SOME POSITIVE RESULTS IN THE CONTEXT OF UNIVERSAL MODELS 3 • For every A ⊆ Nα of size κ and type Γ(x) consistent with the complete theory of (Nα, a)a∈A, Γ(x) is realised in Nα+1. To do the construction at the successor stage α + 1 we simply apply Lemma 2.2 to the model Nα. At the end let M = ακ+ Nα. 2.3 Saturation is not necessary for universality. A weaker notion that still suﬃces is that of a special model: a model M of size κ is a special model if it is the union of an elementary chain Mλ : λ λ∗ such that each λ is a cardinal and Mλ is λ+-saturated. By definition, saturated models are special. The opposite is not true. Relationship between saturation, speciality and universality is given by the following: Theorem 2.4. Every saturated model is special and every special model is universal. Proof. The first sentence follows by definition. Suppose now that M is the union of a specializing chain Mi : i i∗ where Mi+1 is κi-saturated for some cardinals κi increasing to κ, which is the size of M. We may without loss of generality assume that this chain is continuous. Let N be a model of T of size κ, enumerated as {xα : α κ}. By induction on α we choose yα ∈ M such that xα → yα is an elementary embedding. We choose yαs in blocks of κi for i i∗, that is the induction is on i, so that α κi =⇒ yα ∈ Mi. Suppose that yα : α κi have been chosen. Choose yα ∈ Mi+1 for α ∈ [κi, κi+1) by induction on α. Suppose that α κi+1 and yβ for β α have been chosen. We use a modification of Cantor’s idea from the proof of the uniqueness of the rationals: let Γ(x) be the type of xα in (N, xβ)βα. Therefore Γ is a type of size κi+1 in (Mi+1, yβ)βα. By the saturation of Mi+1 we can find yα ∈ Mi+1 which realises this type and the induction continues. 2.4 Theorem 2.5. (Morley and Vaught, [10]) Suppose that κ = 2κ is uncount- able2. Then there is a special model of T of size κ. Proof. If κ = λ+ then by the assumption 2λ = κ and hence by Theorem 2.3 there is a saturated model of T of size κ. Suppose then that κ is a limit cardinal. Our assumptions allow us to choose an increasing sequence κi : i i∗ of infinite cardinals with limit κ, and such that 2κi = κi+1. Then we build an elementary chain Mi : i i∗ by starting with any model M0 of T of size κ0, and applying Lemma 2.2 at successor stages to get a model Mi+1 of size κi+1 which is κi+-saturated. Letting M = ii∗ Mi we obtain a special model as required. 2.5 Conclusion 2.6. Every countable first order theory T has a universal model of size κ for every κ ℵ0 satisfying 2κ = κ. Results presented above can be found in Chapter V of [1]. We also quote a selection of theorems which show that some assumptions on the kind of theories and on cardinal arithmetic are necessary for this conclusion. Theorem 2.7. (1) (Hausdorff, [5]) There exists a saturated linear order of size κ ℵ0 iff κ = κκ. (2) (Shelah, see [7] for a proof) Suppose that V is a model of GCH in which κ is a regular cardinal, and let G be V -generic for the Cohen forcing which adds 2 The role of uncountability here is that we need κ to be larger than the size of T .

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