论文标题
顺序分析的特征
Characterizations of ordinal analysis
论文作者
论文摘要
序数分析是一项研究计划，其中递归序物被分配给公理理论。根据传统观点，序数分析衡量理论的强度。然而，随之而来的力量概念是什么？在本文中，我们提出了解决这个问题的序数分析的抽象特征。 首先，我们将序数分析描述为$σ^1_1 $ - 可定义的分区和$π^1_1 $ sound Theories，即，如果两个理论具有相同的证明理论序列，则两个理论是等效的。我们表明，如果两者同时，则没有等价关系$ \ equiv $比序数分析分区更好：（1）$ t \ equiv u $ ness $ t $ and $ u $ a $ cles cluct of Same clos of Same $π^1_1 $ sensences; （2）对于每个集合的$ u $ u $ u $的$ t \ equiv t+u $，实际上，没有这样的对等关系可以使序数分析分区所没有的单一区别。 其次，我们将序数分析描述为算术可定义的订购，以及$π^1_1 $ sound理论，即，如果$ t $ $ t $的$ t <u $的订单小于$ u $ $ u $的证明理论的序列。测量理论强度的标准方法是一致性强度和$π^0_1 $定理的包含。我们介绍了这些概念的类似物 - $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，并证明它们与序数分析引起的订单相吻合。
Ordinal analysis is a research program wherein recursive ordinals are assigned to axiomatic theories. According to conventional wisdom, ordinal analysis measures the strength of theories. Yet what is the attendant notion of strength? In this paper we present abstract characterizations of ordinal analysis that address this question. First, we characterize ordinal analysis as a partition of $Σ^1_1$-definable and $Π^1_1$-sound theories, namely, the partition whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation $\equiv$ is finer than the ordinal analysis partition if both: (1) $T\equiv U$ whenever $T$ and $U$ prove the same $Π^1_1$ sentences; (2) $T\equiv T+U$ for every set $U$ of true $Σ^1_1$ sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make. Second, we characterize ordinal analysis as an ordering on arithmetically-definable and $Π^1_1$-sound theories, namely, the ordering wherein $T< U$ if the proof-theoretic ordinal of $T$ is less than the proof-theoretic ordinal of $U$. The standard ways of measuring the strength of theories are consistency strength and inclusion of $Π^0_1$ theorems. We introduce analogues of these notions -- $Π^1_1$-reflection strength and inclusion of $Π^1_1$ theorems -- in the presence of an oracle for $Σ^1_1$ truths, and prove that they coincide with the ordering induced by ordinal analysis.
