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Logic and Combinatorics
Edited by: Stephen G. Simpson
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Contemporary Mathematics
1987; 394 pp; softcover
Volume: 65
Reprint/Revision History:
reprinted 1990
ISBN-10: 0-8218-5052-0
ISBN-13: 978-0-8218-5052-7
List Price: US$56
Member Price: US$44.80
Order Code: CONM/65
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In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Gödel. Because of the diversity of the lines of research that have begun to shed light on these issues, there was a need for a comprehensive overview which would tie the lines together.

This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California. With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of "unprovable theorems and fast-growing functions." Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Gödel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level.

Table of Contents

  • V. M. Abrusci -- Dilators, generalized Goodstein sequences, independence results: A survey
  • V. M. Abrusci, J.-Y. Girard, and J. Van de Wiele -- Some uses of dilators in combinatorial problems, part I
  • B. Ackman and J. Owings -- Cross products of Souslin trees
  • V. Bergelson -- Ergodic Ramsey theory
  • A. Blass -- Ultrafilters related to Hindman's finite unions theorem and its extensions
  • A. R. Blass, J. L. Hirst, and S. G. Simpson -- Logical analysis of some theorems of combinatorics and topological dynamics
  • J. E. Baumgartner and A. Hajnal -- A remark on partition relations for infinite ordinals, with an application to finite combinatorics
  • S. H. Brackin -- A summary of "On Ramsey-type Theorems and Their Provability in Weak Formal Systems"
  • W. Buchholz and S. Wainer -- Provably computable functions and the fast growing hierarchy
  • F. van Engelen, A. W. Miller, and J. Steel -- Rigid Borel sets and better quasiorder theory
  • P. Erdös -- Some problems on finite and infinite graphs
  • H. Friedman, N. Robertson, and P. D. Seymour -- The metamathematics of the graph minor theorem
  • N. Hindman -- Summable ultrafilters and finite sums
  • M. Loebl and J. Matoušek -- On undecidability of the weakened Kruskal theorem
  • J. Nešetřil and R. Thomas -- Well quasi orderings, long games, and a combinatorial study of undecidability
  • M. Okada and G. Takeuti -- On the theory of quasi ordinal diagrams
  • P. Pudlák -- Improved bounds to the length of proofs of finitistic consistency statements
  • J.-P. Ressayre -- Non standard universes with strong embeddings, and their finite approximations
  • S. G. Simpson -- Unprovable theorems and fast-growing functions
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