Let ℱ be the class of complete, finitely axiomatizable
ω-categorical theories. It is not known whether there are
simple theories in ℱ. We prove three results of the form: if T∈
ℱ has a sufficently well-behaved definable set J, then T is not
simple. (In one case, we actually prove that T has the strict order
property.) All of our arguments assume that the definable set J
satisfies the Mazoyer hypothesis, which controls how an element
of J can be algebraic over a subset of the model. For every known
example in ℱ, there is a definable set satisfying the Mazoyer
hypothesis.