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