Unprovability of sharp versions of Friedman's sine-principle

Author:
Andrey Bovykin

Journal:
Proc. Amer. Math. Soc. **135** (2007), 2967-2973

MSC (2000):
Primary 03F30, 03F99; Secondary 05D10

DOI:
https://doi.org/10.1090/S0002-9939-07-08933-2

Published electronically:
May 8, 2007

MathSciNet review:
2317975

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Abstract | References | Similar Articles | Additional Information

Abstract: For every and every function of one argument, we introduce the statement : ``for all , there is such that for any set of rational numbers, there is of size such that for any two -element subsets and in , we have

**1.**Bovykin, A. (2005). Model-theoretic treatment of threshold results for PH. Manuscript, downloadable from`http://logic.pdmi.ras.ru/andrey/research.html`.**2.**Carlucci L., Lee G., Weiermann, A. (2005). Classifying the phase transition threshold for regressive Ramsey functions.*Submitted*.**3.**Friedman, H. (2002). A posting in the internet forum FOM. June 8, 2002.`http://www.cs.nyu.edu/pipermail/fom/2002-June/005584.html`**4.**N. I. Fel′dman,*The approximation of certain transcendental numbers. I. Approximation of logarithms of algebraic numbers*, Izvestiya Akad. Nauk SSSR. Ser. Mat.**15**(1951), 53–74 (Russian). MR**0039768****5.**Kanamori, A., McAloon, K. (1987). On Gödel incompleteness and finite combinatorics.*Annals of Pure and Applied Logic*, 33, pp. 23-41. MR**0870685 (88i:03095)****6.**Ketonen, J., Solovay, R. (1981). Rapidly growing Ramsey Functions.*Annals of Mathematics*(ser 2) 113, pp. 267-314.MR**0607894 (84c:03100)****7.**Kojman, M., Lee, G., Omri, E., Weiermann, A. (2005). Sharp thresholds for the phase transition between primitive recursive and ackermannian Ramsey numbers.*Submitted*.**8.**Lee, G. (2005). Phase transitions in axiomatic thought. Ph.D. Thesis, University of Münster.**9.**K. Mahler,*On the approximation of 𝜋*, Nederl. Akad. Wetensch. Proc. Ser. A. 56=Indagationes Math.**15**(1953), 30–42. MR**0054660****10.**Paris, J., Harrington, L. (1977). A mathematical incompleteness in Peano arithmetic.*Handbook for Mathematical Logic*, North-Holland.**11.**Georges Rhin and Carlo Viola,*The group structure for 𝜁(3)*, Acta Arith.**97**(2001), no. 3, 269–293. MR**1826005**, https://doi.org/10.4064/aa97-3-6

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Additional Information

**Andrey Bovykin**

Affiliation:
Steklov Mathematical Institute, Fontanka 27, St. Petersburg, Russia; Liverpool University, Liverpool, United Kingdom

Email:
andrey@logic.pdmi.ras.ru

DOI:
https://doi.org/10.1090/S0002-9939-07-08933-2

Keywords:
Unprovable combinatorial statements,
irrationality measure of $\pi$,
dynamical system,
Paris-Harrington Principle,
Kanamori-McAloon Principle.

Received by editor(s):
June 7, 2006

Published electronically:
May 8, 2007

Communicated by:
Julia Knight

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.