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Ordinal notation systems corresponding to Friedman's linearized well-partial-orders with gap-condition
[article]

2015
*
arXiv
*
pre-print

In this article we investigate whether the addition-free theta functions form a canonical

arXiv:1505.01359v1
fatcat:qvtn4hwggnckxeykoz7w5iwcaq
*notation**system*for the*linear*versions of*Friedman's**well*-*partial*-*orders**with*the so-called*gap*-*condition*over ...*To*this end, we determine the*order*type of the*notation**systems*for addition-free theta functions in terms of*ordinals*less than ε_0. ... The second author wants*to*thank his funding organization Fellowship of the Research Foundation -Flanders (FWO). ...##
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Ordinal notation systems corresponding to Friedman's linearized well-partial-orders with gap-condition

2017
*
Archive for Mathematical Logic
*

In this article we investigate whether the following conjecture is true or not: does the addition-free theta functions form a canonical

doi:10.1007/s00153-017-0559-2
fatcat:sxfnvmpdbncehppwji2a4qeaja
*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. ... The second author wants*to*thank his funding organization Fellowship of the Research Foundation -Flanders (FWO). ...##
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An order-theoretic characterization of the Howard-Bachmann-hierarchy
[article]

2015
*
arXiv
*
pre-print

In this article we provide an intrinsic characterization of the famous Howard-Bachmann

arXiv:1411.4481v2
fatcat:tjomsazqjrglzo7ldwkug6wdlu
*ordinal*in terms of a natural*well*-*partial*-*ordering*by showing that this*ordinal*can be realized as a maximal*order*... 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 Π^1_1-comprehension ... The authors also wish*to*thank Leszek Ko lodziejczyk for his helpful comments and Ryota Akiyoshi for his fruitful discussions*with*the first author. ...##
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An order-theoretic characterization of the Howard–Bachmann-hierarchy

2016
*
Archive for Mathematical Logic
*

In this article we provide an intrinsic characterization of the famous Howard-Bachmann

doi:10.1007/s00153-016-0515-6
fatcat:ztl4ki4chncabiflrdvu2ypn6q
*ordinal*in terms of a natural*well*-*partial*-*ordering*by showing that this*ordinal*can be realized as a maximal*order*... 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 Π 1 1 -comprehension. ... The authors also wish*to*thank Leszek Kołodziejczyk for his helpful comments and Ryota Akiyoshi for his fruitful discussions*with*the first author. ...##
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From Kruskal's theorem to Friedman's gap condition
[article]

2020
*
arXiv
*
pre-print

Harvey

arXiv:2003.02714v1
fatcat:t2nxv6zkbbcdribr6njxd7vwaa
*Friedman's**gap**condition*on embeddings of finite labelled trees plays an important role in combinatorics (proof of the graph minor theorem) and mathematical logic (strong independence results). ... In the present paper we show that the*gap**condition*can be reconstructed from a small number of*well*-motivated building blocks: it arises via iterated applications of a uniform Kruskal theorem. ... Due*to*T 0 (X) ∼ = X it is clear that T 0 preserves*well**partial**orders*. If T n is a normal WPO-dilator, then so is M • T n . ...##
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Phase transitions in Proof Theory

2010
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Discrete Mathematics & Theoretical Computer Science
*

International audience Using standard methods of analytic combinatorics we elaborate critical points (thresholds) of phase transitions from provability

doi:10.46298/dmtcs.2771
fatcat:4waoq43rjbgnlnm4rddfqzujxy
*to*unprovability of arithmetical*well*-*partial*-*ordering*... We are also grateful*to*Jan Christoph Schlage-Puchta and the AofA'10 referees for useful comments and suggestions. ... Acknowledgements We are grateful*to*the John Templeton Foundation and the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (B 60-62). ...##
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Ordinal Notation
[article]

2018
*
arXiv
*
pre-print

We introduce a framework for

arXiv:1610.04633v3
fatcat:hl5i7zjgonhg5nyvwteh7r6ibu
*ordinal**notation**systems*, present a family of strong yet simple*systems*, and give many examples of*ordinals*in these*systems*. ... While much of the material is conjectural, we include*systems**with*conjectured strength beyond second*order*arithmetic (and plausibly beyond ZFC), and prove*well*-foundedness for some weakened versions. ... Another possibility is*to*use such a*to*fill in the*gaps*in the*notation*. The*notation**system*can just as*well*be defined above an arbitrary*ordinal*. ...##
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A Computation of the Maximal Order Type of the Term Ordering on Finite Multisets
[chapter]

2009
*
Lecture Notes in Computer Science
*

Moreover we discuss an approach

doi:10.1007/978-3-642-03073-4_50
fatcat:plh52xlfjbcmfdzcs3qdd7fcrm
*to*compute maximal*order*types of*well*-*partial**orders*which are related*to*tree embeddings. ... We give a sharpening of a recent result of Aschenbrenner and Pong about the maximal*order*type of the term*ordering*on the finite multisets over a wpo. ... The authors is grateful*to*the referees who provided helpful suggestions which led*to*an improved exposition. ...##
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Well Quasi-Orders in Computer Science (Dagstuhl Seminar 16031)

2016
*
Dagstuhl Reports
*

of

doi:10.4230/dagrep.6.1.69
dblp:journals/dagstuhl-reports/Goubault-Larrecq16
fatcat:o3uwzu5ptfavfes6kgexpi4a6q
*Well*quasi-*orders*(known as the Wqo-Theory) and several fields of Computer Science (Verification and Termination of Infinite-State*Systems*, Automata and Formal Languages, Term Rewriting and Proof Theory ... Wqo-Theory is a highly developed part of Combinatorics*with*ever-growing number of applications in Mathematics and Computer Science, and*Well*quasiorders are going*to*become an important unifying concept ... intuitively) for*ordinal**notations*: itcomputes the maximal*order*types of the*ordinal**notation**systems*which*correspond**to*the tree*well*quasi*orderings*in Kruskal's tree theorem. 3. ...##
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The reverse mathematics of wqos and bqos
[article]

2019
*
arXiv
*
pre-print

Thus ACA 0 proves that if (T , T ) is wqo then the

arXiv:1707.08365v5
fatcat:ym22ayu7ibhdtfrqnf2pi5ko44
*system*of*ordinal**notations*is a*well*-*order*. ...*order*on the*ordinal**notation**system*) whenever T 0 T T 1 . ...##
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Impredicativity and Trees with Gap Condition: A Second Course on Ordinal Analysis
[article]

2022
*
arXiv
*
pre-print

Specifically, we analyze parameter-free Π^1_1-comprehension and show that it cannot prove the extended Kruskal theorem due

arXiv:2204.09321v2
fatcat:ga2536tgkzbitmdzmllicndfwe
*to*Harvey Friedman (not even for two labels). ... These lecture notes introduce central notions of impredicative*ordinal*analysis, such as the Bachmann-Howard*ordinal*and the method of collapsing, which transforms uncountable proof trees into countable ... Iterating Kruskal's theorem:*Friedman's**gap**condition*In this section, we show that iterated applications of Kruskal's theorem lead*to**partial**orders**with*a certain '*gap**condition*', which is due*to*Harvey ...##
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What's so special about Kruskal's theorem and the ordinal Γo? A survey of some results in proof theory

1991
*
Annals of Pure and Applied Logic
*

The central theme of this paper is a powerful theorem due

doi:10.1016/0168-0072(91)90022-e
fatcat:lrq545sb5zdlffrobtqqxhvr4q
*to*Kruskal, the "tree theorem", as*well*as a "finite miniaturization" of Kruskal's theorem due*to*Harvey Friedman. ... These*orderings*play a crucial role in proving the termination of*systems*of rewrite rules and the correctness of Knuth-Bendix completion procedures. ... Acknowledgment: I wish*to*thank Robert Constable, Thierry Coquand, Nachum Dershowitz, Jean-Yves Girard, Pierre Lescanne, Anil Nerode, Mitsu Okada, Wayne Snyder, Rick Statman, and Gabriel Stolzenberg, for ...##
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Analytic combinatorics, proof-theoretic ordinals, and phase transitions for independence results

2005
*
Annals of Pure and Applied Logic
*

Finally, we indicate how regularity properties of

doi:10.1016/j.apal.2005.05.012
fatcat:dxkq6aj46ffmhkupj6mmhrqazq
*ordinal*count functions can be used*to*prove logical limit laws. ... We define the*ordinals*below ε 0 in non-logical terms and we survey a selection of recent results about the analytic combinatorics of these*ordinals*. ... (13) They are maximal*linear*extensions of natural*well**partial**orders*[67] . ...##
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THE PREHISTORY OF THE SUBSYSTEMS OF SECOND-ORDER ARITHMETIC

2017
*
The Review of Symbolic Logic
*

notes and publications of Hilbert and Bernays, (iii) the uncertainty as

doi:10.1017/s1755020316000411
fatcat:gqspk7gft5hmvms6krycngjlnq
*to*the constructive status of principles equivalent*to*Weak König's Lemma, and (iv) the large-scale intellectual backdrop*to*arithmetical ... This paper presents a systematic study of the prehistory of the traditional subsystems of second-*order*arithmetic that feature prominently in the reverse mathematics program promoted by Friedman and Simpson ... Acknowledgments This paper has been measurably improved by us having had the opportunity*to*present versions of it at the following events: Computability Theory and Foundations of Mathematics at the Tokyo ...##
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The Prehistory of the Subsystems of Second-Order Arithmetic
[article]

2016
*
arXiv
*
pre-print

This paper presents a systematic study of the prehistory of the traditional subsystems of second-

arXiv:1612.06219v1
fatcat:2cn2jny6jrhbjdfa5frim5kooa
*order*arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. ... lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as*to*the constructive status of principles equivalent*to*Weak K\"onig's Lemma, and (iv) the large-scale intellectual backdrop ... Acknowledgments This paper has been measurably improved by us having had the opportunity*to*present versions of it at the following events: Computability Theory and Foundations of Mathematics at the Tokyo ...
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