Notes on local re ection principles

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 ex pressing the soundness of T that is the statement that every sentence provable in T is true More precisely if ProvT x denotes the canonical 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 x ProvT pA x q A x A x is a formula 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 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 the classes n or n of the arithmetical hierarchy B n denotes the class of all 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 axiomatiz ability of the theories of the hierarchy etc We also obtain optimal conservation results for partial local re ection principles The corresponding questions for uni form re ection principles are well known and easy but are resolved in a rather