Finitely Axiomatizable ω-categorical Theories and the Mazoyer Hypothesis
Journal of Symbolic Logic
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
David Lippel. 2005. "Finitely Axiomatizable ω-Categorical Theories and the Mazoyer Hypothesis" Journal of Symbolic Logic 70 (2): 460-472.