4 MIRNA
DZAMONJAˇ
κ++
Cohen subsets to V . Then in V [G], in power
κ+,
there is no universal graph,
linear order, or generally model of a complete first order T which is unstable in κ.
(3) (Kojman-Shelah, [7]) Suppose κ is a regular cardinal and there is some
cardinal λ such that λ+ κ 2λ. Then there is no universal linear order of
cardinality κ.
Note that if T is stable in κ then it follows by an argument analogous to that
of Theorem 2.8 below that T has a saturated model of size κ.
2.2. Countable universal models. Countable universal models are abun-
dant in mathematics. For example, in addition to the examples we mentioned in
the Introduction there are examples in topology: any countable dense subset of
the Urysohn space is countably universal in the class of metric spaces with almost
isometric embedding. In model theory however, the theory of countable models
is different from the theory of the uncountable ones - as is most strikingly wit-
nessed by the well developed classification theory using the number of pairwise
non-isomorphic models [11], versus the still unresolved Vaught conjecture about
countable models. In the theory of universal models it is similarly the case that
countable models have a special role. Specifically, the methods presented in the rest
of this paper tend to apply only to uncountable models. However, this is not the
case with the concept of saturated models, which makes perfect sense in the case of
κ = ℵ0, see Definition 2.1. In fact, the concept of saturation was first introduced in
the countable case, by Vaught. Exactly the same proof as in the uncountable case
applies to show that saturated models are universal. For example, a well known
example of a countable universal model is the linear order of the rationals, and the
rationals are an example of a saturated model. In the case of countable models
there is a syntactic characterisation of the existence of saturated models, due to
Vaught in [17]. It also appears as Theorem 2.3.7 in [1].
Theorem 2.8. (Vaught, [17]) T has a countable saturated model iff for every
n ω, T has only countably many complete types in n-variables.
Proof. In the forward direction, let M be a countable saturated model of T .
Then for every n and complete type Γ in n variables, Γ is realised by an n-tuple in
M. Since there are only countably many such tuples and none of them can realise
more than one complete type, the conclusion follows.
In the other direction, we shall expand T to a maximal consistent theory T

in
an expanded language, such that any countable model of T

gives us a saturated
model of T . Let us form the language
L∗
by adding countably many new constant
symbols {cn : n ω} to L. Let {ϕn : n ω} be an enumeration of all sentences in
L∗.
For every m-element subset Y of {cn : n ω}, the complete types Γ(x) of T
in LY are in one-to-one correspondence with the complete types Σ(x0, . . . xm−1, x)
of T in L, and hence there are only countably many of them by the assumption.
We let {Γn(x) : n ω} enumerate all such types Γ(x), where Y ranges over all
finite subsets of {cn : n ω}.
By induction on n ω we choose an increasing sequence Tn of consistent
theories with T0 = T , such that
: (1) each Tn contains only finitely many symbols from
L∗
\ L,
: (2) either ϕn Tn+1 or ¬ϕn Tn+1,
: (3) if ϕn Tn+1 and ϕn = (∃x)ψ(x), then ψ(c) Tn+1 for some c L∗ \L,
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