SOME POSITIVE RESULTS IN THE CONTEXT OF UNIVERSAL MODELS 5
: (4) if Γn(x) is consistent with Tn, then Γn(c) Tn+1 for some c
L∗
\ L.
The induction is straightforward and we obtain that T

=

Tn is a maximal
consistent theory in
L∗.
Let M be a countable model of it, therefore by (3),
M = (M, an)nω where M = {an : n ω}. M is clearly a model of T and (4)
guarantees that it is saturated.
2.8
Of course, saturation is not a neccessary condition for universality: ω + Q is a
universal countable linear order but is not a saturated model. We shall not develop
this topic further in this article and from now on we shall only consider uncountable
models.
3. Changing the cardinal arithmetic
When we leave the realm of GCH and its remnants we are more or less left
with universes which we construct with forcing and where instances of GCH are
violated by the construction of the extension. Theorem 2.7(2) shows that if we
are not careful about how we do this, we shall end up basically with no universal
models of any sort. Theorem 2.7(3) shows that for certain theories such as linear
orders, no matter how careful we are, if we violate GCH sufficiently (including
making

=
2κ+
), the universality number at
κ+
will jump to the largest possible
value of
2κ+
. We shall see below that for certain other theories, for example theory
of graphs, it is possible to violate GCH as much as we like and still keep the
universality number low. This indicates that the ability of having a small universal
number in ‘reasonable’ forcing extensions in which the relevant instances of GCH
are violated is a property of the theory itself, which is not possessed by all theories.
In fact, in a series of papers, e.g. [7], [8], [6], [15], [16], [3] the thesis claiming the
connection between the complexity of a theory and its amenability to the existence
of universal models, has been pursued. In [4] we introduced the following definition,
which formalised these notions:
Definition 3.1. We say that a theory T is amenable iff whenever λ is an
uncountable cardinal satisfying
λλ
= λ and

=
λ+,
while θ satisfies cf(θ)
λ+,
there is a
λ+-cc
( λ)-closed forcing notion that forces

to be θ and the
universality number of T at
λ+
to be smaller than θ.
Localising this definition at a particular λ we define what is meant by theories
that are amenable at λ.
Connected to this definition there is a somewhat technical definition of high
non-amenability (see Definition 0.3 of [4]). We shall not quote the definition but
state only that high non-amenability of T implies that T is not amenable in the sense
of Definition 3.1, and that the theory of dense linear order with no endpoints is a
prototypical example of a highly non-amenable T . We should also comment that the
amenability/high non-amenability is envisioned as a dividing in the classification
theory of unstable theories. The exact syntactic properties that correspond to this
line have not been found yet, but is known that the property SOP4 implies high non-
amenability and simplicity implies amenability. We shall not go into these model-
theoretic considerations at this point but refer the reader to the articles mentioned
above. Here we shall simply be concerned to give examples and techniques which
apply to amenable theories. For simplicity in presentation we shall concentrate on
the case of amenability at λ = ℵ1 but we warn the reader that there are some
caveats in looking only at this case-we shall indicate them below.
5 5
Previous Page Next Page