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).