Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbert-style programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength of theories, or better to prove

We study the hierarchy of re ection principles obtained by restricting the full local re ection schema to the classes of the arithmetical hierarchy. Optimal conservation results w.r.t. the arithmetical complexity for such principles are obtained. Re ection principles, for an arithmetical theory T, are formal schemata expressing the soundness of T, that is, the statement that \every sentence provable 0 in T is true". More precisely, if ProvT (x) denotes the canonical 1 provability predicate for T, then the local re ection principle for T is the schema ProvT (pAq) ! A� A is a sentence� and uniform re ection principle is the schema 8x (ProvT (pA(_x)q) ! A(x))� A(x) isaformula: We denote local and uniform re ection principles respectively RfnT and RFNT. Other natural forms of re ection turn out to be equivalent to one of these two (cf also [8]). Partial re ection principles are obtained from local and uniform schemata by imposing a restriction that the formula A may only range over a certain subclass; of the class of T-sentences (formulas). Such schemata will be denoted RfnT (;) and RFNT (;), respectively, and for; one usually takes one of 0 0 the classes n or n of the arithmetical hierarchy. B ( 0 n) denotes the class of all 0 boolean combinations of n sentences. In this note we consider some basic questions concerning the hierarchy of partial local re ection principles: the collapse of this hierarchy, nite axiomatizability of the theories of the hierarchy, etc. We also obtain optimal conservation results for partial local re ection principles. The corresponding questions for uniform re ection principles are well-known and easy, but are resolved in a rather 1 RFN u