abstract

Suppose L is a relational language and P in L is a unary predicate. If M is
an Lstructure then P(M) is the Lstructure formed as the substructure of M
with domain {a: M models P(a)}. Now suppose T is a complete first order theory
in L with infinite models. Following Hodges, we say that T is relatively
lambdacategorical if whenever M, N models T, P(M)=P(N), P(M)= lambda then
there is an isomorphism i:M> N which is the identity on P(M). T is relatively
categorical if it is relatively lambdacategorical for every lambda. The
question arises whether the relative lambdacategoricity of T for some lambda
>T implies that T is relatively categorical.
In this paper, we provide an example, for every k>0, of a theory T_k and an
L_{omega_1 omega} sentence varphi_k so that T_k is relatively
aleph_ncategorical for n < k and varphi_k is aleph_ncategorical for n