Notes on Local Reeection Principles
نویسنده
چکیده
We study the hierarchy of reeection principles obtained by restricting the full local reeection schema to the classes of the arithmetical hierarchy. Optimal conservation results w.r.t. the arithmetical complexity for such principles are obtained. Reeection principles, for an arithmetical theory T, are formal schemata expressing the soundness of T, that is, the statement that \every sentence provable in T is true". More precisely, if Prov T (x) denotes the canonical 0 1 provability predicate for T, then the local reeection principle for T is the schema Prov T (pAq) ! A; A is a sentence; and uniform reeection principle is the schema 8x (Prov T (pA(_ x)q) ! A(x)); A(x) is a formula: We denote local and uniform reeection principles respectively Rfn T and RFN T. Other natural forms of reeection turn out to be equivalent to one of these two (cf also 8]). Partial reeection 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 Rfn T (?) and RFN T (?), respectively, and for ? one usually takes one of the classes 0 n or 0 n of the arithmetical hierarchy. B((0 n) denotes the class of all boolean combinations of 0 n sentences. In this note we consider some basic questions concerning the hierarchy of partial local reeection 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 reeection principles. The corresponding questions for uniform reeection principles are well-known and easy, but are resolved in a rather 1
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