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_{k-1}

TL;DR

This paper constructs examples showing that categoricity in certain cardinalities for first-order theories and L_{omega_1 omega} sentences can stop at a specific aleph_k, despite holding for smaller alephs, highlighting nuanced limits of categoricity transfer.

Contribution

It provides explicit examples demonstrating that categoricity in smaller cardinalities does not necessarily extend to larger ones in the context of first-order and infinitary theories.

Findings

Categoricity can hold for all aleph_n with n<k but fail at aleph_k.

Examples show that categoricity in smaller cardinals does not imply categoricity at larger cardinals.

The results clarify the boundaries of categoricity transfer in model theory.

Abstract

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.