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.

根据 LD Beklemishev 的结果，\(\Sigma _1\)集合规则的嵌套应用的层次结构超过任何\(\Pi _2\)公理化基础理论，将初等算术扩展至其第一级。我们证明这个结果一般不能扩展到任意量词复杂度的基础理论。事实上，给定任何递归可枚举的真实\(\Pi _2\)句子集S，我们构造一个声音\((\Sigma _2 \! \vee \! \Pi _2)\)公理化理论T扩展S使得\(\Sigma _1\)的嵌套应用程序的层次结构- 集合规则超过T是正确的。我们的构造使用了 L. Kristiansen 获得的次递归度理论的一些结果。