It is generally accepted that H. Friedman’s gap condition is closely related to iterated collapsing functions from ordinal analysis. But what precisely is the connection? We offer the following answer: In a previous paper we have shown that the gap condition arises from an iterative construction on transformations of partial orders. Here we show that the parallel construction for linear orders yields familiar collapsing functions. The iteration step in the linear case is an instance of a general construction that we call ‘Bachmann–Howard derivative’. In the present paper, we focus on the unary case, i.e., on the gap condition for sequences rather than trees and, correspondingly, on addition-free ordinal notation systems. This is partly for convenience, but it also allows us to clarify a phenomenon that is specific to the unary setting: As shown by van der Meeren, Rathjen and Weiermann, the gap condition on sequences admits two linearizations with rather different properties. We will see that these correspond to different recursive constructions of sequences.

人们普遍认为，H. Friedman 的间隙条件与序数分析中的迭代折叠函数密切相关。但究竟是什么联系？我们提供以下答案：在之前的一篇论文中，我们已经证明间隙条件是由偏序变换的迭代构造产生的。在这里，我们展示了线性订单的并行构造产生了熟悉的折叠函数。线性情况下的迭代步骤是我们称之为“巴赫曼-霍华德导数”的一般构造的一个实例。在本文中，我们关注一元情况，即序列而不是树的间隙条件，以及相应的无加法序数符号系统。这部分是为了方便，但它也使我们能够澄清一元设置特有的现象：如 van der Meeren、Rathjen 和 Weiermann 所示，序列上的间隙条件允许两个具有相当不同性质的线性化。我们将看到这些对应于序列的不同递归构造。