Lecture 11: Excellence implies Categoricity: A

The following assertion is an exercise in Lecture 4. Let ψ be a complete sentence in Lω1,ω in a countable language L. Then there is a countable language L ′ extending L and a first order L′-theory T such that reduct is a 1-1 map from the atomic models of T onto the models of ψ. So in particular, any complete sentence of Lω1,ω can be replaced (for spectrum purposes) by considering the atomic models of a first order theory. This section is indirectly based on [?, ?, ?], where most of the results were originally proved. But our exposition owes a great deal to [?, ?, ?]. Recall that a model M is atomic if every finite sequence in M realizes a principal type over the empty set. Thus if T is א0-categorical every model of T is atomic.