Categoricity over $P$ for first order $T$ or categoricity for $\phi\in{\rm L}_ {\omega_ 1\omega}$ can stop at $\aleph_ k$ while holding for $\aleph_ 0,\cdots,\aleph_ {k-1}$

by Hart and Shelah. [HaSh:323]
Israel J Math, 1990
Suppose L is a relational language and P in L is a unary predicate. If M is an L-structure then P(M) is the L-structure 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 lambda-categorical 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 lambda-categorical for every lambda . The question arises whether the relative lambda-categoricity 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_n-categorical for n < k and varphi_k is aleph_n-categorical for n<k but T_k is not relatively beth_k-categorical and varphi_k is not beth_k-categorical.

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