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Sets: naive, axiomatic and applied: a basic compendium with exercises for use in set theory for non logicians, working and teaching mathematicians and students

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Bibliographic Details
Main Authors:	Dalen, Dirk van 1932- (Author), Doets, Kees (Author), De Swart, Harrie 1944- (Author)
Format:	Book
Language:	English Dutch
Published:	Oxford [u.a.] Pergamon Press 1978
Series:	International series in pure and applied mathematics 106
Subjects:	Ensembles, Théorie axiomatique des Ensembles, Théorie des Verzamelingen (wiskunde) Axiomatic set theory Set theory Mengenlehre
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001475871&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Physical Description:	XVII, 342 S. graph. Darst.
ISBN:	0080211666 0080230474

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Record in the Search Index

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adam_text	CONTENTS Preface i x Acknowledgements xiii Introduction xv Chapter I. Naifve Set Theory 1. Some important sets and notations 1 Natural numbers, integers, rationals, reals, singleton. 2. Equality of sets 3 Extensionality axiom. 3. Subsets 4 4. The NaVve Comprehension principle and the empty set 6 5. Union, intersection and relative complement 9 Complement, de Morgan s laws. 6. Power set 18 7. Unions and intersections of families 21 Greatest and smallest set with property #, disjoint, pairwise disjoint. 8. Ordered pairs 29 Unordered pairs, ordered n tuples. 9. Cartesian product 32 10. Relations 36 Binary relation, converse, composition, identity relation, domain, range, reflexive, symmetric, transitive. 11. Equivalence relations 41 Equivalence class, representative, partition, quotient set, z, Q, 12. Real numbers 51 Chains of segments, Dedekind cut, Cauchy sequence. 13. Functions (mappings) 56 v vi Contents Injection, surjection, bijection, identity mapping, characteristic function, equality of functions, composition, inverse, restriction, structure, isomorphism, cartesian product of a family. 14. Orderings 78 Lexicographic ordering, partial ordering, diagram, minimal element, smallest element, lower bound, infimum, total(linear) ordering, well ordering, immediate successor, principle of transfinite induction, initial segment. 15. Equivalence I cardinality) 92 , diagonal method, , denumerable, countable, Cantor Bernstein, uncountable. 16. Finite and infinite 107 Axiom of choice, choice function, Dedekind infinite. 17. Denumerable sets 114 Hilbert Hotel, closure under union and product, Ramsey s theorem. 18. Uncountable sets 125 The continuum, P({$) , continuum hypothesis. 19. The paradoxes 130 Russell s paradox, the set of all sets, the separation principle. 20. The set theory of Zermelo Fraenkel (ZF) 134 Formal language, axioms, natural numbers, e minimal. 21. Peano s arithmetic 146 Axioms, definition by recursion. Chapter II. Axiomatic Set Theory 1. The axiom of regularity 150 Motivation, consistency. 2. Induction and Recursion 154 Well foundedness, transitive closure, induction principles, definition by recursion, representation theorem for well founded extensional structures. 3. Ordinal numbers 163 Von Neumann s ordinals. 4. The cumulative hierarchy 166 Rank, partial universes. 5. Ordinal arithmetic 172 Contents vii Addition, multiplication, exponentiation, Cantor s normal form. 6. Normal operations 178 Cofinality, regularity, fixed points, derivative. 7. The reflection principle 185 8. Initial numbers 188 Hartogs function, weakly inaccessible, well ordering of ORXOR and consequences. 9. The axiom of choice 192 Motivation, equivalents, two proofs of the well ordering theorem, Zorn s Lemma, GCH, DC, CC. 10. Cardinal numbers 202 Motivation, possible definitions, alephs, addition, multiplication, exponentiation, regularity, cofinality, K6nigls inequality. 11. Models 214 Purpose and meaning, conceptual difficulties, standard models and absoluteness, natural models, role of the axioms of infinity and substitution; strongly inaccessible, reflections from a strongly inaccessible ordinal, non finite axiomatizability, hereditarily ; role of the axioms of sum , and powerset, hereditarily finite. 12. Measurable cardinals 227 Higher infinities via reflection, measurables giving rise to reflexive situations, ultrapowers. Chapter III. Applications 1. Filters 237 Free filter, ultrafilter. 2. Boolean algebra 240 Axioms, representation theorems, duality. 3. Order types 251 Back and forth method, type of Q, ordinals, sum, product. 4. Inductive definitions 260 Minimal fixed point,length, Cantor Bendixson. 5. Applications of the axiom of choice 266 Basis of a vectorspace, compactness of products (Tychonov), non measurable sets (Vitali), algebraic closure (Steinitz). 6. The Borel hierarchy 273 viii Contents FQ, G , universal sets, hierarchy, separation, reduction. 7. Trees 295 Ordinals, continuity, infinity lemma. 8. The axiom, of Determinateness (AD) 311 Games, strategy, CC, AD ¦ 1 AC, Lebesgue measure on g. Appendix 321 Symbol* 326 Literature 329 Index 331 Other Titles in the Series 341
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author	Dalen, Dirk van 1932- Doets, Kees De Swart, Harrie 1944-
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ctrlnum	(OCoLC)2463590 (DE-599)BVBBV002245765
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dewey-hundreds	500 - Natural sciences and mathematics
dewey-ones	511 - General principles of mathematics
dewey-raw	511.3/2
dewey-search	511.3/2
dewey-sort	3511.3 12
dewey-tens	510 - Mathematics
discipline	Mathematik
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isbn	0080211666 0080230474
language	English Dutch
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physical	XVII, 342 S. graph. Darst.
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series	International series in pure and applied mathematics
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spellingShingle	Dalen, Dirk van 1932- Doets, Kees De Swart, Harrie 1944- Sets: naive, axiomatic and applied a basic compendium with exercises for use in set theory for non logicians, working and teaching mathematicians and students International series in pure and applied mathematics Ensembles, Théorie axiomatique des Ensembles, Théorie des Verzamelingen (wiskunde) gtt Axiomatic set theory Set theory Mengenlehre (DE-588)4074715-3 gnd
subject_GND	(DE-588)4074715-3
title	Sets: naive, axiomatic and applied a basic compendium with exercises for use in set theory for non logicians, working and teaching mathematicians and students
title_alt	Verzamelingen
title_auth	Sets: naive, axiomatic and applied a basic compendium with exercises for use in set theory for non logicians, working and teaching mathematicians and students
title_exact_search	Sets: naive, axiomatic and applied a basic compendium with exercises for use in set theory for non logicians, working and teaching mathematicians and students
title_full	Sets: naive, axiomatic and applied a basic compendium with exercises for use in set theory for non logicians, working and teaching mathematicians and students Dirk van Dalen ; H. C. Doets ; H. de Swart
title_fullStr	Sets: naive, axiomatic and applied a basic compendium with exercises for use in set theory for non logicians, working and teaching mathematicians and students Dirk van Dalen ; H. C. Doets ; H. de Swart
title_full_unstemmed	Sets: naive, axiomatic and applied a basic compendium with exercises for use in set theory for non logicians, working and teaching mathematicians and students Dirk van Dalen ; H. C. Doets ; H. de Swart
title_short	Sets: naive, axiomatic and applied
title_sort	sets naive axiomatic and applied a basic compendium with exercises for use in set theory for non logicians working and teaching mathematicians and students
title_sub	a basic compendium with exercises for use in set theory for non logicians, working and teaching mathematicians and students
topic	Ensembles, Théorie axiomatique des Ensembles, Théorie des Verzamelingen (wiskunde) gtt Axiomatic set theory Set theory Mengenlehre (DE-588)4074715-3 gnd
topic_facet	Ensembles, Théorie axiomatique des Ensembles, Théorie des Verzamelingen (wiskunde) Axiomatic set theory Set theory Mengenlehre
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001475871&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV001888024
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