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/lp/springer-journals/a-model-theoretic-approach-to-proof-theory-some-combinatorics-d9MmIsAvyf
- Publisher
- Springer International Publishing
- Copyright
- © Springer Nature Switzerland AG 2019
- ISBN
- 978-3-030-28920-1
- Pages
- 1 –37
- DOI
- 10.1007/978-3-030-28921-8_1
- Publisher site
- See Chapter on Publisher Site
Abstract
[In this chapter we give some material about combinatorics of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-large sets and their partitions. In the first two sections we prove classical Ramsey–type results. In later sections we introduce ordinals up to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _0$$\end{document} and we work out the so–called Hardy hierarchy of functions. This hierarchy determines a notion of a finite set of natural numbers being \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}–large, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha <\varepsilon _0$$\end{document}. We prove some partition properties for this notion of largeness. We point out that this type of results is just a technical refinement of results worked out by Ketonen and Solovay.]
Published: Sep 27, 2019