AI 中文总结
本研究在算术丘奇论题下细化二阶算术的解析分层，得到更精细区分量词交替的范式，并以此证明Frittaion分离定理中的选择假设不可省略。
AI 中文摘要
我们在算术丘奇论题（即每个自然数集都是算术的这一断言）下，研究二阶算术上解析分层的一种细化形式。该论题在完全标准模型中为假，但在由算术集构成的ω模型中自然成立。在$\textsf{ACA}_0^\textsf{*}+\textsf{ACT}$系统中，二阶量词可替换为对算术集编码的量化，由此得到的范式能比通常的解析分层更精细地区分一阶与二阶量词的交替层级。作为应用，我们利用这些范式解答了Frittaion提出的关于二阶算术中一致反射片段的问题：其分离定理中的选择假设不能被简单省略。
英文摘要
We study a refinement of the analytic hierarchy over second-order arithmetic under the arithmetical Church thesis, the assertion that every set of natural numbers is arithmetical. The thesis is false in the full standard model, but it is naturally satisfied in the $ω$-model consisting of the arithmetical sets. Over $\mathsf{ACA}_0^\ast+\mathsf{ACT}$, second-order quantifiers can be replaced by quantification over codes for arithmetical sets, and this gives normal forms which distinguish first-order and second-order quantifier alternations more finely than the usual analytic hierarchy. As an application, we use these normal forms to answer a question of Frittaion on fragments of uniform reflection in second-order arithmetic: the choice assumption in his separation theorem cannot simply be omitted.
