Semi-honest subrecursive degrees and the collection rule in arithmetic

By a result of L.D. Beklemishev, 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. We prove 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, we construct a sound \((\Sigma _2 \! \vee \! \Pi _2)\)-axiomatized theory T extending S such that the hierarchy of nested applications of the \(\Sigma _1\)-collection rule over T is proper. Our construction uses some results on subrecursive degree theory obtained by L. Kristiansen.