SOME POSITIVE RESULTS IN THE CONTEXT OF UNIVERSAL MODELS 3
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 =
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 : 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
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 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.
2.4
Theorem 2.5. (Morley and Vaught, [10]) Suppose that κ =

is uncount-
able2.
Then there is a special model of T of size κ.
Proof. If κ =
λ+
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
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 = 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
2The
role of uncountability here is that we need κ to be larger than the size of T .
3 3
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