eBook ISBN:  9780821876558 
Product Code:  CONM/65.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9780821876558 
Product Code:  CONM/65.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 

Book DetailsContemporary MathematicsVolume: 65; 1987; 394 ppMSC: Primary 03; Secondary 05
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

Articles

V. Michele Abrusci — Dilators, generalized Goodstein sequences, independence results: a survey [ MR 891240 ]

V. M. Abrusci, J.Y. Girard and J. Van de Wiele — Some uses of dilators in combinatorial problems. I [ MR 891241 ]

Bruce Ackman and Jim Owings — Cross products of Souslin trees [ MR 891242 ]

Vitaly Bergelson — Ergodic Ramsey theory [ MR 891243 ]

Andreas Blass — Ultrafilters related to Hindman’s finiteunions theorem and its extensions [ MR 891244 ]

Andreas R. Blass, Jeffry L. Hirst and Stephen G. Simpson — Logical analysis of some theorems of combinatorics and topological dynamics [ MR 891245 ]

J. E. Baumgartner and A. Hajnal — A remark on partition relations for infinite ordinals with an application to finite combinatorics [ MR 891246 ]

Stephen H. Brackin — A summary of “On Ramseytype theorems and their provability in weak formal systems” [ MR 891247 ]

Wilfried Buchholz and Stan Wainer — Provably computable functions and the fast growing hierarchy [ MR 891248 ]

Fons van Engelen, Arnold W. Miller and John Steel — Rigid Borel sets and better quasiorder theory [ MR 891249 ]

P. Erdős — Some problems on finite and infinite graphs [ MR 891250 ]

Harvey Friedman, Neil Robertson and Paul Seymour — The metamathematics of the graph minor theorem [ MR 891251 ]

Neil Hindman — Summable ultrafilters and finite sums [ MR 891252 ]

Martin Loebl and Jiří Matoušek — On undecidability of the weakened Kruskal theorem [ MR 891253 ]

Jaroslav Nešetřil and Robin Thomas — Well quasiorderings, long games and a combinatorial study of undecidability [ MR 891254 ]

Mitsuhiro Okada and Gaisi Takeuti — On the theory of quasiordinal diagrams [ MR 891255 ]

P. Pudlák — Improved bounds to the length of proofs of finitistic consistency statements [ MR 891256 ]

J.P. Ressayre — Nonstandard universes with strong embeddings, and their finite approximations [ MR 891257 ]

Stephen G. Simpson — Unprovable theorems and fastgrowing functions [ MR 891258 ]


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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 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.

Articles

V. Michele Abrusci — Dilators, generalized Goodstein sequences, independence results: a survey [ MR 891240 ]

V. M. Abrusci, J.Y. Girard and J. Van de Wiele — Some uses of dilators in combinatorial problems. I [ MR 891241 ]

Bruce Ackman and Jim Owings — Cross products of Souslin trees [ MR 891242 ]

Vitaly Bergelson — Ergodic Ramsey theory [ MR 891243 ]

Andreas Blass — Ultrafilters related to Hindman’s finiteunions theorem and its extensions [ MR 891244 ]

Andreas R. Blass, Jeffry L. Hirst and Stephen G. Simpson — Logical analysis of some theorems of combinatorics and topological dynamics [ MR 891245 ]

J. E. Baumgartner and A. Hajnal — A remark on partition relations for infinite ordinals with an application to finite combinatorics [ MR 891246 ]

Stephen H. Brackin — A summary of “On Ramseytype theorems and their provability in weak formal systems” [ MR 891247 ]

Wilfried Buchholz and Stan Wainer — Provably computable functions and the fast growing hierarchy [ MR 891248 ]

Fons van Engelen, Arnold W. Miller and John Steel — Rigid Borel sets and better quasiorder theory [ MR 891249 ]

P. Erdős — Some problems on finite and infinite graphs [ MR 891250 ]

Harvey Friedman, Neil Robertson and Paul Seymour — The metamathematics of the graph minor theorem [ MR 891251 ]

Neil Hindman — Summable ultrafilters and finite sums [ MR 891252 ]

Martin Loebl and Jiří Matoušek — On undecidability of the weakened Kruskal theorem [ MR 891253 ]

Jaroslav Nešetřil and Robin Thomas — Well quasiorderings, long games and a combinatorial study of undecidability [ MR 891254 ]

Mitsuhiro Okada and Gaisi Takeuti — On the theory of quasiordinal diagrams [ MR 891255 ]

P. Pudlák — Improved bounds to the length of proofs of finitistic consistency statements [ MR 891256 ]

J.P. Ressayre — Nonstandard universes with strong embeddings, and their finite approximations [ MR 891257 ]

Stephen G. Simpson — Unprovable theorems and fastgrowing functions [ MR 891258 ]