We say that the formula φ(x;y) :=
φ(x1,…,xn;y1,…,ym)
has the order property in the model M if there
are an infinite sequence a0, a1, … of
n-tuples from M and
an infinite sequence b0, b1, … of m-tuples
from M so that
φ(ai;bj) ⇔ i ≤ j
We say that φ(x;y); has the order property (relative to the theory T) if
it has the order property in some model of T. A theory T is
stable if and only if no formula has the
order property relative to T.
Starting from an instance of the order property
(M, φ(x;y),
{ai | i ∈ &omega}, {bi | j ∈ &omega})
setting
ci := ai bi
and
ψ(u1, …, um+n;v1, …, vm+n)
:= φ(u1, …, un;vn+1,
…, vm+n)
then &psi defines an infinite linear order on
the set of tuples {ci | i ∈ &omega }.