Walks on Ordinals and Their Characteristics (eBook)

Stevo Todorcevic (Autor)

eBook Download: PDF

2007 | 2007
VI, 324 Seiten
Springer Basel (Verlag)
978-3-7643-8529-3 (ISBN)

Infos

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.

Inhalt

Contents 6
Introduction 8
1.1 Walks and the metric theory of ordinals 8
1.2 Summary of results 17
1.3 Prerequisites and notation 24
1.4 Acknowledgements 25
Walks on Countable Ordinals 26
2.1 Walks on countable ordinals and their basic characteristics 26
2.2 The coherence of maximal weights 36
2.3 Oscillations of traces 47
2.4 The number of steps and the last step functions 54
Metric Theory of Countable Ordinals 61
3.1 Triangle inequalities 61
3.2 Constructing a Souslin tree using 64
3.3 A Hausdor. gap from 69
3.4 A general theory of subadditive functions on 72
3.5 Conditional weakly null sequences based on subadditive functions 83
Coherent Mappings and Trees 97
4.1 Coherent mappings 97
4.2 Lipschitz property of coherent trees 101
4.3 The global structure of the class of coherent trees 114
4.4 Lexicographically ordered coherent trees 130
4.5 Stationary C-lines 134
The Square-bracket Operation on Countable Ordinals 139
5.1 The upper trace and the square-bracket operation 139
5.2 Projecting the square-bracket operation 145
5.3 Some geometrical applications of the square- bracket operation 150
5.4 A square-bracket operation from a special Aronszajn tree 158
5.5 A square-bracket operation from the complete binary tree 163
General Walks and Their Characteristics 167
6.1 The full code and its application in characterizing Mahlo cardinals 167
6.2 The weight function and its local versions 180
6.3 Unboundedness of the number of steps 184
Square Sequences 192
7.1 Square sequences and their full lower traces 192
7.2 Square sequences and local versions of 200
7.3 Special square sequence and the corresponding function p 207
7.4 The function p on successors of regular cardinals 210
7.5 Forcing constructions based on p 218
7.6 The function p on successors of singular cardinals 225
The Oscillation Mapping and the Square-bracket Operation 237
8.1 The oscillation mapping 237
8.2 The trace filter and the square-bracket operation 247
8.3 Projections of the square-bracket operation on accessible cardinals 255
8.4 Two more variations on the square-bracket operation 261
Unbounded Functions 274
9.1 Partial square-sequences 274
9.2 Unbounded subadditive functions 276
9.3 Chang’s conjecture and È2 280
9.4 Higher dimensions and the continuum hypothesis 286
Higher Dimensions 292
10.1 Stepping-up to higher dimensions 292
10.2 Chang’s conjecture as a 3-dimensional Ramsey-theoretic statement 297
10.3 Three-dimensional oscillation mapping 301
10.4 Two-cardinal walks 308
Bibliography 316
Index 324

Produkt-Details

• Erscheint lt. Verlag: 15.12.2007
• Reihe/Serie: Progress in Mathematics
• Reihe/Serie: Progress in Mathematics
• Zusatzinfo: VI, 324 p.
• Verlagsort: Basel
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: Algebra • coherent mapping • combinatorics • countable ordinals • Function • mapping • Mathematics • ordinal • Random Walk • square sequence • Theorem • Topology
• ISBN-10: 3-7643-8529-4 / 3764385294
• ISBN-13: 978-3-7643-8529-3 / 9783764385293