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21
DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
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Cited by 416 (115 self)
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An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between DavenportSchinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
Accessible independence results for Peano arithmetic
 Bulletin of the London Mathematical Society
, 1982
"... Recently some interesting firstorder statements independent of Peano Arithmetic (P) have been found. Here we present perhaps the first which is, in an informal sense, purely numbertheoretic in character (as opposed to metamathematical or combinatorial). The methods used to prove it, however, are c ..."
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Cited by 39 (0 self)
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Recently some interesting firstorder statements independent of Peano Arithmetic (P) have been found. Here we present perhaps the first which is, in an informal sense, purely numbertheoretic in character (as opposed to metamathematical or combinatorial). The methods used to prove it, however, are combinatorial. We also give another independence result (unashamedly combinatorial in character) proved by the same methods. The first result is an improvement of a theorem of Goodstein [2]. Let m and n be natural numbers, n> 1. We define the base n representation of m as follows: First write m as the sum of powers of n. (For example, if m = 266, n = 2, write 266 = 2 8 + 2 3 + 2 1.) Now write each exponent as the sum of powers of n. (For example, 266 = 2 23 + 2 2 + 1 +2 1.) Repeat with exponents of exponents and so on until the representation stabilizes. For example, 266 stabilizes at the representation 2 * +l + 2 2 + l +2 l. We now define the number Gn(m) as follows. If m = 0 set Gn(m) = 0. Otherwise set Gn(m) to be the number produced by replacing every n in the base n representation of m by n +1 and then subtracting 1. (For example, G2(266) = 3 33+1 + 3 3 + 1 +2). Now define the Goodstein sequence for m starting at 2 by So, for example, m0 = m, mx = G2{m0), m2 = G^mJ, m3 = G^m2),.... 266O = 266 = 2 22+1 + 2 2+1 + 2 X = 3 33+1 + 3 3+1 + 2 ~ 1O 38 2662 = 4 44+1 + 4 4+1 + l ~ 10 616 2663 = 5 s5+1 + 5 5+1 ~ 10 10 000. Similarly we can define the Goodstein sequence for m starting at n for any n> 1. THEOREM 1. (i) (Goodstein [2]) Vm 3/c mk = 0. More generally for any m, n> 1 the Goodstein sequence for m starting at n eventually hits zero. (ii) Vm 3k mk = 0 (formalized in the language of first order arithmetic) is not provable
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 ..."
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Cited by 16 (5 self)
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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.
A ModelTheoretic Approach to Ordinal Analysis
 Bulletin of Symbolic Logic
, 1997
"... . We describe a modeltheoretic approach to ordinal analysis via the finite combinatorial notion of an #large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in no ..."
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Cited by 11 (3 self)
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. We describe a modeltheoretic approach to ordinal analysis via the finite combinatorial notion of an #large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first and secondorder arithmetic. x1. Introduction. Two of proof theory's defining goals are the justification of classical theories on constructive grounds, and the extraction of constructive information from classical proofs. Since Gentzen, ordinal analysis has been a major component in these pursuits, and the assignment of recursive ordinals to theories has proven to be an illuminating way of measuring their constructive strength. The traditional approach to ordinal analysis, which uses cutelimination procedures to transfor...
Proof Interpretations and the Computational Content of Proofs. Draft of book in preparation
, 2007
"... This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question ..."
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Cited by 9 (1 self)
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This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question
A Ramsey Theorem in BoyerMoore Logic
 Journal of Automated Reasoning
, 1995
"... We use the BoyerMoore Prover, Nqthm, to verify the ParisHarrington version of Ramsey's Theorem. The proof we verify is a modification of the one given by Ketonen and Solovay. The Theorem is not provable in Peano Arithmetic, and one key step in the proof requires ffl 0 induction. x0. Introduction. ..."
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Cited by 8 (1 self)
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We use the BoyerMoore Prover, Nqthm, to verify the ParisHarrington version of Ramsey's Theorem. The proof we verify is a modification of the one given by Ketonen and Solovay. The Theorem is not provable in Peano Arithmetic, and one key step in the proof requires ffl 0 induction. x0. Introduction. The most wellknown formalizations of finite mathematics are PA (Peano Arithmetic) and PRA (Primitive Recursive Arithmetic). In both, the "intended" domain of discourse is the set of natural numbers. PA is formalized in standard firstorder logic, and contains the induction schema, which can apply to arbitrary firstorder formulas. The logic of PRA allows only quantifierfree formulas, which are thought of as being universally quantified, and PRA has the induction scheme for quantifierfree formulas, expressed as a proof rule. Also, for each primitive recursive function f , PRA contains a function symbol for f and has the recursive definition of f as an axiom. Clearly, PRA is much weaker tha...
Ordinal Arithmetic with List Structures
 In Logical Foundations of Computer Science
, 1992
"... We provide a set of "natural" requirements for wellorderings of (binary) list structures. We show that the resultant ordertype is the successor of the first critical epsilon number. The checker has to verify that the process comes to an end. Here again he should be assisted by the programmer givi ..."
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Cited by 5 (0 self)
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We provide a set of "natural" requirements for wellorderings of (binary) list structures. We show that the resultant ordertype is the successor of the first critical epsilon number. The checker has to verify that the process comes to an end. Here again he should be assisted by the programmer giving a further definite assertion to be verified. This may take the form of a quantity which is asserted to decrease continually and vanish when the machine stops. To the pure mathematician it is natural to give an ordinal number. In this problem the ordinal might be (n \Gamma r)! 2 + (r \Gamma s)! + k. A less highbrow form of the same thing would be to give the integer 2 80 (n \Gamma r) + 2 40 (r \Gamma s) + k. Alan M. Turing (1949) 1 Introduction A riddleconsider the Lisplike function f , f(a) = a f(b) = b Research supported in part by the National Science Foundation under Grants CCR90 07195 and CCR9024271. f(cons(x; y)) = 8 ? ? ? ? ? ? ? ? ? ? ! ? ? ? ? ? ? ? ? ? ?...
The Witness Function Method and Provably Recursive Functions of Peano
 Logic, Methodology and Philosophy of Science IX
, 1994
"... This paper presents a new proof of the characterization of the provably recursive functions of the fragments I# n of Peano arithmetic. The proof method also characterizes the # k definable functions of I# n and of theories axiomatized by transfinite induction on ordinals. The proofs are complete ..."
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This paper presents a new proof of the characterization of the provably recursive functions of the fragments I# n of Peano arithmetic. The proof method also characterizes the # k definable functions of I# n and of theories axiomatized by transfinite induction on ordinals. The proofs are completely prooftheoretic and use the method of witness functions and witness oracles.
Unprovability of sharp versions of Friedman’s sineprinciple
 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 135
, 2007
"... For every n ≥ 1 and every function F of one argument, we introduce the statement SPn F:“forall m, thereisNsuch that for any set A = {a1,a2,...,aN} of rational numbers, there is H ⊆ A of size m such that for any two nelement subsets ai1
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Cited by 5 (2 self)
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For every n ≥ 1 and every function F of one argument, we introduce the statement SPn F:“forall m, thereisNsuch that for any set A = {a1,a2,...,aN} of rational numbers, there is H ⊆ A of size m such that for any two nelement subsets ai1 <ai2 < ·· · <ain and ai1 <ak2 < ·· · <akn in H, wehave sin(ai1 · ai2 ···ain) − sin(ai1 · ak2 ···akn)  <F(i1)”. We prove that for n ≥ 2 and any function F(x) eventually dominated by ( 2 3)log(n−1) (x) n+1, the principle SPF is not provable in IΣn. In particular, the statement ∀nSPn ( 2 3)log(n−1) is not provable in Peano Arithmetic. In dimension 2, the result is: IΣ1 does not prove SP2 F,whereF(x)=(2 3) A−1 (x) √ x and A−1 is the inverse of the Ackermann function.