Notes on Set TheorySpringer Science & Business Media, 2006 M06 15 - 278 páginas The axiomatic theory of sets is a vibrant part of pure mathematics, with its own basic notions, fundamental results, and deep open problems. It is also viewed as a foundation of mathematics so that "to make a notion precise" simply means "to define it in set theory." This book gives a solid introduction to "pure set theory" through transfinite recursion and the construction of the cumulative hierarchy of sets, and also attempts to explain how mathematical objects can be faithfully modeled within the universe of sets. In this new edition the author has added solutions to the exercises, and rearranged and reworked the text to improve the presentation. |
Contenido
| 1 | |
EQUINUMEROSITY | 7 |
PARADOXEs AND AXIOMS | 19 |
ARE SETS ALL THERE Is? | 33 |
Cardinal numbers 42 Structured sets 44 Problems for Chapter | 45 |
Recursion Theorem 53 Addition and multiplication 58 Pigeonhole | 67 |
Point Theorem 76 About topology 79 Graphs 82 Problems for Chapter 6 | 83 |
WELL ORDERED SETS | 89 |
Analytic pointsets 141 Perfect Set Theorem 144 Borel sets 147 | 154 |
Transitive classes 161 Basic Closure Lemma 162 The grounded | 165 |
Mostowski Collapsing Lemma 170 Consistency and independence results | 171 |
Ordinal recursion 182 Ordinal addition and multiplication 183 | 187 |
Problems for Chapter 12 190 The operation 10 194 Strongly inaccessible cardinals | 197 |
Countable dense linear orderings 208 The archimedean property 210 Nested | 204 |
interval property 213 Dedekind cuts 216 Existence of the real numbers | 217 |
APPENDIX B AXIOMS AND UNIVERSES | 225 |
rability Ofwell ordered sets 99 Wellfoundedness Of go 100 Hartogs Theorem | 100 |
CHOICES | 109 |
Zorns Lemma 114 Countable Principle of Choice ACN 114 Axiom VII | 119 |
Problems for Chapter 9 | 130 |
Riegers Theorem 233 Antifoundation Principle AFA 238 Bisimulations 239 | 245 |
| 271 | |
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Términos y frases comunes
A I B atoms Axiom of Choice Axiom of Replacement axiomatic Baire space basic bijection binary bisimulation Borel Cantor cardinal number Cauchy Chapter compute construct Continuum Hypothesis contradiction countable countable sets define definite condition definite operation Domain(p equinumerosity equinumerous Exercise exists exactly find first fixed point Fixed Point Theorem function f graph G grounded set hence hypothesis implies inductive poset infinite initial segment initial similarity injection isomorphism Least Fixed Point least upper bound Lemma limn mapping mathematical monotone natural numbers non-empty objects ordered field ordered set ordinal partial function partial ordering pointset powerset Principle Problem PROOF properties proposition prove real numbers Recursion Theorem Rieger universe satisfies the identities seg(x sequence set theory specific subset Suppose surjection TC(A topological transitive transitive set trivial unique wellordering ZDC+AC Zermelo Zermelo universe