1. Introduction
1.1. Walks and the metric theory of ordinals ............... 1
1.2. Summary of results ................................... 10
1.3. Prerequisites and notation ........................... 17
1.4. Acknowledgements ..................................... 18
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 ρ .................. 58
3.3. A Hausdorff gap from ρ ............................... 63
3.4. A general theory of subadditive functions on ω1 ...... 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 ρ ............ 195
7.3. Special square sequence and the corresponding
function ρ .......................................... 202
7.4. The function ρ on successors of regular cardinals ... 205
7.5. Forcing constructions based on ρ .................... 213
7.6. The function ρ 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 Θ2 ........................... 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
Bibliography .................................................. 313
Index ......................................................... 321
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