Zeilschr. f. math. Logik und Grundlagen d. Malh. Bd. Z i , S. 4 1 1 - 4 1 8 ( 1 9 8 1 )

A CLASS OF KO-CATEGORICAL THEORIES

by ANAND PILLAY in London (Great Britain)

Introduction We define a class of s,-categorical theories which we call very simple theories. We

then show that such theories are o-stable with finite 0 1 ~ and that they are not finitely axiomatisable. We also prove that these theories have finite fundamental order (in the sense of POIZAT [4)). We finally observe as an example that an No-categorical theory which is o-stable with aT = 2 is simple and not finitely axiomatisable.

Our motivation is the attempt to give a formalisation of the notion of an N,-cate- gorical theory for which the isomorphisms between its countable models are essentially obtained from arbitrary 1 - 1 onto maps between infinite sets. Here, however, we have only an approximation to such a formalisation, in so far as we abstract from the case in which there are nontrivial dependence relations. Thus for example the theory of (Z,,)”’ will not be very simple, although it is clearly simple.

In general one observes that an No-categorical theory is either K,-categorical because of lack of structure (e.g. theory of equality) or because of presence of structure (e.g. dense linear orders). In the first case only set theory is needed to get the isomorphisms between the models, whereas in the second case some nontrivial back-and-forth argu- ment is needed. The latter sort of theory is clearly more interesting. In particular, can such a theory be stable? However, we are here interested rather in the former sort of theory.

Notation here is standard. Any theory we talk about will be countable and complete.

1. Trees

In order to facilitate our definition of very simple theories we first define a certain class of trees and their automorphisms.

Rcoall that w denotes the set of sequences of natural numbers of length at most m. We will be interested in trees P for which there is m < (11 such that P wCTIL. (I-Iow- ever the lexicographic order of the tree will not concern us.) If P is such a tree and 7 E P then we denote by P, the set of a such that f a E P. q 4 (T means that rj is an initial segment of (T.

for some m) which also satisfy the following conditions: (i) P is infinite, (ii) P is closed under initial scg- ments, (iii) if 7 E P and f ( i ) E P then f ( j ) E P for all j < i, (iv) if q E P and $“i> E P for all i < (0 (or equivalently for infinitely many i, by (iii)) then P,,-<l> = P,,-(,> for all i , j < o.

If y~ E P and f ( i ) E P for only finitely many i we say that q has finite branching. Otherwise 7 has infinite branching. If 17 E P and has no extension in P we call 11 an endpoint of P. If P E K then by (i) a t least one element of P has infinite branching. If n < (0 then PI denotes the set of elements of I’ of length n and P S I ) the set of de- nients of P of length a t most n.

Drf in i t i on 1. R is the class of treesP as above (subsets of

412 ANAND PILLAY

We now fix some tree P in K . We want to define a certain class of maps from P to itself which we will call good maps. Suppose that P 5 cosiiz. We will define by in- duction for n 5 m a class F,, of maps which have domain PSI1. Let F, be the singleton consisting of just the map taking the empty sequence to itself. Suppose now that F,, is defined. For each u E P,, which is not an endpoint let C, be the set of permutations of (I) if c has infinite branching, and otherwise (if (I has finite branching) let G , consist just of the identity map on OJ. Now let H be the set of choice functions h such that for u E Ptl which is not an endpoint h, = h(o) E G,. Now for each f in F,, and h in H let f' = f ' ( f , h) be the map withdomain PStr+l which is defined as follows: if u E P z " then f'(o) = /(a), and if o = qn(i), where q E Pi', then f'(o) = j(q)nh,,(i). Now F,,,, is the class of maps f ' ( f , h) above as f and h range over F,, and H respectively.

For P in K and the Fi:s as defined here we have the following:

Lemma 2. Let f E F,, and o E PSI1. Then f(o) E P , lhfa) = Ih(f(o)) and P, = P,(,). Proof. By induction on n. Suppose the lemma is true for n. Let f E F,L+l, and

o E PS1l+l. We may assume that u E Pll+l, for if not then by the definition of F,,+l there is fl in Fit such that f (o) = fl(a) and we use the induction hypothesis. So let o = qn( i ) , where q is in Pn. By definition there is fl in F, and h E H (where H is as above) such that f(o) = fl(q)"h,(i). By induction hypothesis,

(*I f l ( r l ) E Prl and P,, = Pflc,,, . If 7 hasfinite branching then h,,(i) = i , thus from (*) f(o) E Pn+l. Moreover P, = P,-(,> = = Prl(,,, - < 1 > = Pfco,, using (*). On the other hand, if q has infinite branching then by (*) 80 has fl(q), and thus f(o) = fl(q)"h,,(i) E P+l. Also, using Definition 1 (iv), P, = Pqyl> = Pfl(,,)-<c> = Pflcs,-hnc,, = P f o . So the lemma is proved.

Now let m be the height of P. Putting F = F , we see from Lemma 2 that f is a class of maps from P to itself which preserve length. We call the maps in F good maps of P. In fact, it is not difficult to see that F is a group of automorphisms of the structure (P, -a, P J I ) ~ ~ ~ This can be proved by induction as in the proof of Lemma 2. Thus we can talk of the orbits of P relative to F . Thus 7 and o are in the same orbit if there is a good map f such that f(q) = o.

We state the following lemma without proof.

Lemma 3. Let P be a tree in K . Let X and Y be subsets of P such that for all q in X and o in Y , q and o are incomparable (with respect to -a). Let f be a good map of P such that f ( X ) E X and f ( Y ) g Y and let X' be a subset of X . Then there is a good map g of P such that g(q) = q for all q in X' and g(o) = f(a) for all o in Y .

If X is a subset of P then the set F x of good maps which (pointwise) fix X is a group of automorphisms of P.

Corol lary 4. Suppose P is in K and X is a finite subset of P. Then the number of orbits in P relative to Fx is finite.

Proof. Let m be the height of P. We show by induction on n 5 m that Pn is divided into only finitely many orbits relative to F-y. Suppose true for n < m. To prove it for n + 1 it is enough to show that for each 7 in PI the set {f(q)n<i): f E Fy and f ( ~ ) ~ ( i ) E P} is divided into finitely many orbits relative to F x . Remember that P,, = Prc,,, for all good f. If 7 has finite branching then f(q"(i)) = f ( ~ ) ~ ( i ) for all f in FA\ and we clearly finish. So suppose that 7 has infinite branching.

A CLASS OF R~-CATEGORICAL THEORIES 413

Case 1. There is no x E X such that r j -a x. Then also for cvery f E Fx there is no x E X such that f (q) 4 x. Remembering the original definition of the class of good maps, thereis for each f E FAY and i , j < o some good map g such that g(r f ' ( i ) ) = = f(r)"( j) and g(o) = /(a) for all o of length 6 rL. Let us denote by S the orbit of r j under maps in F,\ and by S' the set of elements of P which are comparable to elements in S. Then S' is closed under g. If x E X n S' then by case hypothesis x is of length 5 1% and is thus fixed by g. Let X ' be the subset of X consisting of those elements not in 8'. Then by Lemma 3 there is a good map g' which fixes X ' and agrees with g on AS'. Thus g' E F y and g'(f(i)) = f(rj)"(i). This shows that all elements of the form f(rj)"(i) for f E FA and i < w are in the same orbit relative to F.\.

Case 2. There is x E X and r j u x. Then clearly f ( r j ) = q for all f in F\ . So we just need look a t the number of orbits that the set {rj"(i): i < w} is divided into relative to F-, . As X is finite there is a finite set W w such that there is x E X with rj"(i) 4 x iff i E W . By a similar argument to the one in Case 1 we can find for each i, j not in W a map g in FA such that g(r j " ( i ) ) = q"(i). Thus again there are a finite number of orbits.

The following is an easy consequence of the previous two results:

Corol lary 5 . Let r j E P, where P is in K . Then there i s r < w such that if X is a u a} finite set of elements of P each of which is incomparable to r j then the set {o E P :

is divided iiito r orbits relative to F s (the good maps of P which f i x X ) .

2. Very simple models and Morley rank

Let M be a model in a countable first order language. By a definable stib.set of X wc mean a subset of M that is definable in M by a first order formula possibly with para- meters in M .

Defin i t ion 6. We say that M is a very simple model if there is a tree P in K and for each r j in P a definable subset A,, of M such that (i) A, , is all of M (the universe of AT), (ii) if q is an endpoint of P then A, is a singleton, which we write as {a,), (iii) if q is in P and is not an endpoint then {A , , -< (>: rj"(i) E P } in a collection of dairwise disjoint sets whose union is A,,, (iv) if f is a good map of P then the mapping f defined by f(a,,) = af(,, for an endpoint rj of P is an automorphism of M .

Remark . By (i), (ii) and (iii) every element of M is of the form a, for some end- point r j of P. Thus (iv) makes sense. Also if / is as in (iv) then for any r j in P the image of A,, under f is clearly Af( , , . Note that M must be a countably infinite model.

Def in i t ion 7. A theory T is said to be very simple if there is a very simple model M such that T is Th(M).

Examples . The obvious example of a very simple theory is the theory of a finite number of nested equivalence relations El, E,, . . . , E,, , each of which has infinitely many infinite classes and such that E,+l infinitely refines E , . In this case the cor- responding tree is just wsn+1.

However by means of finite branching one can express some simple relations of dependence. For instance consider the theory which says that R and Q are subsets of the universe which are disjoint and whose union is the universe and that f is a bijection between R and Q. The corresponding tree is the set of r j in o>c2 such that rj(0) is arbi-

414 ANAND PILLAY

trary and ~ ( 1 ) is 0 or 1. Then if we list R"' as {aL: i < w}, we have A,,, = { a , , f (aL)> for each i < w, A<,,o) = {a,> and A(, , ,> = ({(ai)>.

F a c t 8. A theory T i s No-categorical if and only if there is a model M of T such that for any finite sequence ii of elements in M, Th(M, ii) has only a finite number of 1-types.

We note that this follows easily from the usual RYLL-NARDZEWSKI characterisation.

P ropos i t i on 9. If T i s very simple then T is No-categorical.

Proof . Let M be a very simple model of T and let P be the tree corresponding to ~ 1 1 according to Definition 6. Let 6 be a finite sequence of elements of M. Then a is a, , , a,,, . . ., a,,% for some endpoints ql, qz, . . . , q,L of P. By Corollary 4 the endpoints of P are divided into a finite number of orbits relative to good maps which fix ql, . . . , T , ~ . Thus by Definition 6(iv) M is divided into finitely many orbits relative to automorphisms which fix ii (pointwise). But if x is mapped to y by an automorphism of ill which fixes 6 then x and y have the same type over 6 in M . Thus finitely many distinct 1-types over 6 are realised in M from which it follows that Th(M, 6) has only finitely many 1-types. So T is No-categorical.

We recall the definition of Morley rank for formulae. If M is a modcl and 9 is a formula with parameters in M then we identify pl with the subset of M which it defines. Let M be an w-saturated model of T. We define a rank on the definable subsets of M . Rank(A) 2 0 if A is nonempty. Rank(A) 2 a + 1 if there are definable subsets A, of A for i < (I) such that A, and A, are disjoint for i + j and Rank(A,) 2 01 for all i. For 6 limit Rank(A) 2 6 if Rank(A) 2 a for all LY < 6. T is said to be w-stable if Rank(M) is defined. In this case 01r is defined to be Rank(M) + 1.

Propos i t i on 10. Let T be a very simple theory. Then T is o-stable with finite 0 1 ~ .

Proof . Let M be the very simple model of T . Then as T is No-categorical M is already w-saturated. We show that Rank(M) is finite. Let P be the tree in K which corresponds to M. We show by downward induction on length of q in P that the dcfin- able subset A,, of M has finite rank. If lh(7) = m = height of P then A , is a singleton and so clearly Rank(A,) = 0. Now assume that lh(7) = n. We may assume that 7 is not an endpoint of P. If 7 has finite branching then A,, is the union of {A,,-<,): i < k } for some finite k. By induction hypothesis Rank(A,-(,>) is finite for each i. It is easy to see that in this case Rank(A,) is also finite. So let us now assume that rl has infinite branching. For each i , j there is a good map of P taking f ( i ) to f ( j ) . Thus from Definition 6 there is an automorphism of M which maps A,,-<,) onto A,,-<,>. By induction hypothesis and as rank is clearly preserved by automorphism there is k < w such that Rank(A,-(,,) = k for all i < co. As the Avn(,) partition A,, it follows that Rank(A,) 2 k + 1. We will show that Rank(A,) is exactly k + 1. Suppose for contradiction that Rank(A,) > k + 1. Then there are definable subsets of M, say X , for i < w, such that X, j, and Rank(X,) 2 k + 1 for all i . We first observe that for each i there are an infinite number of j such that X , n (A,-(,)) is nonempty. For if not, then as the A,-<,, partition A,, X, would be a subset of the union of finitely many of the A,-(,, . But this implies that Rank(X,) = k , which is a contradiction. Now consider some X , . Let X, be defined by a formula with parameters from the finite set S, say. Let jl, j z , . . . , j r be such that X n A,, 5 i r ] . From Lemma 3 and Definition 6 i t follows that for each j, j' which are

A, for all i , X , n X, = 0 for i

A CLASS OF N,,-CATEUOHICAL THEORIES 416

different from jl, . . . , j r there is an automorphism of M which is the identity on S and which maps A,-<,) onto A,,-( , , ) . The subset X , is mapped onto itself under this automorphism. Thus we see that in fact

(*) for each i there are cofinitely many j such that X , n A,-<,> is nonempty. Now as for each j, j ' there is an automorphism taking A,-(,) onto A,-<,,) we have by Corollary 5 and Definition 6 the following :

(**) there is d < cr) such that for each j < 0 : if S is a finite set of elements of ill none of which are in A,-(J) then A,- ( , ) is decomposed into a t most d orbits by automorphisms of M wlaich fix S .

Now consider X, , X, , . . . , X d . Let S be a finite set which contains all the parameters appearing in the formulae defining these X , . By (*) there is j such that S n A,, : J ) is empty and X , n A,-(,) is nonempty for i = 0 , 1, . . . , d. Now let i,, i, 5 d , a E X , , and 6 E X l 2 . Then, as X,l n XtZ = 8, a and 6 have different types over 8. Thus no automorphism of M which fixes S can take a to 6 . Thus we have at least d + 1 orbits in A,-<,) under automorphisms of M which fix S. This contradicts (**). Thus this proves that Rank(A,) = k + 1. Thus Rank(A,) is finite for all 7 in P and so we have that rank is finite in the universe of M , which proves the proposition.

3. The fundamental order

The fundamental order of a theory was defined by B. POIZAT [4] and also appears in [l]. We consider a theory T and all the complete 1-types over models of T. For a 1-type p over M of T we define Class@) to be {p(x, fj) in the language of T : there is 6 in M such that ~ ( z , 6) E p ] . We then define Class(p) 5 Class(y) if Class@) 2 Class(y) (where p and y may be types over different models). (In the context of the funda- mental order the notions of forking and of superstability have a very natural expres- sion.) It is known that in the case where T is stable and No-categorical then the Morley rank of a type p over a model corresponds exactly to the foundation rank of Class(p) in the above defined order (i.e. in the fundamental order). Thus if T is very simple then by the result of the previous section we know that the height of the fundamental order is finite. Here we show that the fundamental order of T is finite, i.e. that there are a finite number of classes of types over models of T .

We remark that to determine the possible classes of types over models of an K,- categorical theory it is enough to look a t the types over the countable model of the theory. This is because for a 1-type p over a model N there is clearly a countable ele- mentary substructure M of N such that Class(p r M ) = Class@).

From now on we assume (both for simplicity and also because this is the case that we are interested in) that T is KO-categorical and M is the countable model of T. We recall some other facts about Morley rank.

F a c t 11. Let A be a definable subset of M with Rank(A) = a. Then there is d < w such that A can be partitioned into at most d definable subsets of M each of rank a. The least such d is called Degree(A).

Def in i t ion 12. If p is a (complete) 1-type over M then Rank(p) is the least a such that oc = Rank(p) for some formula p in p .

416 ANAND PILLAY

F a c t 13. Let p be a 1-type over M of rank a. Then there is a formula p in p of rank a and degree 1 such that for any formula x with parameters in M , x E p if and only if Rankig, A i x ) < a.

Defin i t ion 14. Let p(x, 5 ) be a formula in L ( T ) (the pure language of T ) . We say that p(x, 8) fixes the type of ij if there is a complete type (over 0 ) p such that

We remark that by No-categoricity any definable subset of M is defined by a for- mula p(z, 6) with ti in M where p(x, 8) fixes the type of 8.

We remark also that if p(x, jj) fixes the type of jj and if p(x, a) and p(x, 6’) are con- sistent (where ii, a‘ are in M ) then Rank(p(x, a)) = Rank(p(x, a’)) and Degree(p(x, 6)) = = Degree(p(x, a’)). This is because 6 and ii’ will have the same type in M and thus as M is homogeneous there is an automorphism of M taking 6 to 6’. But automorphisms clearly preserve rank and degree.

By means of this remark, given a formula p(x, jj) of L(T) which fixes the type of jj, we talk by abuse of language, of the rank and degree of p(x, g), when what we really mean is the rank and degree of p(x, 6) for some in M for which p(x, 6) is consistent.

Lcmma 16. Let T he t+,-categorical and co-stable with a1 finite, say n. Then T has finite fundamental order if and only if for each m < n there are formulae q;”(x, jjy), pr(z. jj?), . . ., p3(x, 8:) of L(T) of rank m and each fixing the type of the appro- priate 4 such that if p is a type over M (the countable model of T ) and Rank(p) = m, where m < n, then there is r$(x, gy) which is in Class(p) for some 1 5 i

Proof. We just prove the right to left direction. So assume the condition on the right hand side. By Fact 11 and Definition 12 we can assume also that all the for- mulae py have degree 1. Now for any type p over M we have Rank(p) < n. So it vill be enough to show that if p and p‘ are types of rank m < n and e;,’”(x, a), @(x, 6’) are in p and p’ respectively for some j 5 i,,, and 6 , a’ are in M with lh(ii) = lh(a’) = = lh(y;’”) then Class(p) = Class(p’). So assume this condition. Let x(x, 2) be in Class(p). So there is 8 in M with ~ ( x , 6) E p. As Rank(p) = m and by Fact 13 Rank(q(x, a) A

A i x ( x , 6 ) ) < m. There is an automorphism of M taking ti to a’ (as they have the same type). Under this automorphism 6 is taken to some 6‘. Then clearly Rank(q(z, ii‘) A

A i x ( x , 6’)) < m and so again by Fact 13 ~ ( x , 8’) E p’. So x(x, z ) E Class@’). Thus Class@) Class(p,’). Similarly Class@’) E Class(p), so the lemma is proved.

Lemma 16. Let M be a very simple modd and let P be a corresponding tree in K for M . Theii there are a finite number of formulae pl(x, gl), q2(x, &), . . . , pr(x, j j r ) (in the lan- p a p of Th(M)) each of which fixes the type of the respective j j l such that if (T E P then there is j 5 r and b in M such that A , is defined by pJ(x, 6) i n M .

Proof . Easy by using the facts about good maps and the No-categoricity of Th(M).

Def in i t ion 17. Let y and x be formule (possibly with parameters). Then y cax

Propos i t i on 18. Let T be a very simple theory. Then T has finite fundamental order,

Proof . By Propositions 9 and 10 we can use Lemma 15. Let pl, y Z , . . . , pr be as given by Lemma 16. Let aT = n. We show by induction on m < n that if x is a formula (with parameters in M ) such that Rank(X) = m and Degree(X) = 1 then there is

by(”> D) + P ( a .

i,,,.

means that Rank(y A i x ) < a.

A CLASS OP KO-CATEGORICAL THEORIES 41 7

j 5 r with Rank(p,) = m (in the generalised sense) and Ci in M such that x yJ(x, a). By Fact 13 and Lemma 15 this will be enough. So let x be a formula with parameters in M of rank m + 1 and degree 1, where m + 1 < n. By definition of rank there are formulae 0, for j < (0 of rank m and degree 1 which are disjoint and are subsets of x. By induction hypothesis for each j there is j' 5 r and b, in M such that 6, criL cp,.(x, 6,). piow consider the set of (T in the tree P such that Rank(A,) = m + 1 and there is j < OJ such that pJ,(x, 6,) c A, .

Case 1. There are only finitely many such a. Then for some such a there are in- finitely many j < w such that pJp(x, 6,) c A,. As Rank(X) = 7n + 1 and Degree(x) = 1 it is then clear that x c lnfl A , . But by Lemma 16 A, is defined by a formula p,(x, 6) for some i 5 r . So we finish.

Case 2. There are infinitely many such a. Then by looking a t the tree P it is clear that there is q in P with infinite branching such that for infinitely many i there is j with pJ,(x, b,) c A,-(,). Pick such an i for which A,-( ,> contains none of the parameters appearing in x. Let pJ,(x, 6,) which is in this A,-( ,> be say A, . Then as Rank(A,) < < Rank@,-,,,), there are infinitely many pairwise disjoint images of A , (which are subsets of A,-,,,) under automorphisms of M which map x onto itself (by Lemma 3). This shows that x c n L + l A,-(,) and so we finish as before.

4. Non finite axiomatisability We show here that a very simple theory is not finitely axiomatisable. We use the

methods as in MAKOWSKY [2], namely EHRENFEUCHT-FRAISSE games. We recall that if M , and Ml are models (in the same language) then M , and MI are said to be n-equiv- alent (in the E.-F. game) if for every f : n -+ 2 ,

Vxo E J f f ( o , 3yo E M,-fco, * . * VzrL-1 E Mf(n-1) 3 ~ n - I E Ml-f(n-1) (a,, a,, . . ., E ( b o , bl , . . ., Ll)

where a , = x, if f(i) = 0, and a , = yI if f ( i ) = 1, and "similarly" for the b,'s. We write M , 3 M I . Then we know the following :

F a c t 19. Suppose that T i s a theory in a pure relational language and that T i s finitely axiomatisable. Then there is n < co such that if M i s a model of T and M =,, N then fl i s a model of T .

Now let T be a very simple theory and let M be a very simple model of T and P the corresponding tree. (We can assume that T is in a relational language.) Let n < (o. We define a subtree P,, of P as follows : ( ) is in PI,. If q E P,, and q has infinite branch- ing in P, then qn(0), q"<l), . . . , q"(n - 1) are all in P,. If ? is in P,, and 7 has finite branching in P then for all i q"( i ) E P implies that q"(i) E P,,.

Now suppose that the height of P is m. We show that lor all r 5 m(P,)'' is n equiv- alent to P5' via good maps (Le. maps in F , in the obvious sense). Suppose we know it for r . Now let, for example, qo be in P S r f 1 . We may assume that q, is in Pr+l. Then qo is q"( i ) for some q in P' and i < cr). By induction hypothesis there is q' in (P,,). and good map taking q to q'. If q has infinite branching then there is a good map taking qn( i ) to qfn(0), the latter which is in If q has finite branching then so has q' and there is a good map taking qn(i) to qIn(i), the latter again in (Pn)r+l. Carrying on in this way for r go's we see that PST+l is n-equivalent to (P,,)s'+1 via good maps. So P is n-equivalent to P,, . This is clearly still true if we restrict the plays of the game to picking endpoints. Note that PI, is a finite tree. Let M,, be the sub-

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418 ANANU PILLAY

structure of M consisting of those a,, in A4 such that q is (endpoint) in P, l . So M,, is finite. Finally note that if q,, . . . , v , , - ~ is taken to q;, . . . , q;L-l by a good map of P then a,,”, . . .,a,, ,,-, is isomorphic to a,,o,, . . ., a,,,, ;.

Thus we have shown:

Lemma 20. Por pach 72 < ( I ) , M f,, M , , , where M is very simple and X,, is as defiaed

Propos i t ion 21. If 1’ is very simple then T is not finitely axiomatisable. Proof. As ,1/1 is infinite and MI, is finite they cannot be elementarily equivalcnt.

in the abow paragraph.

So it follows from Fact 19 and the completeness of T .

5. Example. The ease aT = 2

The structure of s-stable theories T with LX, = 2 is quite straightforward and ib already known, e.g. MORTIMER [3], although it does not appear to have bpen stated that, if we additionally assume u,-categoricity, then we have non-finite axiomatis- ability. We state the structure theorem here for the u,-categorical case. We assume knowledge of the dimension theory of strongly minimal sets (i.e. sets of rank 1 and degree l ) , and the notion of algebraic closure.

Propos i t ion 22. Xicppose that T is u,-categorical and w-stable with LX? = 2 . Thpn T is simple, i.e. if M is the countable model of 1’ then there is some expansion M‘ of M by finitely many constants such that in M’ there are a finite number of pairwisp disjoint sets A , , . . ., A , of rank 1 and degree 1, definable in M’ udhout parameters such that A1 = cl( U ,4,), each A , has infinite dimension, and if X , , Y , are bases for - 4 , , for

i = 1, . . . , d , and for each i f L i s any bijection from X , to Y , , theii f = U f , extends to

Proof. As LY, is 2 we already know that there are a finite number of sets of rank 1 and degree 1 which cover M for all but a finite part. By adding constants to the lan- guage where neczssary we refine the choice of such sets to A, , . . . , Ad whose closure IS M , and such that if i 5 d, a , b E A , , .? E U A, and a E cl(6 A c ) then already a E cl(b).

C o r o l l a r y 23. Ld T be N,-categoricaZ and w-stable with m g = 2. Then T is not finitely axiomatisa ble.

Proof. Let M’ be the expansion of M as in Proposition 22. It is enough to show that Th(M’) is not finitely axiomatisable. Let A , , A , , . . . , Ad be as in Proposition 22. Then by K,-categoricity there are integers k,, k,, . . ., k, such that if a is in iM then there are ii, in A, , . . . , a c l in A d with Iii,l 5 k, for i = 1 , . . . , d such that a E cl(6, u 6, u , . . u ad) . Then as in MAKOWSKY by given n < (9, for each i = 1, . . . , d choose an independent subset of A , of cardinality 12 . k,, say X:, and put M,, = cl( U AT:,). Then A’,, is finite and M

15 I

an automorphiswb of $1 (or, equivalently, is elementary). 12d

J + l

I Cr l M I , .

References [l] LASCAR, D., and B. POIZAT, An introduction to forking. J. Symb. Logic 44 (1979), 330-350. 121 MAKOWSKY, J., Categoricit,y and finite axiornstisability. Diplomarbeit, ETH Zurich 1971. [3] MOKTINER, M., Doctoral thesis. University of London 1974. 141 POIZAT, B., DBvistion des types. Doctors1 dissertation, Universit6 de Paris VI, 1977.

(Eingegangen am 8. Januar 1980)