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29
An application of graphical enumeration to PA
 Journal of Symbolic Logic
, 2003
"... For α less than ε0 let Nα be the number of occurrences of ω in the Cantor normal form of α. Further let n  denote the binary length of a natural number n, let nh denote the htimes iterated binary length of n and let inv(n) be the least h such that nh ≤ 2. We show that for any natural number h ..."
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Cited by 13 (3 self)
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For α less than ε0 let Nα be the number of occurrences of ω in the Cantor normal form of α. Further let n  denote the binary length of a natural number n, let nh denote the htimes iterated binary length of n and let inv(n) be the least h such that nh ≤ 2. We show that for any natural number h first order Peano arithmetic, PA, does not prove the following sentence: For all K there exists an M which bounds the lengths n of all strictly descending sequences 〈α0,..., αn 〉 of ordinals less than ε0 which satisfy the condition that the Norm Nαi of the ith term αi is bounded by K + i  · ih. As a supplement to this (refined Friedman style) independence result we further show that e.g. primitive recursive arithmetic, PRA, proves that for all K there is an M which bounds the length n of any strictly descending sequence 〈α0,..., αn 〉 of ordinals less than ε0 which satisfies the condition that the Norm Nαi of the ith term αi is bounded by K +i· inv(i). The proofs are based on results from proof theory and techniques from asymptotic analysis of Polyastyle enumerations. Using results from Otter and from Matouˇsek and Loebl we obtain similar characterizations for finite bad sequences of finite trees in terms of Otter’s tree constant 2.9557652856.... ∗ Research supported by a HeisenbergFellowship of the Deutsche Forschungsgemeinschaft. † The main results of this paper were obtained during the authors visit of T. Arai in
EQUIVALENCE BETWEEN FRAÏSSÉ’S CONJECTURE AND JULLIEN’S THEOREM.
, 2004
"... We say that a linear ordering L is extendible if every partial ordering that does not embed L can be extended to a linear ordering which does not embed L either. Jullien’s theorem is a complete classification of the countable extendible linear orderings. Fraïssé’s conjecture, which is actually a th ..."
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Cited by 7 (4 self)
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We say that a linear ordering L is extendible if every partial ordering that does not embed L can be extended to a linear ordering which does not embed L either. Jullien’s theorem is a complete classification of the countable extendible linear orderings. Fraïssé’s conjecture, which is actually a theorem, is the statement that says that the class of countable linear ordering, quasiordered by the relation of embeddablity, contains no infinite descending chain and no infinite antichain. In this paper we study the strength of these two theorems from the viewpoint of Reverse Mathematics and Effective Mathematics. As a result of our analysis we get that they are equivalent over the basic system of RCA0+Σ1 1IND. We also prove that Fraïssé’s conjecture is equivalent, over RCA0, to two other interesting statements. One that says that the class of well founded labeled trees, with labels from {+, −}, and with a very natural order relation, is well quasiordered. The other statement says that every linear ordering which does not contain a copy of the rationals is equimorphic to a finite sum of indecomposable linear orderings. While studying the proof theoretic strength of Jullien’s theorem, we prove the extendibility of many linear orderings, including ω2 and η, using just ATR0+Σ1 1IND. Moreover, for all these linear orderings, L, we prove that any partial ordering, P, which does not embed L has a linearization, hyperarithmetic (or equivalently ∆1 1) in P ⊕ L, which does not embed L.
Ordinal Systems
 SETS AND PROOFS
, 2001
"... Ordinal systems are structures for describing ordinal notation systems, which extend the more predicative approaches to ordinal notation systems, like the Cantor normal form, the Veblen function and the Schütte Klammer symbols, up to the BachmannHoward ordinal. oeordinal systems, which are natu ..."
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Cited by 6 (1 self)
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Ordinal systems are structures for describing ordinal notation systems, which extend the more predicative approaches to ordinal notation systems, like the Cantor normal form, the Veblen function and the Schütte Klammer symbols, up to the BachmannHoward ordinal. oeordinal systems, which are natural extensions of this approach, reach without the use of cardinals the strength of the theories for transfinitely iterated inductive definitions ID oe in an essentially predicative way. We explore the relationship with the traditional approach to ordinal notation systems via cardinals and determine, using "extended Schütte Klammer symbols", the exact strength of oeordinal systems.
Choice principles in constructive and classical set theories
 POHLERS (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2002
, 2002
"... The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models ..."
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Cited by 4 (3 self)
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The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.
On Fraïssé’s conjecture for linear orders of finite Hausdorff rank
, 2007
"... We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is ϕ2(0), the first fixed point of the εfunction. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in ACA + 0 + “ϕ2(0) is wellordered ..."
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Cited by 3 (2 self)
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We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is ϕ2(0), the first fixed point of the εfunction. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in ACA + 0 + “ϕ2(0) is wellordered” and, over RCA0, implies ACA ′ 0 + “ϕ2(0) is wellordered”.
An Inductive Version of NashWilliams’ MinimalBadSequence Argument for Higman’s Lemma
 IN P. CALLAGHAN, E.AL., TYPES FOR PROOFS AND PROGRAMS, LECTURE NOTES IN COMPUTER SCIENCE 2277
, 2001
"... Higman’s lemma has a very elegant, nonconstructive proof due to NashWilliams [NW63] using the socalled minimalbadsequence argument. The objective of the present paper is to give a proof that uses the same combinatorial idea, but is constructive. For a two letter alphabet this was done by Coquan ..."
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Higman’s lemma has a very elegant, nonconstructive proof due to NashWilliams [NW63] using the socalled minimalbadsequence argument. The objective of the present paper is to give a proof that uses the same combinatorial idea, but is constructive. For a two letter alphabet this was done by Coquand and Fridlender [CF94]. Here we present a proof in a theory of inductive definitions that works for arbitrary decidable well quasiorders.