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Proceedings of the American Mathematical Society

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Independence of Ramsey theorem variants using $\varepsilon _0$

Authors: Harvey Friedman and Florian Pelupessy
Journal: Proc. Amer. Math. Soc. 144 (2016), 853-860
MSC (2010): Primary 03F30; Secondary 03F15, 03D20, 05C55
Published electronically: June 26, 2015
MathSciNet review: 3430859
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Abstract: We show that Friedman’s finite adjacent Ramsey theorem is unprovable in Peano Arithmetic, and give a new proof of the unprovability of the Paris–Harrington theorem in $\mathrm {PA}$. We also determine the status of these theorems for each dimension. It is to be noted that the finite adjacent Ramsey theorem for dimension $d$ is equivalent to the Paris–Harrington theorem for dimension $d+1$.

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Additional Information

Harvey Friedman
Affiliation: Department of Mathematics, Ohio State University, Room 754, Mathematics Building, 231 West 18th Avenue, Columbus, Ohio 43210
MR Author ID: 69465

Florian Pelupessy
Affiliation: Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan

Keywords: Finite adjacent Ramsey, Paris–Harrington, Hydra battles, Ramsey theory, unprovability, independence, Peano Arithmetic, Cantor normal form, Hardy hierarchy, fundamental sequences
Received by editor(s): February 8, 2013
Received by editor(s) in revised form: January 19, 2015
Published electronically: June 26, 2015
Additional Notes: This research was partially supported by the John Templeton Foundation grant ID #36297 and an Ohio State University Presidential Research Grant. The opinions expressed here are those of the author and do not necessarily reflect the views of the John Templeton Foundation.
Communicated by: Mirna DĹľamonja
Article copyright: © Copyright 2015 American Mathematical Society