# A Model–Theoretic Approach to Proof TheorySome Combinatorics

A Model–Theoretic Approach to Proof Theory: Some Combinatorics [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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Model–Theoretic Approach to Proof TheorySome Combinatorics

Part of the Trends in Logic Book Series (volume 51)
Springer Journals — Sep 27, 2019
36 pages

/lp/springer-journals/a-model-theoretic-approach-to-proof-theory-some-combinatorics-d9MmIsAvyf
Publisher
Springer International Publishing
© 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