Walks on ordinals and their characteristics / Stevo Todorcevic.

Format:

Book

Author/Creator:

Todorcevic, Stevo.

Series:

Progress in mathematics (Boston, Mass.) ; v. 263.

Progress in mathematics ; v. 263

Language:

English

Subjects (All):

Numbers, Ordinal.

Number theory.

Physical Description:

vi, 324 pages : illustrations ; 24 cm.

Place of Publication:

Basel ; Boston : Birkhäuser, 2007.

Summary:

The analysis of the characteristics of walks on ordinals is a powerful new technique for building mathematical structures, developed by the author over the last twenty years. This is the first book-length exposition of this method. Particular emphasis is placed on applications which are presented in a unified and comprehensive manner and which stretch across several areas of mathematics such as set theory, combinatorics, general topology, functional analysis, and general algebra. The intended audience for this book are graduate students and researchers working in these areas interested in mastering and applying these methods.

Contents:

1.1 Walks and the metric theory of ordinals 1

1.3 Prerequisites and notation 17

2 Walks on Countable Ordinals

2.1 Walks on countable ordinals and their basic characteristics 19

2.2 The coherence of maximal weights 29

2.3 Oscillations of traces 40

2.4 The number of steps and the last step functions 47

3 Metric Theory of Countable Ordinals

3.1 Triangle inequalities 55

3.2 Constructing a Souslin tree using [rho] 58

3.3 A Hausdorff gap from [rho] 63

3.4 A general theory of subadditive functions on [omega] 66

3.5 Conditional weakly null sequences based on subadditive functions 77

4 Coherent Mappings and Trees

4.1 Coherent mappings 91

4.2 Lipschitz property of coherent trees 95

4.3 The global structure of the class of coherent trees 108

4.4 Lexicographically ordered coherent trees 124

4.5 Stationary C-lines 128

5 The Square-bracket Operation on Countable Ordinals

5.1 The upper trace and the square-bracket operation 133

5.2 Projecting the square-bracket operation 139

5.3 Some geometrical applications of the square-bracket operation 144

5.4 A square-bracket operation from a special Aronszajn tree 152

5.5 A square-bracket operation from the complete binary tree 157

6 General Walks and Their Characteristics

6.1 The full code and its application in characterizing Mahlo cardinals 161

6.2 The weight function and its local versions 174

6.3 Unboundedness of the number of steps 178

7 Square Sequences

7.1 Square sequences and their full lower traces 187

7.2 Square sequences and local versions of [rho] 195

7.3 Special square sequence and the corresponding function [rho] 202

7.4 The function [rho] on successors of regular cardinals 205

7.5 Forcing constructions based on [rho] 213

7.6 The function [rho] on successors of singular cardinals 220

8 The Oscillation Mapping and the Square-bracket Operation

8.1 The oscillation mapping 233

8.2 The trace filter and the square-bracket operation 243

8.3 Projections of the square-bracket operation on accessible cardinals 251

8.4 Two more variations on the square-bracket operation 257

9 Unbounded Functions

9.1 Partial square-sequences 271

9.2 Unbounded subadditive functions 273

9.3 Chang's conjecture and [Theta subscript 2] 277

9.4 Higher dimensions and the continuum hypothesis 283

10 Higher Dimensions

10.1 Stepping-up to higher dimensions 289

10.2 Chang's conjecture as a 3-dimensional Ramsey-theoretic statement 294

10.3 Three-dimensional oscillation mapping 298

10.4 Two-cardinal walks 305.

Notes:

Includes bibliographical references (pages [313]-320) and index.

ISBN:

3764385286

9783764385286

OCLC:

166357947