Rich Models

We define a rich model to be one which contains a proper elementary substructure isomorphic to itself. Existence, non-structure, and categoricity theorems for rich models are proved. We show that a countable theory with few rich models is categorical in Ki. We also consider a stronger notion of richness, and in discussing it prove: "If T is an unstable theory, then for any saturated model M of T of cardinality > |Z>(X)| there is an elementary chain of length u; of models isomorphic to M, whose union is not Ni-saturated." We propose here to examine a class of models which we shall call rich models. Definition 1 A model M is said to be rich if it contains a proper elementary substructure N such that N = M. Clearly, any rich model is infinite. Also, if T is a theory, then any universal model of T is rich (this is most easily seen by noting that if in the definition we replace "substructure" by "extension" then we have defined the same concept). What if T has no universal models of a particular cardinality A? The existence of a rich model of cardinality A is then a little less obvious. One method to prove existence is to consider some extended language containing a lot of set theory, a unary predicate (for a model of T) and a function symbol (for an isomorphism * Supported by a grant from the NSF to a proper elementary substructure of the predicate). A suitably large special model of this extended theory, and the downward Lowenheim Skolem theorem then complete the construction. The perceptive reader will wonder why we have left the details of this construction so vague—read on. Universal models have additional properties: Definition 2 A model M is said to be weakly a-rich (a an ordinal) if it contains a strict chain of length a consisting of proper elementary substructures Np -< M (/? < a) such that Np^M for all /? < a. Now it is again easily seen that a universal model M for T is weakly a-rich for all a < ||M||. The models produced by the construction outlined above need not have this property. But it is significant that we did not require the chain in the definition above to be continuous, and of course for an arbitrary universal model, there is no reason to suppose that it should be. Instead of banging our heads against a succession of walls, let us now introduce the most appropriate definitions of richness, and see what we can say. Perhaps surprisingly, it turns out that by making the conditions more stringent, we are led to much more basic proofs of the essential results. Definition 3 A model M is said to be a-rich (a an ordinal) if it contains a strict continuous chain of length a consisting of proper elementary substructures Np -< M (/? < a) such that Np S M for all /3 < a. We say that M is very rich if it is a-rich for every a < ||M|| Now that we have the right definition, it is time to begin proving theorems. Theorem 4 Suppose that T is a theory which has infinite models. Then for every A > max(No> |r |) there is a very rich model MofT such that \\M\\ = A. Proof: We may assume, without loss of generality, that T has built in Skolem functions. The Skolem hull M of a chain X of indiscernibles, order isomorphic to A is very rich. To see this, note that for each /? < A we can find a continuous strictly increasing sequence X{ (i < A) of subsets of X, each of which is order-isomorphic to X. Applying the Skolem functions yields a chain of submodels Mtwithin M having the required properties. • We presented the proof above, to outline the very simple idea that answering questions about the existence of very rich models often involves similar questions concerning other structures, only without the requirement that the appropriate substructures are necessarily elementary. This idea is also the foundation of the next result. Theorem 5 Suppose that T is not superstable, and A > \T\ is regular. Then for all fi > A, T has 2 very rich models of cardinality fi, none elementarily embeddable in another. Proof: For the cognoscenti we remark only that the proof will show that the standard set of 2 not mutually embeddable models have this property, and that the argument is essentially the same as the one above. First we consider the case // = A. We see from [2] VIII.2 (or less technically in [1]) that 2 models, none elementarily embeddable in another are constructed as Ehrenfeucht-Mostowski hulls of sets of indiscernibles (for a Skolemization of T). These indiscernibles have the structure of trees X = W AU5 where S is a stationary subset of A contained in the set {6 < A : ciS = a;}. The set S is construed as a subset of ^A consisting of some fixed increasing sequences j]$ which converge to 6 for each 6 E S. No generality is lost by assuming that these sequences rjs themselves consist of successors of limit ordinals (which amounts to restricting the original set to those ordinals which are multiples of a;). To avoid overcomplicating our notation, we will identify the indiscernibles with the corresponding trees. For such X and M = EM(X)y set Xo = (A {( 42 : C < A C a limit ordinal}) U S. It is clear that Xo is isomorphic to X as a tree, (the isomorphism being induced by a bijection between A and A — {£ + 2 : £ < A £ a limit ordinal} ) and that the isomorphism may be chosen so that S and all the branches corresponding to the sequences rjs are fixed. Since these trees are of height u; the isomorphism preserves the height of each node. Further, beginning with Xo we may also construct a continuous increasing chain of subtrees of X, each isomorphic to X, of arbitrary length a < A. This is accomplished by forming a descending chain of subsets Ai (i < a) of {£ -f 2 : £ < A £ a limit ordinal} and then setting Clearly, the Ehrenfeucht-Mostowski hulls of these subtrees witness the fact that M is very rich. If A < fj, then our task is easier. This time the models may be constructed as Ehrenfeucht-Mostowski hulls of indiscernibles X UY where X is as above and y = «>(/i -A) . The entire construction can then be carried out inside Y, and since X and Y are disjoint we do not even have to worry about branches converging to 5. • Corollary 6 / / \i> A, and M are as above then there is an order preserving embedding from V(fi) to the set {N : N -< M and N S M). This embedding can be chosen to preserve arbitrary intersections. Proof: All notation is as above. We give the proof only for the case p = A, since again p < A is similar but easier. Let A={C + 2 : C < A < a limit ordinal}. Take an infinite subset B C A such that \A — B\ = A and define an order reversing injection from V{\) to V(A) by taking any bijection / from A to AB, and, for T C A define: