Categoricity over P for first order T or categoricity for phi in L_{omega_1 omega} can stop at aleph_k while holding for aleph_0, ..., aleph_{k1}
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<k but T_k is not relatively beth_kcategorical and varphi_k is not beth_kcategorical.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 1990
 arXiv:
 arXiv:math/9201240
 Bibcode:
 1992math......1240H
 Keywords:

 Mathematics  Logic
 EPrint:
 Israel J. Math. 70 (1990), 219235