Hilbert versus Hindman
type: Item
instance of: scholarly article
title: Hilbert versus Hindman
zbMATH Open document ID: 1239.03038
author: Jeffry L. Hirst
published in: Archive for Mathematical Logic
publication date: 2012-02-10T00:00:00Z
review text: It is well known that Hindman's theorem can (equivalently) be formulated in the following finite union form: Let \(f: \mathbb{N^{<N}} \to k\) be a finite coloring of the finite subsets of \(\mathbb{N}\). Then there exists an infinite sequence of finite subsets \((X_i)_{i\in\mathbb{N}}\) such that the sets \(X_i\) are unmeshed, i.e., for all \(i,j\) either \(\max(X_i) < \min(X_{j})\) or \(\max(X_j) < \min(X_i)\), and that there is a \(c\) such that for all finite sets \(F\) we have \(f(\bigcup_{i\in F} X_i) = c\). In this paper the author considers the variant of this theorem where `unmeshed' is relaxed to `distinct'. He shows that this variant is equivalent to the infinite pigeonhole principle and thus to the \(\Pi^0_1\)-bounded collection principle (\(\mathrm{B}\Pi^0_1\)). This shows that this variant of Hindman's theorem is finitistic in the sense of Hilbert, while Hindman's theorem is not.
Mathematics Subject Classification ID: 03F35
Mathematics Subject Classification ID: 03B30
zbMATH DE Number: 6005023
zbMATH Keywords: reverse mathematics
zbMATH Keywords: pigeonhole principle
zbMATH Keywords: conservation
zbMATH Keywords: finitism
zbMATH Keywords: Hilbert's program
reviewed by: Alexander P. Kreuzer
MaRDI profile type: Publication
full work available at URL: s00153-011-0257-4
OpenAlex ID: W2015534989
cites work: Fragments of arithmetic
cites work: Partial realizations of Hilbert's program
cites work: Q3395521
DOI: 10.1007/S00153-011-0257-4
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description: scientific article; zbMATH DE number 6005023