categorical in if and only if T is categorical in every X > K . Special cases of this theorem have been proved by Rowbottom [15] and ~essa~r'e(131. The general theorem has been proven by Shelah [171. In [4] Morley conjectured that if K(T) =A> and T is A categorical in A then T has a model with power less Hthan . Shelah (181 has proven this conjecture in the case = A. Harnik and Ressayre [14] prove the following theorem. T is categorical in K~>K (T), if and only if every set C K 0 of power 0' which is a proper subset of a model , has a prime extension which is an elementary submodel of a . The notion of stability and many other concepts related to categoricity are explored by Shelah in [18]. BIBLIOGRAPHY 80 IN Je T. Baldwin and A. H. Lachlan, On Strongly Minimal Sets, to appear Journal of Symbolic Logic. 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