For every A 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 =
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 : λ λ∗ such that each λ is a cardinal and is
λ+-saturated. By definition, saturated models are special. The opposite is not
true. Relationship between saturation, speciality and universality is given by the
Theorem 2.4. Every saturated model is special and every special model is
Proof. The first sentence follows by definition. Suppose now that M is the
union of a specializing chain Mi : 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 M such that
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 =⇒ Mi.
Suppose that : α κi have been chosen. Choose Mi+1 for α
[κi, κi+1) by induction on α. Suppose that α κi+1 and 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 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 Mi+1
which realises this type and the induction continues.
Theorem 2.5. (Morley and Vaught, [10]) Suppose that κ =

is uncount-
Then there is a special model of T of size κ.
Proof. If κ =
then by the assumption

= 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
of infinite
cardinals with limit κ, and such that
= κi+1. Then we build an elementary chain
Mi : 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
Letting M =
Mi we obtain a special model as required.
Conclusion 2.6. Every countable first order theory T has a universal model
of size κ for every κ ℵ0 satisfying = 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 =
(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
role of uncountability here is that we need κ to be larger than the size of T .
3 3
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