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

Couverture
Springer Science & Business Media, 2003 - 536 pages

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 a oegenetica 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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Table des matières

Beginnings
15
1 Inaccessibility
16
2 Measurability
22
3 Constructibility
28
4 Compactness
36
5 Elementary Embeddings
44
6 lndescribability
57
Partition Properties
69
19 Iterated Ultrapowers
244
20 Inner Models of Measurability
261
21 Embeddings 0 and 0
277
Strong Hypotheses
297
22 Supercompactness
298
23 Extendibility to Inconsistency
311
24 The Strongest Hypotheses
325
25 Combinatorics of Pky
340

7 Partitions and Trees
70
8 Partitions and Structures
85
9 Indiscernibles and 0
99
Forcing and Sets of Reals
113
10 Development of Forcing
114
11 Lebesgue Measurability
132
12 Descriptive Set Theory
145
13 𝚷¹₁ Sets and 𝚺¹₂ Sets
162
14 𝚺¹₂ Sets and Sharps
178
15 Sharps and 𝚺¹₃ Sets
192
Aspects of Measurability
209
16 Saturated Ideals I
210
17 Saturated Ideals II
220
18 Prikry Forcing
234
26 Extenders
352
Determinacy
367
27 Infinite Games
368
28 AD and Combinatorics
383
29 Prewellorderings
403
30 Scales and Projective Ordinals
417
3l Det𝜶𝚷¹₁
437
32 Consistency of AD
450
Chart of Cardinals
472
Appendix
473
Indexed References
483
Subject Index
531
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