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A classification of rapidly growing Ramsey functions

Author: Andreas Weiermann
Journal: Proc. Amer. Math. Soc. 132 (2004), 553-561
MSC (2000): Primary 03F30; Secondary 03D20, 03C62, 05D10
Published electronically: August 19, 2003
MathSciNet review: 2022381
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Abstract: Let $f$ be a number-theoretic function. A finite set $X$ of natural numbers is called $f$-large if $card(X)\geq f(min(X))$. Let $PH_f$ be the Paris Harrington statement where we replace the largeness condition by a corresponding $f$-largeness condition. We classify those functions $f$ for which the statement $PH_f$ is independent of first order (Peano) arithmetic $PA$. If $f$ is a fixed iteration of the binary length function, then $PH_f$ is independent. On the other hand $PH_{\log^*}$ is provable in $PA$. More precisely let $f_\alpha(i):= {\lvert i \rvert}_{H_\alpha^{-1}(i)}$where $\mid i\mid_h$ denotes the $h$-times iterated binary length of $i$ and $H_\alpha^{-1}$ denotes the inverse function of the $\alpha$-th member $H_\alpha$of the Hardy hierarchy. Then $PH_{f_\alpha}$ is independent of $PA$ (for $\alpha\leq \varepsilon_0$) iff $\alpha=\varepsilon_0$.

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Additional Information

Andreas Weiermann
Affiliation: Institut für Mathematische Logik und Grundlagenforschung, der Westfälischen Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany
Address at time of publication: Department of Mathematics, P.O. Box 80.010, 3508 TA Utrecht, The Nederlands

Keywords: Paris Harrington theorem, rapidly growing Ramsey functions, independence results, fast growing hierarchies, Peano arithmetic
Received by editor(s): February 21, 2002
Received by editor(s) in revised form: July 4, 2002, and September 26, 2002
Published electronically: August 19, 2003
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2003 American Mathematical Society

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