Title | ||
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What's So Special About Kruskal's Theorem and the Ordinal Gamma0? A Survey of Some Results in Proof Theory |
Abstract | ||
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This paper consists primarily of a survey of results of Harvey Friedman about some proof-theoretic aspects of various forms of Kruskal's tree theorem, and in particular the connection with the ordinal GAMMA-0. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Veblen hierarchies, some subsystems of second-order logic, slow-growing and fast-growing hierarchies including Girard's result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to Kruskal, the 'tree theorem', as well as a 'finite miniaturization' of Kruskal's theorem due to Harvey Friedman. These versions of Kruskal's theorem are remarkable from a proof-theoretic point of view because they are not provable in relatively strong logical systems. They are examples of so-called 'natural independence phenomena', which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Godel. Kruskal's tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of Knuth-Bendix completion procedures. There is also a close connection between a certain infinite countable ordinal called GAMMA-0 and Kruskal's theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence. |
Year | Venue | DocType |
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1991 | ANNALS OF PURE AND APPLIED LOGIC | Journal |
Volume | Issue | ISSN |
53 | 3 | 0168-0072 |
Citations | PageRank | References |
10 | 0.95 | 0 |
Authors | ||
1 |
Name | Order | Citations | PageRank |
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Jean H. Gallier | 1 | 749 | 111.86 |