Bachmann-Howard Derivatives

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.

Minimal bad sequences are necessary for a uniform Kruskal theorem

“…In addition to Higman's lemma and Kruskal's theorem for finite ordered trees, our uniform result covers the case of finite rooted trees without an order between children (let W (X) = M (X) be the set of finite multisets with elements from X) and the case of finite rooted or ordered trees with vertex labels from a given wpo Z (take W (X) = Z × M (X) and W (X) = Z × Seq(X), respectively). In [9,13] the construction of T W is relativized, so that it yields a further transformation X → T W (X), rather than a single order T W as in the present paper. This makes it possible to iterate the construction, i.e., to repeat it with X → T W (X) at the place of W .…”