The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings / Edition 2

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This is the softcover reprint of the very popular hardcover edition. The theory of large cardinals is currently a broad mainstream of modern set theory, the main area of investigation for the analysis of the relative consistency of mathematical propositions and possible new axioms for mathematics. The first of a projected multi-volume series, this book provides a comprehensive account of the theory of large cardinals from its beginnings and some of the direct outgrowths leading to the frontiers of contemporary research. A "genetic" approach is taken, presenting the subject in the context of its historical development. With hindsight the consequential avenues are pursued and the most elegant or accessible expositions given. With open questions and speculations provided throughout the reader should not only come to appreciate the scope and coherence of the overall enterprise but also become prepared to pursue research in several specific areas by studying the relevant sections.
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Product Details

  • ISBN-13: 9783540003847
  • Publisher: Springer-Verlag New York, LLC
  • Publication date: 7/29/2003
  • Series: Springer Monographs in Mathematics
  • Edition description: REV
  • Edition number: 2
  • Pages: 564
  • Product dimensions: 6.14 (w) x 9.21 (h) x 1.31 (d)

Table of Contents

Ch. 1 Beginnings
1 Inaccessibility 16
2 Measurability 22
3 Constructibility 28
4 Compactness 36
5 Elementary Embeddings 44
6 Indescribability 57
Ch. 2 Partition Properties
7 Partitions and Trees 70
8 Partitions and Structures 85
9 Indiscernibles and 0[superscript #] 99
Ch. 3 Forcing and Sets of Reals
10 Development of Forcing 114
11 Lebesgue Measurability 132
12 Descriptive Set Theory 145
13 [Pi][subscript 1][superscript 1] Sets and [Sigma][subscript 2][superscript 1] Sets 162
14 [Sigma][subscript 2][superscript 1] Sets and Sharps 178
15 Sharps and [Sigma][subscript 3][superscript 1] Sets 192
Ch. 4 Aspects of Measurability
16 Saturated Ideals I 210
17 Saturated Ideals II 220
18 Prikry Forcing 234
19 Iterated Ultrapowers 244
20 Inner Models of Measurability 261
21 Embeddings, 0[superscript #], and 0[actual symbol not reproducible] 277
Ch. 5 Strong Hypotheses
22 Supercompactness 298
23 Extendibility to Inconsistency 311
24 The Strongest Hypotheses 325
25 Combinatorics of [actual symbol not reproducible] 340
26 Extenders 352
Ch. 6 Determinacy
27 Infinite Games 368
28 AD and Combinatorics 383
29 Prewellorderings 403
30 Scales and Projective Ordinals 417
31 Det([alpha]-[Pi][subscript 1][superscript 1]) 437
32 Consistency of AD 450
Chart of Cardinals 472
Appendix 473
Indexed References 483
Subject Index 531
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