Semi-honest subrecursive degrees and the collection rule in arithmetic
The paper is devoted to the application of the subrecursive degree theory to the study of fragments of Peano Arithmetic. It is connacted with the result of L.D. Beklemishev who showed that the hierarchy of nested applications of the \(\Sigma_{1}\)-collection rule over any \(\Pi_{2}\)-axiomatizable base theory extending Elementary Arithmetic collapses to its first level. In the paper under review it is proved that this result cannot in general be extended to base theories of arbitrary quantifier complexity. In fact, given any recursively enumerable set of true \(\Pi_{2}\)-sentences, S, a sound \(\Sigma_{2} \vee \Pi_{2}\)- axiomatized theory T extending S is constructed such that the hierarchy of nested applications of the \(\Sigma_{1}\)-collection rule over T is proper. The construction uses some results on subrecursive degree theory obtained by L. Kristiansen.
scientific article; zbMATH DE number 7790935
scientific article