论文标题
阶级理论和二阶算术中的非紧密度
Non-tightness in class theory and second-order arithmetic
论文作者
论文摘要
如果t（用同一语言）的t（用相同语言）的不同演绎封闭的扩展不可能是双重解释，则理论t是紧密的。许多经过充分研究的基础理论都很紧密，包括PA [Visser2006]，ZF，Z2和KM [ENAYAT2017]。在本文中，我们将Enayat的调查扩展到后两个理论的子系统。我们证明，限制Z2和KM的理解模式提供了非紧密的理论。具体而言，我们表明GB和ACA0每个都接受了不同的双交易所扩展，并且通过添加Sigma^1_k渗透，对于它们的扩展也相同，因为K <= 1。这些结果提供了Z2和KM的张力，以最小的方式表征了Z2和KM。
A theory T is tight if different deductively closed extensions of T (in the same language) cannot be bi-interpretable. Many well-studied foundational theories are tight, including PA [Visser2006], ZF, Z2, and KM [enayat2017]. In this article we extend Enayat's investigations to subsystems of these latter two theories. We prove that restricting the Comprehension schema of Z2 and KM gives non-tight theories. Specifically, we show that GB and ACA0 each admit different bi-interpretable extensions, and the same holds for their extensions by adding Sigma^1_k-Comprehension, for k <= 1. These results provide evidence that tightness characterizes Z2 and KM in a minimal way.
