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  1. An order-theoretic characterization of the Howard–Bachmann-hierarchy.Jeroen Van der Meeren, Michael Rathjen & Andreas Weiermann - 2017 - Archive for Mathematical Logic 56 (1-2):79-118.
    In this article we provide an intrinsic characterization of the famous Howard–Bachmann ordinal in terms of a natural well-partial-ordering by showing that this ordinal can be realized as a maximal order type of a class of generalized trees with respect to a homeomorphic embeddability relation. We use our calculations to draw some conclusions about some corresponding subsystems of second order arithmetic. All these subsystems deal with versions of light-face Π11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPi ^1_1$$\end{document}-comprehension.
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  • Ordinal notation systems corresponding to Friedman’s linearized well-partial-orders with gap-condition.Michael Rathjen, Jeroen Van der Meeren & Andreas Weiermann - 2017 - Archive for Mathematical Logic 56 (5-6):607-638.
    In this article we investigate whether the following conjecture is true or not: does the addition-free theta functions form a canonical notation system for the linear versions of Friedman’s well-partial-orders with the so-called gap-condition over a finite set of n labels. Rather surprisingly, we can show this is the case for two labels, but not for more than two labels. To this end, we determine the order type of the notation systems for addition-free theta functions in terms of ordinals less (...)
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  • Bachmann–Howard derivatives.Anton Freund - 2023 - Archive for Mathematical Logic 62 (5):581-618.
    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 (...)
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