**Contemporary Mathematics**

Volume: 65;
1987;
394 pp;
Softcover

MSC: Primary 03;
Secondary 05

**Print ISBN: 978-0-8218-5052-7
Product Code: CONM/65**

List Price: $63.00

AMS Member Price: $50.40

MAA Member Price: $56.70

**Electronic ISBN: 978-0-8218-7655-8
Product Code: CONM/65.E**

List Price: $59.00

AMS Member Price: $47.20

MAA Member Price: $53.10

# Logic and Combinatorics

Share this page *Edited by *
*Stephen G. Simpson*

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

## Logic and Combinatorics

- Table of Contents vii8 free
- Preface ix10 free
- Dilators, Generalized Goodstein Sequences, Independence Results: A Survey 114 free
- Some Uses of Dilators in Combinatorial Problems, Part I 2538
- Cross Products of Souslin Trees 5568
- Ergodic Ramsey Theory 6376
- Ultrafilters Related to Hindman's Finite Unions Theorem and its Extensions 89102
- Logical Analysis of Some Theorems of Combinatorics and Topological Dynamics 125138
- A Remark on Partition Relations for Infinite Ordinals, With an Application to Finite Combinatorics 157170
- A Summary of "On Ramsey-type Theorems and Their Provability in Weak Formal Systems" 169182
- Provably Computable Functions and the Fast Growing Hierarcy 179192
- Rigid Borel Sets and Better Quasiorder Theory 199212
- Some Problems on Finite and Infinite Graphs 223236
- The Metamathematics of the Graph Minor Theorem 229242
- Summable Ultrafilters and Finite Sums 263276
- On Undecidability of the Weakened Kruskal Theorem 275288
- Well Quasi Orderings, Long Games, and a Combinatorial Study of Undecidability 281294
- On the Theory of Quasi Ordinal Diagrams 295308
- Improved Bounds to the Length of Proofs of Finitistic Consistency Statements 309322
- Non Standard Universes with Strong Embeddings, and Their Finite Approximations 333346
- Unprovable Theorems and Fast-Growing Functions 359372