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Hilbert versus Hindman

Jeffry L. Hirst

Archive for Mathematical Logic 51 (1-2):123-125 (2012)

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(1 other version)Hindman’s theorem for sums along the full binary tree, $$\Sigma ^0_2$$ Σ 2 0 -induction and the Pigeonhole principle for trees. [REVIEW]Daniele Tavernelli & Lorenzo Carlucci - 2022 - Archive for Mathematical Logic 61 (5-6):827-839.details We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the resulting principle is equivalent to Σ20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma ^0_2$$\end{document}-induction over RCA0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {RCA}_0$$\end{document}. The proof uses the equivalence of this Hindman-type theorem with the Pigeonhole Principle for trees TT1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) Download Export citation Bookmark
Linear extensions of partial orders and reverse mathematics.Emanuele Frittaion & Alberto Marcone - 2012 - Mathematical Logic Quarterly 58 (6):417-423.details We introduce the notion of τ-like partial order, where τ is one of the linear order types ω, ω, ω + ω, and ζ. For example, being ω-like means that every element has finitely many predecessors, while being ζ-like means that every interval is finite. We consider statements of the form “any τ-like partial order has a τ-like linear extension” and “any τ-like partial order is embeddable into τ” . Working in the framework of reverse mathematics, we show that these (...) Download Export citation Bookmark 3 citations
(1 other version)Hindman’s theorem for sums along the full binary tree, $$\Sigma ^0_2$$ Σ 2 0 -induction and the Pigeonhole principle for trees. [REVIEW]Lorenzo Carlucci & Daniele Tavernelli - 2022 - Archive for Mathematical Logic 61 (5):827-839.details We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the resulting principle is equivalent to \-induction over \. The proof uses the equivalence of this Hindman-type theorem with the Pigeonhole Principle for trees \ with an extra condition on the solution tree. Download Export citation Bookmark
A weak variant of Hindman’s Theorem stronger than Hilbert’s Theorem.Lorenzo Carlucci - 2018 - Archive for Mathematical Logic 57 (3-4):381-389.details Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over \ to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey’s Theorem for pairs and strictly stronger than Hirst’s Hilbert’s Theorem. The lower bound (...) Download Export citation Bookmark 2 citations

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