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

Front Cover
Springer Science & Business Media, Nov 23, 2008 - Mathematics - 538 pages
0 Reviews
Reviews aren't verified, but Google checks for and removes fake content when it's identified
The higher in?nite refers to the lofty reaches of the in?nite cardinalities of set t- ory as charted out by large cardinal hypotheses. These hypotheses posit cardinals that prescribe their own transcendence over smaller cardinals and provide a sup- structure for the analysis of strong propositions. As such they are the rightful heirs to the two main legacies of Georg Cantor, founder of set theory: the extension of number into the in?nite and the investigation of de?nable sets of reals. The investigation of large cardinal hypotheses is indeed a mainstream of modern set theory, and they have been found to play a crucial role in the study of de?nable sets of reals, in particular their Lebesgue measurability. Although formulated at various stages in the development of set theory and with different incentives, the hypotheses were found to form a linear hierarchy reaching up to an inconsistent extension of motivating concepts. All known set-theoretic propositions have been gauged in this hierarchy in terms of consistency strength, and the emerging str- ture of implications provides a remarkably rich, detailed and coherent picture of the strongest propositions of mathematics as embedded in set theory. The ?rst of a projected multi-volume series, this text provides a comp- hensive account of the theory of large cardinals from its beginnings through the developments of the early 1970’s and several of the direct outgrowths leading to the frontiers of current research.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

0 Preliminaries
1
20
6
21
7
Beginnings
15
Partition Properties
68
Forcing and Sets of Reals
113
Aspects of Measurability
208
Inner Models of Measurability
261
25 Combinatorics of PKy
343
26 Extenders
352
Deterrninacy 27 Infinite Games
368
28 AD and Combinatorics
383
29 Prewellorderings
403
30 Scales and Projective Ordinals
430
31 Dem111
437
32 Consistency of AD
450

Embeddings 0 and 0T
277
Strong Hypotheses
297
22 Supercompactness
298
23 Extendibility to Inconsistency
311
24 The Strongest Hypotheses
325
Chart of Cardinals
472
Appendix
473
Indexed References
483
Subject Index
531
Copyright

Other editions - View all

Common terms and phrases

Bibliographic information