Contemporary Mathematics 1987; 394 pp; softcover Volume: 65 Reprint/Revision History: reprinted 1990 ISBN10: 0821850520 ISBN13: 9780821850527 List Price: US$59 Member Price: US$47.20 Order Code: CONM/65
 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 AMSIMSSIAM 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 fastgrowing 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 lengthofproof 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 firstyear 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 Ramseytype 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 fastgrowing functions
