Another incompleteness result for Hoare's logic

It is known (Bergstra and Tucker (1982) J. Comput. System Sci. 25, 217) that if the Hoare rules are complete for a first-order structure %plane1D;49C;, then the set of partial correctness assertions true over %plane1D;49C; is recursive in the first-order theory of %plane1D;49C;. We show that the converse is not true. Namely, there is a first-order structure %plane1D;49E; such that the set of partial correctness assertions true over %plane1D;49E; is recursive in the theory of %plane1D;49E;, but the Hoare rules are not complete for %plane1D;49E;