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Walks on Ordinals and Their Characteristics 2007 ed. [Hardback]

  • Formāts: Hardback, 324 pages, height x width: 235x155 mm, weight: 661 g, VI, 324 p., 1 Hardback
  • Sērija : Progress in Mathematics 263
  • Izdošanas datums: 17-Sep-2007
  • Izdevniecība: Birkhauser Verlag AG
  • ISBN-10: 3764385286
  • ISBN-13: 9783764385286
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Walks on Ordinals and Their Characteristics 2007 ed.Palielināt
  • Formāts: Hardback, 324 pages, height x width: 235x155 mm, weight: 661 g, VI, 324 p., 1 Hardback
  • Sērija : Progress in Mathematics 263
  • Izdošanas datums: 17-Sep-2007
  • Izdevniecība: Birkhauser Verlag AG
  • ISBN-10: 3764385286
  • ISBN-13: 9783764385286
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783764385286.html
Keywords:
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.

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.
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 p on successors of regular cardinals
205
7.5 Forcing constructions based on ρ
213
7.6 The function p 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