Bachmann-Howard Derivatives

TL;DR

This paper explores the Bachmann-Howard derivative, a construction related to ordinal analysis, showing how it connects gap conditions and collapsing functions, especially in the unary case for sequences.

Contribution

It introduces the Bachmann-Howard derivative as a general construction linking gap conditions to collapsing functions in ordinal analysis, focusing on the unary case.

Findings

The gap condition on sequences admits two different linearizations.

The Bachmann-Howard derivative provides a unified framework for these constructions.

The unary case clarifies phenomena specific to sequence-based gap conditions.

Abstract

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.