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Well-Partial-Orders and Ordinal Notation Systems

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Well-Partial-Orders and Ordinal Notation Systems Faculty of Sciences Department of Mathematics Connecting the Two Worlds: Well-partial-orders and Ordinal Notation Systems Jeroen Van der Meeren Supervisor: Prof. Dr. Andreas Weiermann Dissertation submitted in fulfillment of the requirements for the degree of Doctor (Ph.D.) in Sciences, Mathematics 2015 Copyright. The author and the supervisor give the authorization to consult and to copy parts of this work for personal use only. Any other use is limited by the laws of copyright. Permission to reproduce any material contained in this work should be obtained from the author. This does not include the Ghent University logo on the front page, which remains under full copyright of Ghent University. Das Unendliche hat wie keine andere Frage von jeher so tief das Gem¨utdes Menschen bewegt; das Unendliche hat wie kaum eine andere Idee auf den Verstand so anregend und fruchtbar gewirkt; das Unendliche ist aber auch wie kein anderer Begriff so der Aufkl¨arungbed¨urftig. From time immemorial, the infinite has stirred men's emotions more than any other question. Hardly any other idea has stimu- lated the mind so fruitfully. Yet, no other concept needs clarifi- cation more than it does. - David Hilbert, Uber¨ das Unendliche (On the infinite ) [39] Preface Kruskal claims in his now-classical 1972 paper [47] that well-partial-orders are among the most frequently rediscovered mathematical objects. Well- partial-orders have applications in many fields outside the theory of orders: computer science, proof theory, reverse mathematics, algebra, combinatorics, etc. The maximal order type of a well-partial-order characterizes that order's strength. Moreover, in many natural cases, a well-partial-order's maximal order type can be represented by an ordinal notation system. However, there are a number of natural well-partial-orders whose maximal order types and corresponding ordinal notation systems remain unknown. Prominent exam- ples are Friedman's well-partial-orders of trees with the gap-embeddability relation [76]. The main goal of this dissertation is to investigate a conjecture of Weier- mann [86], thereby addressing the problem of the unknown maximal order types and corresponding ordinal notation systems for Friedman's well-partial- orders [76]. Weiermann's conjecture concerns a class of structures, a typical member of which is denoted by T (W ), each are ordered by a certain gap- embeddability relation. The conjecture indicates a possible approach towards determining the maximal order types of the structures T (W ). Specifically, Weiermann conjectures that the collapsing functions #i correspond to maxi- mal linear extensions of these well-partial-orders T (W ), hence also that these collapsing functions correspond to maximal linear extensions of Friedman's famous well-partial-orders. For a more detailed overview and summary of the dissertation, we refer to the introductory Sections 1.1 and 1.3. ii Preface Acknowledgements The doctoral dissertation you are reading is the result of a four-year journey through the world of logic and proof theory. During this trip, I met many nice people and I am very grateful for the enrichment they gave to my life. They are with too many to sum up, but I definitely want to mention the following persons. First of all, I want to thank my supervisor Prof. Andreas Weiermann for his superb guidance and support during these four years. Andreas, thank you for your time and help. Secondly, I want to thank the other members of my exam committee, Prof. Marnix Van Daele, Prof. Lev Gordeev, Prof. Michael Rathjen, Prof. Monika Seisenberger, Dr. Paul Shafer, Prof. Leo Storme, and Prof. Jasson Vindas. Lev, for your comments and remarks on my dissertation. Michael, for our very fruitful discussions during my visits at Leeds University and for being an splendid co-author. Monika, for being an excellent host when I was visiting Swansea University and for your enthusiasm for my research. I want to thank my current and former colleagues at Ghent University. Es- pecially Korneel Debaene, Karsten Naert, Florian Pelupessy, Erik Rijcken, Sam Sanders, Bert Seghers, Paul Shafer, and Lawrence Wong. Paul and Flo- rian, for being close friends and for our nice discussions on logic. Lawrence and Erik, for being perfect office mates and for creating a nice environment for my research. Korneel, Karsten and Bert, for being good friends and co- students during my time at Ghent University. You were all very supportive. Additionally, I want to thank Leo Storme, Hans Vernaeve, Jasson Vindas and Andreas Weiermann for giving me the opportunity to be the assistant of their courses. I also want to thank my research foundation FWO (Research Foundation - Flanders) for giving me the opportunity to do a Ph.D. in logic. Furthermore, I want to thank all the people I met during conferences, work- iv Acknowledgements shops and research visits. You are with too many to mention. Afraid to forget someone, let me restrict to only those persons that I visited: Michael Rath- jen, Toshiyasu Arai, Ryota Akiyoshi, Kazuyuki Tanaka, Monika Seisenberger, Anton Setzer, Arnold Beckmann, Leszek Ko lodziejczyk, Konrad Zdanowski, Zofia Adamowicz, Alberto Marcone, Emanuele Frittaion, Thomas Forster and Henry Towsner. I want to thank my parents and brother for their support during my studies, especially when I was still living at home. Without them, I would never have been able to become the person that I am today. Furthermore, I want to thank all my friends and family, here and abroad. And last, but definitely not least, I want to thank my girlfriend Sanne. For her love, her immense patience and support. Thanks. Jeroen Van der Meeren May 2015 Contents Preface i Acknowledgements iii 1 Introduction 1 1.1 Historical background . .1 1.1.1 Ordinal notation systems . .1 1.1.2 Well-partial-orderings . .6 1.2 Preliminaries . .7 1.2.1 Notations . .7 1.2.2 Partial orders, linear orders, well-orders and ordinals .7 1.2.3 Ordinal notation systems below Γ0 ........... 10 1.2.4 Ordinal notation systems going beyond the limit of predicativity . 12 1.2.5 Ordinal notation systems without addition . 19 1.2.6 Theories and reverse mathematics . 26 1.2.7 Well-partial-orderings . 29 1.2.8 Examples of and constructors on well-partial-orderings 32 1.2.9 Well-partial-orders with gap-condition . 40 1.2.10 Tree-constructors and Weiermann's conjecture . 42 1.3 Overview and summary of the dissertation and possible appli- cations . 59 2 Unstructured trees 63 2.1 Introduction . 63 2.2 Lower bound . 65 3 Capturing the big Veblen number 69 3.1 Introduction . 69 3.2 Finite multisets of pairs . 70 3.3 Finite sequences of pairs . 83 4 Using one uncountability: the Howard-Bachmann number 95 4.1 Introduction . 95 4.2 Approaching Howard-Bachmann . 98 4.3 Obtaining Howard-Bachmann . 99 5 Capturing the gap-trees with two labels 111 5.1 Introduction . 111 wgap 5.2 Upper bound for o(T2 ).................... 118 wgap 5.3 Lower bound for o(T2 )..................... 130 gap 5.4 Maximal order type of Tn .................... 136 6 Independence results 139 6.1 Introduction . 139 6.2 Impredicative theories . 140 6.2.1 Lower bounds . 142 6.2.2 Upper bounds . 146 6.3 Independence results . 148 1 − 6.3.1 Independence results for ACA0 + (Π1-CA0) ...... 148 6.3.2 A general approach . 151 1 0 − 6.3.3 Independence results for RCA0 + (Π1(Π3)-CA0) .... 152 7 The linearized version 153 7.1 Introduction . 153 7.2 Known results . 156 7.3 From an order-theoretical view . 158 7.3.1 Maximal linear extension of gap-sequences with one and two labels . 158 7.3.2 The order type of (Tn[0]; <) with n > 2 . 160 7.3.3 Binary #-functions . 175 7.4 From a reverse mathematical point of view . 179 A Nederlandstalige samenvatting 185 A.1 Inleiding . 185 A.2 De resultaten . 187 Bibliography 191 Index 197 Chapter 1 Introduction 1.1 Historical background In order that the reader can situate this dissertation in its context, we present a brief overview of ordinal notation systems and well-partial-orders. One can find more complete surveys and overviews of these subjects in the literature (e.g. [20,47]). 1.1.1 Ordinal notation systems Well-orders and ordinal notation systems have been studied for their own order-theoretic and combinatorial interests. Additionally, also their appli- cations in proof-theoretic investigations of formal systems are studied [13, 31, 59, 60, 71, 79]. In this dissertation, we especially focus on the orderings themselves. At the end of the 19th century, Cantor extended the natural numbers into the transfinite by defining ordinals (also called ordinal numbers). It enabled him to study the order of such infinite numbers. In 1908, Veblen [84] intro- duced new fast growing functions on the class of ordinals by his techniques of derivation (i.e. enumerating fixed points of monotonic increasing continuous functions) and iteration. Veblen's techniques of derivation and iteration can be seen as a generalization of Cantor's normal form. Veblen's work gives rise to ordinal representation systems for specific ordinals. Hence, in some way, Veblen's (or Cantor's) work can be considered to be the starting point 2 Section 1.1. Historical background of ordinal notation systems and also of the notorious natural well-ordering problem [20], which is known to be extremely difficult. The natural well- ordering problem is a conceptual question about when a representation of a well-ordering is considered natural. Veblen's article [84] yields the Veblen hierarchy, which is nowadays well-known among most proof-theorists. It consists of a family of functions 'α on ordinals, where α is also an ordinal number.
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