### Model Companions of $T_{\rm Aut}$ for stable $T$

by Baldwin and Shelah. [BlSh:759]

Notre Dame J Formal Logic, 2001

Let T be a complete first order theory in a countable
relational language L . We assume relation symbols have been added
to make each formula equivalent to a predicate. Adjoin a new unary
function symbol sigma to obtain the language L_sigma ;
T_sigma is obtained by adding axioms asserting that sigma is
an L-automorphism. We provide necessary and sufficient conditions
for T_Aut to have a model companion when T is stable. Namely,
we introduce a new condition: T admits obstructions, and
show that T_Aut has a model companion iff and only if
T does not admit obstructions. This condition is weakening
of the finite cover property: if a stable theory T has the finite
cover property then T admits obstructions.

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