Results 1 
4 of
4
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
Abstract

Cited by 5 (2 self)
 Add to MetaCart
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.
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.
SHARP PHASE TRANSITION THRESHOLDS FOR THE PARIS HARRINGTON RAMSEY NUMBERS FOR A FIXED DIMENSION
"... Abstract. This article is concerned with investigations on a phase transition which is related to the (finite) Ramsey theorem and the ParisHarrington theorem. For a given numbertheoretic function g, letRd c (g)(k) betheleast natural number R such that for all colourings P of the delement subsets ..."
Abstract
 Add to MetaCart
Abstract. This article is concerned with investigations on a phase transition which is related to the (finite) Ramsey theorem and the ParisHarrington theorem. For a given numbertheoretic function g, letRd c (g)(k) betheleast natural number R such that for all colourings P of the delement subsets of {0,...,R − 1} with at most c colours there exists a subset H of {0,...,R − 1} such that P has constant value on all delement subsets of H andsuchthat the cardinality of H is not smaller than max{k, g(min(H))}. If g is a constant function with value e, thenRd c (g)(k) is equal to the usual Ramsey number Rd c (max{e, k}); and if g is the identity function, then Rd c (g)(k) is the corresponding ParisHarrington number, which typically is much larger than Rd c (k). In this article we give for all d ≥ 2 a sharp classification of the functions g for which the function m ↦ → Rd m(g)(m) grows so quickly that it is no longer provably total in the subsystem of Peano arithmetic, where the induction scheme is restricted to formulas with at most (d − 1)quantifiers. Such a quick growth will in particular happen for any function g growing at least as fast as i ↦ → ε · log(···(log ( i) ···)(whereε>0 is fixed) but not for the function (d−1)−times
1 The strength of Ramsey theorem for coloring relatively large sets
"... Abstract—We characterize the computational content and the prooftheoretic strength of a Ramseytype theorem for bicolorings of socalled exactly large sets. An exactly large set is a set X ⊂ N such that card(X) = min(X) + 1. The theorem we analyze is as follows. For every infinite subset M of N, f ..."
Abstract
 Add to MetaCart
Abstract—We characterize the computational content and the prooftheoretic strength of a Ramseytype theorem for bicolorings of socalled exactly large sets. An exactly large set is a set X ⊂ N such that card(X) = min(X) + 1. The theorem we analyze is as follows. For every infinite subset M of N, for every coloring C of the exactly large subsets of M in two colors, there exists and infinite subset L of M such that C is constant on all exactly large subsets of L. This theorem is essentially due to Pudlàk and Rödl and independently to Farmaki. We prove that — over Computable Mathematics — this theorem is equivalent to closure under the ω Turing jump (i.e., under arithmetical truth). Natural combinatorial theorems at this level of complexity are rare. Our results give a complete characterization of the theorem from the point of view of Computable Mathematics and of the Proof Theory of Arithmetic. This nicely extends the current knowledge about the strength of Ramsey Theorem. We also show that analogous results hold for a related principle based on the Regressive Ramsey Theorem. In addition we give a further characterization in terms of truth predicates over Peano Arithmetic. We conjecture that analogous results hold for larger ordinals. I.