论文标题
正常功能和最大订单类型
Normal functions and maximal order types
论文作者
论文摘要
井部分订单的转换通过最大秩序类型的概念在序中引起函数。在文献中的大多数示例中，这些函数不是正常的，与正常函数在序数分析和可计算理论相关的工作中起的核心作用形成鲜明对比。本文旨在解释这一现象。为此，我们研究了一类丰富的订单转换类别，这些转换被称为$ \ mathsf {wpo} $ - 扩张器。根据本文的第一个主要结果，$ \ mathsf {wpo} $ - 扩张器在满足相当限制的条件时会诱导正常功能，我们称之为强的正态性。此外，相反的含义也存在，对于表现良好的$ \ mathsf {wpo} $ - 扩张器。 Strong normality also allows us to explain another phenomenon: by previous work of Freund, Rathjen and Weiermann, a uniform Kruskal theorem for $\mathsf{WPO}$-dilators is as strong as $Π^1_1$-comprehension, while the corresponding result for normal dilators on linear orders is equivalent to the much weaker principle of $Π^1_1$-induction.作为我们的第二个主要结果，我们证明〜$π^1_1 $诱导等效于统一的kruskal定理，用于$ \ mathsf {wpo} $ - 扩张器，它们非常正常。
Transformations of well partial orders induce functions on the ordinals, via the notion of maximal order type. In most examples from the literature, these functions are not normal, in marked contrast with the central role that normal functions play in ordinal analysis and related work from computability theory. The present paper aims to explain this phenomenon. In order to do so, we investigate a rich class of order transformations that are known as $\mathsf{WPO}$-dilators. According to a first main result of this paper, $\mathsf{WPO}$-dilators induce normal functions when they satisfy a rather restrictive condition, which we call strong normality. Moreover, the reverse implication holds as well, for reasonably well behaved $\mathsf{WPO}$-dilators. Strong normality also allows us to explain another phenomenon: by previous work of Freund, Rathjen and Weiermann, a uniform Kruskal theorem for $\mathsf{WPO}$-dilators is as strong as $Π^1_1$-comprehension, while the corresponding result for normal dilators on linear orders is equivalent to the much weaker principle of $Π^1_1$-induction. As our second main result, we show~that $Π^1_1$-induction is equivalent to the uniform Kruskal theorem for $\mathsf{WPO}$-dilators that are strongly normal.
