论文标题
强大的证明理论井井有条
A robust proof-theoretic well-ordering
论文作者
论文摘要
众所周知，根据逻辑强度的各种特征，例如一致性强度和$π^0_1 $定理的包含，天然的公理理论是通过逻辑强度预先确定的。尽管这些逻辑强度的概念对于天然理论来说是一致的，但它们通常并不相等。我们研究了这些概念的类似物，例如$π^1_1 $ - 反射强度和$π^1_1 $定理的包含 - 在$σ^1_1 $真相的存在下。在这种情况下，这些概念重合。此外，我们获得了公理理论的真实预订单，并可能会删除“自然”理论的非数学定量。
It is well-known that natural axiomatic theories are pre-well-ordered by logical strength, according to various characterizations of logical strength such as consistency strength and inclusion of $Π^0_1$ theorems. Though these notions of logical strength coincide for natural theories, they are not generally equivalent. We study analogues of these notions -- such as $Π^1_1$-reflection strength and inclusion of $Π^1_1$ theorems -- in the presence of an oracle for $Σ^1_1$ truths. In this context these notions coincide; moreover, we get genuine pre-well-orderings of axiomatic theories and may drop the non-mathematical quantification over "natural" theories.
