Inner Models and Large Cardinals

By: Martin Zeman

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Hardcover | 1 January 2002

At a Glance

Hardcover
384 Pages

Age
22

Grade
17

Dimensions(cm)
23.5 x 17.15 x 1.91

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This volume is an introduction to inner model theory, an area of set theory which is concerned with fine structural inner models reflecting large cardinal properties of the set theoretic universe.

The monograph contains a detailed presentation of general fine structure theory as well as a modern approach to the construction of small core models, namely those models containing at most one strong cardinal, together with some of their applications. The final part of the book is devoted to a new approach encompassing large inner models which admit many Woodin cardinals.

The exposition is self-contained and does not assume any special prerequisities, which should make the text comprehensible not only to specialists but also to advanced students in Mathematical Logic and Set Theory.

Industry Reviews

Insgesamt ist Zeman ein ausgezeichnetes Lehrbuch uber Kernmodelltheorie gelungen. Es ist sehr gut zum Selbststudium geeignet.
H.-D. Donder, Jahresbericht der Deutschen Mathematiker-Vereinigung 106, 3-2004

This elegant volume provides a systematic introduction to the theory of fine structural inner models of set theory. [...] The author can fairly claim to have systematized the theory through to strong cardinals [...]. Mathematical Reviews

The book is a well-written and self-contained introduction to the theory of fine structural inner models, an area of set theory concerned with questions about models of set theory. [...] The author gives an introduction to the techniques of this theory; so the book can serve as a textbook even for graduate students. Zentralblatt fur Mathematik

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Preface	p. v
Fine Structure	p. 1
Acceptable J-Structures	p. 1
The [Sigma subscript 1]-Projectum	p. 6
Downward Extension of Embeddings Lemmata	p. 8
Upward Extension of Embeddings Lemma	p. 12
Iterated Projecta	p. 18
[Sigma]*-Relations	p. 20
[Sigma superscript (n) subscript l]-Embeddings	p. 23
Substitution and Good Functions	p. 26
Standard Parameters	p. 33
Two Applications to L	p. 36
More on Downward Extensions of Embeddings	p. 38
Witnesses and Solidity	p. 41
Notes	p. 45
Extenders and Coherent Structures	p. 47
Extenders	p. 47
The Hypermeasure Representation of Extenders	p. 54
Amenability	p. 56
Coherent Structures	p. 58
Extendibility	p. 61
Strong Cardinals	p. 68
Notes	p. 70
Fine Ultrapowers	p. 71
The *-Ultrapower Construction	p. 71
Some Special Preservation Properties	p. 82
When F Is Close to M	p. 85
Extendibility	p. 89
k-Ultrapowers	p. 93
Pseudoultrapowers	p. 96
Notes	p. 108
Mice and Iterability	p. 109
Premice	p. 109
Iterations	p. 114
Copying and the Dodd--Jensen Lemma	p. 119
Comparison Process	p. 127
Some Iterability Criteria	p. 131
Bicephali	p. 142
Notes	p. 145
Solidity and Condensation	p. 146
Cores and Coiterations	p. 146
The Solidity Theorem	p. 150
Consequences of Solidity	p. 155
The Canonical Ordering of Mice	p. 160
Condensation Lemma	p. 163
Upwards Extensions to Premice	p. 167
Notes	p. 174
Extender Models	p. 175
Extender Models and Iterations	p. 175
The Canonical Ordering of Weasels	p. 178
Universality	p. 183
The Model K[superscript c]	p. 186
0[superscript ++]	p. 198
Weak Covering	p. 203
Notes	p. 211
The Core Model	p. 212
Inductive Definition of K	p. 212
Steel's Definition of K	p. 214
The Existence of K	p. 218
Embeddings of K and Generic Absoluteness	p. 230
Weak Covering for K	p. 235
Notes	p. 249
One Strong Cardinal	p. 251
Premice	p. 251
Properties of Mice	p. 258
Extender Models up to One Strong Cardinal	p. 269
Notes	p. 279
Overlapping Extenders	p. 280
Premice and Iteration Trees	p. 281
Copying and the Dodd--Jensen Property	p. 299
Solidity and Condensation	p. 318
Uniqueness of Well-Founded Branches	p. 345
Towards the Ultimate Model K[superscript c]	p. 355
Notes	p. 358
Bibliography	p. 359
Index	p. 365
Table of Contents provided by Syndetics. All Rights Reserved.

Inner Models and Large Cardinals

At a Glance

Hardcover

Industry Reviews

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