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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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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. ..."
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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.