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

Michael Rathjen, Jeroen Van der Meeren, Andreas Weiermann
2015 arXiv   pre-print
In this article we investigate whether 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  ...  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).  ... 
arXiv:1505.01359v1 fatcat:qvtn4hwggnckxeykoz7w5iwcaq

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  
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.  ...  The second author wants to thank his funding organization Fellowship of the Research Foundation -Flanders (FWO).  ... 
doi:10.1007/s00153-017-0559-2 fatcat:sxfnvmpdbncehppwji2a4qeaja

An order-theoretic characterization of the Howard-Bachmann-hierarchy [article]

Jeroen Van der Meeren, Michael Rathjen, Andreas Weiermann
2015 arXiv   pre-print
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  ...  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.  ... 
arXiv:1411.4481v2 fatcat:tjomsazqjrglzo7ldwkug6wdlu

An order-theoretic characterization of the Howard–Bachmann-hierarchy

Jeroen Van der Meeren, Michael Rathjen, Andreas Weiermann
2016 Archive for Mathematical Logic  
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  ...  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.  ... 
doi:10.1007/s00153-016-0515-6 fatcat:ztl4ki4chncabiflrdvu2ypn6q

From Kruskal's theorem to Friedman's gap condition [article]

Anton Freund
2020 arXiv   pre-print
Harvey 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 .  ... 
arXiv:2003.02714v1 fatcat:t2nxv6zkbbcdribr6njxd7vwaa

Phase transitions in Proof Theory

Lev Gordeev, Andreas Weiermann
2010 Discrete Mathematics & Theoretical Computer Science  
International audience Using standard methods of analytic combinatorics we elaborate critical points (thresholds) of phase transitions from provability 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).  ... 
doi:10.46298/dmtcs.2771 fatcat:4waoq43rjbgnlnm4rddfqzujxy

Ordinal Notation [article]

Dmytro Taranovsky
2018 arXiv   pre-print
We introduce a framework for 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.  ... 
arXiv:1610.04633v3 fatcat:hl5i7zjgonhg5nyvwteh7r6ibu

A Computation of the Maximal Order Type of the Term Ordering on Finite Multisets [chapter]

Andreas Weiermann
2009 Lecture Notes in Computer Science  
Moreover we discuss an approach 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.  ... 
doi:10.1007/978-3-642-03073-4_50 fatcat:plh52xlfjbcmfdzcs3qdd7fcrm

Well Quasi-Orders in Computer Science (Dagstuhl Seminar 16031)

Jean Goubault-Larrecq, Monika Seisenberger, Victor Selivanov, Andreas Weiermann, Marc Herbstritt
2016 Dagstuhl Reports  
of 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.  ... 
doi:10.4230/dagrep.6.1.69 dblp:journals/dagstuhl-reports/Goubault-Larrecq16 fatcat:o3uwzu5ptfavfes6kgexpi4a6q

The reverse mathematics of wqos and bqos [article]

Alberto Marcone
2019 arXiv   pre-print
Thus ACA 0 proves that if (T , T ) is wqo then the system of ordinal notations is a well-order.  ...  order on the ordinal notation system) whenever T 0 T T 1 .  ... 
arXiv:1707.08365v5 fatcat:ym22ayu7ibhdtfrqnf2pi5ko44

Impredicativity and Trees with Gap Condition: A Second Course on Ordinal Analysis [article]

Anton Freund
2022 arXiv   pre-print
Specifically, we analyze parameter-free Π^1_1-comprehension and show that it cannot prove the extended Kruskal theorem due 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  ... 
arXiv:2204.09321v2 fatcat:ga2536tgkzbitmdzmllicndfwe

What's so special about Kruskal's theorem and the ordinal Γo? A survey of some results in proof theory

Jean H. Gallier
1991 Annals of Pure and Applied Logic  
The central theme of this paper is a powerful theorem due 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  ... 
doi:10.1016/0168-0072(91)90022-e fatcat:lrq545sb5zdlffrobtqqxhvr4q

Analytic combinatorics, proof-theoretic ordinals, and phase transitions for independence results

Andreas Weiermann
2005 Annals of Pure and Applied Logic  
Finally, we indicate how regularity properties of 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] .  ... 
doi:10.1016/j.apal.2005.05.012 fatcat:dxkq6aj46ffmhkupj6mmhrqazq

THE PREHISTORY OF THE SUBSYSTEMS OF SECOND-ORDER ARITHMETIC

WALTER DEAN, SEAN WALSH
2017 The Review of Symbolic Logic  
notes and publications of Hilbert and Bernays, (iii) the uncertainty as 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  ... 
doi:10.1017/s1755020316000411 fatcat:gqspk7gft5hmvms6krycngjlnq

The Prehistory of the Subsystems of Second-Order Arithmetic [article]

Walter Dean, Sean Walsh
2016 arXiv   pre-print
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 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  ... 
arXiv:1612.06219v1 fatcat:2cn2jny6jrhbjdfa5frim5kooa
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