Results 1  10
of
13
A classification of rapidly growing Ramsey functions
 PROC. AMER. MATH. SOC
, 2003
"... Let f be a numbertheoretic function. A finite set X of natural numbers is called flarge if card(X) ≥ f(min(X)). Let PHf be the Paris Harrington statement where we replace the largeness condition by a corresponding flargeness condition. We classify those functions f for which the statement PHf i ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
Let f be a numbertheoretic function. A finite set X of natural numbers is called flarge if card(X) ≥ f(min(X)). Let PHf be the Paris Harrington statement where we replace the largeness condition by a corresponding flargeness condition. We classify those functions f for which the statement PHf is independent of first order (Peano) arithmetic PA.Iffis a fixed iteration of the binary length function, then PHf is independent. On the other hand PHlog ∗ is provable in PA. More precisely let fα(i):=i  H −1 α (i) where  i h denotes the htimes iterated binary length of i and H−1 α denotes the inverse function of the αth member Hα of the Hardy hierarchy. Then PHfα is independent of PA (for α ≤ ε0) iffα = ε0.
Long Finite Sequences
, 2001
"... Let k be a positive integer. There is a longest finite sequence x 1 ,...,x n in k letters in which no consecutive block x i ,...,x 2i is a subsequence of any other consecutive block x j ,...,x 2j . Let n(k) be this longest length. We prove that n(1) = 3, n(2) = 11, and n(3) is incomprehensibly large ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
Let k be a positive integer. There is a longest finite sequence x 1 ,...,x n in k letters in which no consecutive block x i ,...,x 2i is a subsequence of any other consecutive block x j ,...,x 2j . Let n(k) be this longest length. We prove that n(1) = 3, n(2) = 11, and n(3) is incomprehensibly large. We give a lower bound for n(3) in terms of the familiar Ackerman hierarchy. We also give asymptotic upper and lower bounds for n(k). We view n(3) as a particularly elemental description of an incomprehensibly large integer. Related problems involving binary sequences (two letters) are also addressed. We also report on some recent computer explorations of R. Dougherty which we use to raise the lower bound for n(3). TABLE OF CONTENTS 1. Finiteness, and n(1),n(2). 2. Sequences of fixed length sequences. 3. The Main Lemma. 4. Lower bound for n(3). 5. The function n(k). 6. Related problems and computer explorations. 1. FINITENESS, AND n(1),n(2) We use Z for the set of all integers, Z + f...
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
Proof theory of reflection
 Annals of Pure and Applied Logic
, 1994
"... The paper contains proof–theoretic investigations on extensions of Kripke–Platek set theory, KP, which accommodate first order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Πn reflection rules. Th ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
The paper contains proof–theoretic investigations on extensions of Kripke–Platek set theory, KP, which accommodate first order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Πn reflection rules. This leads to consistency proofs for the theories KP + Πn–reflection using a small amount of arithmetic (PRA) and the well–foundedness of a certain ordinal notation system with respect to primitive recursive descending sequences. Regarding future work, we intend to avail ourselves of these new cut elimination techniques to attain an ordinal analysis of Π 1 2 comprehension by approaching Π1 2 comprehension through transfinite levels of reflection. 1
The Realm of Ordinal Analysis
 SETS AND PROOFS. PROCEEDINGS OF THE LOGIC COLLOQUIUM '97
, 1997
"... A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is ma ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of `proof theoretic ordinals' to theories, gauging their `consistency strength' and `computational power'. Ordinaltheoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie "  the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency  technical results in pro...
An application of results by Hardy, Ramanujan and Karamata to Ackermannian functions
 Discrete Mathematics and Computer Science
"... The Ackermann function is a fascinating and well studied paradigm for a function which eventually dominates all primitive recursive functions. By a classical result from the theory of recursive functions it is known that the Ackermann function can be defined by an unnested or descent recursion along ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
The Ackermann function is a fascinating and well studied paradigm for a function which eventually dominates all primitive recursive functions. By a classical result from the theory of recursive functions it is known that the Ackermann function can be defined by an unnested or descent recursion along the segment of ordinals below ω ω (or equivalently along the order type of the polynomials under eventual domination). In this article we give a fine structure analysis of such a Ackermann type descent recursion in the case that the ordinals below ω ω are represented via a Hardy Ramanujan style coding. This paper combines numbertheoretic results by Hardy and Ramanujan, Karamata’s celebrated Tauberian theorem and techniques from the theory of computability in a perhaps surprising way.
The Higher Infinite in Proof Theory
 Logic Colloquium '95. Lecture Notes in Logic
, 1995
"... this paper. The exposition here diverges from the presentation given at the conference in two regards. Firstly, the talk began with a broad introduction, explaining the current rationale and goals of ordinaltheoretic proof theory, which take the place of the original Hilbert Program. Since this par ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
this paper. The exposition here diverges from the presentation given at the conference in two regards. Firstly, the talk began with a broad introduction, explaining the current rationale and goals of ordinaltheoretic proof theory, which take the place of the original Hilbert Program. Since this part of the talk is now incorporated in the first two sections of the BSLpaper [48] there is no point in reproducing it here. Secondly, we shall omit those parts of the talk concerned with infinitary proof systems of ramified set theory as they can also be found in [48] and even more detailed in [45]. Thirdly, thanks to the aforementioned omissions, the advantage of present paper over the talk is to allow for a much more detailed account of the actual information furnished by ordinal analyses and the role of large cardinal hypotheses in devising ordinal representation systems. 2 Observations on ordinal analyses
Brief introduction to unprovability
"... Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. In addition, we obtain a short accessible proof of unprovability of the ParisHarrington Principle. The proof employs old ideas but uses only one colouring and directly extracts the set of indiscernibles from its homogeneous set. We also present modified, abridged statements whose unprovability proofs are especially simple. These proofs were tailored for teaching purposes. The article is intended to be accessible to the widest possible audience of mathematicians, philosophers and computer scientists as a brief survey of the subject, a guide through the literature in the field, an introduction to its modeltheoretic techniques and, finally, a modeltheoretic proof of a modern theorem in the subject. However, some understanding of logic is assumed on the part of the readers. The intended audience of this paper consists of logicians, logicaware mathematicians andthinkers of other backgrounds who are interested in unprovable mathematical statements.