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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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English Dutch |
| Published: |
Oxford [u.a.]
Pergamon Press
1978
|
| Series: | International series in pure and applied mathematics
106 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 342 S. graph. Darst. |
| ISBN: | 0080211666 0080230474 |
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MARC
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| 245 | 1 | 0 | |a Sets: naive, axiomatic and applied |b a basic compendium with exercises for use in set theory for non logicians, working and teaching mathematicians and students |c Dirk van Dalen ; H. C. Doets ; H. de Swart |
| 264 | 1 | |a Oxford [u.a.] |b Pergamon Press |c 1978 | |
| 300 | |a XVII, 342 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a International series in pure and applied mathematics |v 106 | |
| 650 | 4 | |a Ensembles, Théorie axiomatique des | |
| 650 | 4 | |a Ensembles, Théorie des | |
| 650 | 7 | |a Verzamelingen (wiskunde) |2 gtt | |
| 650 | 4 | |a Axiomatic set theory | |
| 650 | 4 | |a Set theory | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-001475871 | ||
Record in the Search Index
| _version_ | 1804116701878419456 |
|---|---|
| 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
|
| any_adam_object | 1 |
| author | Dalen, Dirk van 1932- Doets, Kees De Swart, Harrie 1944- |
| author_GND | (DE-588)133205053 (DE-588)1158181949 (DE-588)120024691 |
| author_facet | Dalen, Dirk van 1932- Doets, Kees De Swart, Harrie 1944- |
| author_role | aut aut aut |
| author_sort | Dalen, Dirk van 1932- |
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| building | Verbundindex |
| bvnumber | BV002245765 |
| callnumber-first | Q - Science |
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| callnumber-search | QA248 QH248 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 150 SK 155 |
| ctrlnum | (OCoLC)2463590 (DE-599)BVBBV002245765 |
| dewey-full | 511.3/2 |
| 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 |
| format | Book |
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| id | DE-604.BV002245765 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T15:42:42Z |
| institution | BVB |
| isbn | 0080211666 0080230474 |
| language | English Dutch |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001475871 |
| oclc_num | 2463590 |
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| physical | XVII, 342 S. graph. Darst. |
| publishDate | 1978 |
| publishDateSearch | 1978 |
| publishDateSort | 1978 |
| publisher | Pergamon Press |
| record_format | marc |
| series | International series in pure and applied mathematics |
| series2 | International series in pure and applied mathematics |
| spelling | Dalen, Dirk van 1932- Verfasser (DE-588)133205053 aut Verzamelingen 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 Oxford [u.a.] Pergamon Press 1978 XVII, 342 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier International series in pure and applied mathematics 106 Ensembles, Théorie axiomatique des Ensembles, Théorie des Verzamelingen (wiskunde) gtt Axiomatic set theory Set theory Mengenlehre (DE-588)4074715-3 gnd rswk-swf Mengenlehre (DE-588)4074715-3 s DE-604 Doets, Kees Verfasser (DE-588)1158181949 aut De Swart, Harrie 1944- Verfasser (DE-588)120024691 aut International series in pure and applied mathematics 106 (DE-604)BV001888024 106 HBZ Datenaustausch application/pdf 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 Inhaltsverzeichnis |
| 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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