Freund, A (2017) Proof lengths for instances of the Paris–Harrington principle. Annals of Pure and Applied Logic, 168 (7). pp. 1361-1382. ISSN 0168-0072
Abstract
As Paris and Harrington have famously shown, Peano Arithmetic does not prove that for all numbers k , m , n there is an N which satisfies the statement PH ( k , m , n , N ) : For any k-colouring of its n-element subsets the set 0 , … , N − 1 has a large homogeneous subset of size ≥m. At the same time very weak theories can establish the Σ 1 -statement ∃ N PH ( k ‾ , m ‾ , n ‾ , N ) for any fixed parameters k , m , n . Which theory, then, does it take to formalize natural proofs of these instances? It is known that ∀ m ∃ N PH ( k ‾ , m , n ‾ , N ) has a natural and short proof (relative to n and k) by Σ n − 1 -induction. In contrast, we show that there is an elementary function e such that any proof of ∃ N PH ( e ( n ) ‾ , n + 1 ‾ , n ‾ , N ) by Σ n − 2 -induction is ridiculously long. In order to establish this result on proof lengths we give a computational analysis of slow provability, a notion introduced by Sy-David Friedman, Rathjen and Weiermann. We will see that slow uniform Σ 1 -reflection is related to a function that has a considerably lower growth rate than F ε 0 but dominates all functions F α with α < ε 0 in the fast-growing hierarchy.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2017 Elsevier B.V. This is an author produced version of a paper published in Annals of Pure and Applied Logic . Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Peano arithmetic; Proof length; Paris–Harrington principle; Finite Ramsey theorem; Slow consistency; Fast growing hierarchy |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 11 Jan 2017 16:48 |
Last Modified: | 06 Jan 2018 01:38 |
Published Version: | https://doi.org/10.1016/j.apal.2017.01.004 |
Status: | Published |
Publisher: | Elsevier |
Identification Number: | 10.1016/j.apal.2017.01.004 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:110453 |