The article is organised as follows: it first presents some classical results from
model theory that apply in ZFC or under the GCH-like assumptions. This is
the content of §2, which is divided to subsections relating to saturated and special
models and countable universal models. Section 3 moves into the realm where CH
is violated and considers the possible existence of universal models in forcing exten-
sions where CH fails, concentrating on ℵ1. This section is divided into subsections
dealing with graphs, triangle-free graphs, linear orders and amenability at ℵ1.
Throughout κ stands for an infinite cardinal. An unattributed T stands for
a theory, which means a complete first order theory with infinite models. For
simplicity in this presentation we restrict ourselves to the case of countable theories.
A type for us is any consistent set of sentences, and a complete type is a maximal
consistent set of sentences. By a universal model of T of size κ we mean a model
in which every other model of T of size κ embeds
2. Some classical results
We present some results on the existence of certain kinds of universal models
for complete first order theories, again restricting to the case of countable theories.
Results presented in this section mostly come from 1960s.
2.1. Saturated and special models.
Definition 2.1. A model M of a theory T is said to be κ-saturated if for
every A M of size κ, the expansion (M, a)a∈A realises every type Γ(x) of the
expanded language which is consistent with the complete theory Th(M, a)a∈A. M
is said to be saturated if it is |M|-saturated.
A generalisation of Cantor’s proof that the rationals are a unique countable
dense linear order with no first or last elements, gives us that saturated models are
universal. See Theorem 2.4 for a detailed statement. The basic theorem about the
existence of saturated models at uncountable cardinals is the following
Lemma 2.2. (Vaught, [10]) Suppose that N is a model T of size 2κ. Then
there is a κ+-saturated extension M of N of size 2κ.
Proof. Note that
and for every A
the language
L∪{ca}a∈A has size κ, therefore the total number of relevant types is
for each such type Σ a new symbol cΣ. We can form the set of sentences Γ consisting
of the elementary diagram of N along with Σ(cΣ) for all relevant Σ. This is a finitely
satisfiable set of sentences, hence it has a model, so it has a model of size 2κ. Let
M be the reduction of this model to the original language.
Provided that we assume some cardinal arithmetic this now gives us the exis-
tence of saturated models in successor cardinals:
Theorem 2.3. (Vaught, [10]) Suppose that κ satisfies

Then there is
a saturated model M of T of size
Proof. Let N be any model of T of size
By induction on α

we choose
models of T so that
N0 = N, Nα+1, =
for δ limit 0,
this context, because of the compactness theorem, we could equivalently require that
every model of infinite size κ embeds into the κ-universal model.
2 2
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