序分析的特征
Characterizations of ordinal analysis
摘要：
序数分析是一种研究程序，其中将递归序数分配给公理理论。根据传统的看法，序分析衡量理论的力量。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，我们的特点序数分析作为一个分区的$Σ^1_1$ -可定义和$Π^1_1$ -健全的理论，即分区，其中两个理论是等价的，如果他们有相同的证明理论序数。我们表明，没有等价关系$\equiv$是更好的顺序分析划分，如果：（1）$T\equiv U$时，$T$和$U$证明相同的$Π^1_1$句子;（2）$T\equiv T+U$的每个集合$U$的真正$Σ^1_1$的句子。事实上，没有这样的等价关系作出一个单一的区别，顺序分析划分不作出。其次，我们将序数分析刻画为算术可定义和$Π^1_1$ -合理理论上的一种排序，即如果$T$的证明论序数小于$U$的证明论序数，则$T< U$的排序。衡量理论强度的标准方法是一致性强度和包含$Π^0_1$定理。我们介绍类似物的这些概念- $Π^1_1$ -反射强度和包含的$Π^1_1$定理-在存在的神谕为$Σ^1_1$真理，并证明它们符合有序分析诱导。
英文摘要：
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.
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