Results 1 
2 of
2
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 ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
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