Instability and theories with few models

Abstract

Some results are obtained concerning n ( T ) n(T) , the number of countable models up to isomorphism, of a countable complete first order theory T. It is first proved that if n ( T ) = 3 n(T) = 3 and T has a tight prime model, then T is unstable. Secondly, it is proved that if n ( T ) n(T) is finite and more than one, and T has few links, then T is unstable. Lastly we show that if T has an algebraic model and has few links, then n ( T ) n(T) is infinite.