The walks on ordinals and analysis of their characteristics is a subject matter started by the author some twenty years ago in order to disprove a particular extension of the Ramsey theorem. A further analysis has shown however that the resulting method is quite useful in detecting critical mathematical objects in contexts where only rough classifications are possible. The book gives a careful and comprehensive account of the method and gathers many of these applications in a unified and comprehensive manner.
Author(s): Stevo Todorcevic
Series: Progress in Mathematics
Edition: 2007
Publisher: Birkhäuser
Year: 2007
Language: English
Pages: 330
Contents......Page 6
1.1 Walks and the metric theory of ordinals......Page 8
1.2 Summary of results......Page 17
1.3 Prerequisites and notation......Page 24
1.4 Acknowledgements......Page 25
2.1 Walks on countable ordinals and their basic characteristics......Page 26
2.2 The coherence ofmaximal weights......Page 36
2.3 Oscillations of traces......Page 47
2.4 The number of steps and the last step functions......Page 54
3.1 Triangle inequalities......Page 62
3.2 Constructing a Souslin tree using ρ......Page 65
3.3 A Hausdorff gap from ρ......Page 70
3.4 A general theory of subadditive functions on ω[sub(1)]......Page 73
3.5 Conditional weakly null sequences based on subadditive functions......Page 84
4.1 Coherentmappings......Page 98
4.2 Lipschitz property of coherent trees......Page 102
4.3 The global structure of the class of coherent trees......Page 115
4.4 Lexicographically ordered coherent trees......Page 131
4.5 Stationary C-lines......Page 135
5.1 The upper trace and the square-bracket operation......Page 140
5.2 Projecting the square-bracket operation......Page 146
5.3 Some geometrical applications of the square-bracket operation......Page 151
5.4 A square-bracket operation from a special Aronszajn tree......Page 159
5.5 A square-bracket operation from the complete binary tree......Page 164
6.1 The full code and its application in characterizing Mahlo cardinals......Page 168
6.2 The weight function and its local versions......Page 181
6.3 Unboundedness of the number of steps......Page 185
7.1 Square sequences and their full lower traces......Page 194
7.2 Square sequences and local versions of ρ......Page 202
7.3 Special square sequence and the corresponding function ρ......Page 209
7.4 The function ρ on successors of regular cardinals......Page 212
7.5 Forcing constructions based on ρ......Page 220
7.6 The function ρ on successors of singular cardinals......Page 227
8.1 The oscillation mapping......Page 240
8.2 The trace filter and the square-bracket operation......Page 250
8.3 Projections of the square-bracket operation on accessible cardinals......Page 258
8.4 Two more variations on the square-bracket operation......Page 264
9.1 Partial square-sequences......Page 278
9.2 Unbounded subadditive functions......Page 280
9.3 Chang's conjecture and Θ[sub(2)]......Page 284
9.4 Higher dimensions and the continuum hypothesis......Page 290
10.1 Stepping-up to higher dimensions......Page 296
10.2 Chang's conjecture as a 3-dimensional Ramsey-theoretic statement......Page 301
10.3 Three-dimensional oscillation mapping......Page 305
10.4 Two-cardinal walks......Page 312
Bibliography......Page 320
Index......Page 328
M......Page 330
W......Page 331
