Basic Set Theory, Volume 13Although this book deals with basic set theory (in general, it stops short of areas where model-theoretic methods are used) on a rather advanced level, it does it at an unhurried pace. This enables the author to pay close attention to interesting and important aspects of the topic that might otherwise be skipped over. Written for upper-level undergraduate and graduate students, the book is divided into two parts. The first covers pure set theory, including the basic notions, order and well-foundedness, cardinal numbers, the ordinals, and the axiom of choice and some of its consequences. The second part deals with applications and advanced topics, among them a review of point set topology, the real spaces, Boolean algebras, and infinite combinatorics and large cardinals. A helpful appendix deals with eliminability and conservation theorems, while numerous exercises supply additional information on the subject matter and help students test their grasp of the material. 1979 edition. 20 figures. |
Contents
The Basic Notions | 3 |
Order and WellFoundedness | 32 |
Cardinal Numbers | 76 |
The Ordinals | 112 |
The Axiom of Choice and Some of its Consequences | 158 |
A Review of Point Set Topology | 199 |
The Real Spaces | 216 |
Boolean Algebras | 244 |
Infinite Combinatorics and Large Cardinals | 289 |
The Eliminability and Conservation Theorems | 357 |
| 367 | |
Additional Bibliography | 376 |
| 383 | |
Corrections and Additions | 393 |
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Common terms and phrases
aleph assume the axiom axiom of choice axiom of replacement axiom schema Baire space basic language bijection branch of length Cantor cf(a choice function class terms class variables closed set closed unbounded subset cofinal coloring complete Boolean algebra continuum hypothesis contradicting countable define denote denumerable Dom(F equinumerous equivalent Exercise existence extended language finite set finite subset follows formula function F hence Hint of proof holds homomorphism inaccessible cardinal induction hypothesis infinite initial segment injection isomorphism least member lemma Let F limit ordinal metric space natural numbers obtain order type ordered cardinal ordered class ordered set pairwise disjoint partially ordered poset prime ideal Proposition prove in ZFC real numbers regular open sets relation Rng(f sequence set theory Souslin statement stationary subset subclass of Q successor ordinal theorem topological space topology ultrafilter weakly compact weakly inaccessible cardinals well-orderable well-ordering ZFGC