Foundations of set theory:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam
North-Holland Publishing Co.
1973
|
| Ausgabe: | 2. revised edition |
| Schriftenreihe: | Studies in logic and the foundations of mathematics
67 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | X, 404 Seiten |
| ISBN: | 0720422701 |
Internformat
MARC
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| 245 | 1 | 0 | |a Foundations of set theory |c Abraham A. Fraenkel, Yehoshua Bar-Hillel, Azriel Levy |
| 250 | |a 2. revised edition | ||
| 264 | 1 | |a Amsterdam |b North-Holland Publishing Co. |c 1973 | |
| 300 | |a X, 404 Seiten | ||
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Datensatz im Suchindex
| _version_ | 1804136096688242688 |
|---|---|
| adam_text | CONTENTS
Preface ix
CHAPTER I. THE ANTINOMIES 1 14
§1. Historical introduction 1
§2. Logical antinomies 5—8
1. Rusell s antinomy 5
2. Cantor s antinomy 7
3. Burali Forti s antinomy 8
§3. Semantical antinomies 8—10
1. Richard s antinomy 8
2. Grelling s antinomy 9
3. The liar 9
§4. General remarks 10
§5. The three crises 12
CHAPTER II. AXIOMATIC FOUNDATIONS OF SET THEORY 15 153
§1. Introduction 15
§2. Some basic notions, equality and extensionality 22
§3. Axioms of comprehension and infinity 30—53
1. The axiom schema of comprehension 30
2. The axiom of pairing. Ordered pairs 32
3. The axioms of union and power set 33
4. The axiom schema of subsets 35
5. Relations, order, functions 41
6. The axiom of infinity 44
7. The axiom schema of replacement 49
§4. The axiom of choice 53—86
1. Formulation of the axiom. Its introduction into mathe¬
matics 53
vi CONTENTS
2. The consistency and the independence of the axiom 58
3. Special (weakened) forms of the axiom 61
4. The existential character of the axiom. Effectivity. Selec¬
tors 67
5. Some typical applications of the axiom 73
6. Mathematicians attitude towards the axiom 80
§5. The axiom of foundation 86—102
1. Introducing the axiom 86
2. Ordinal numbers 91
3. Well founded sets 93
4. Cardinal numbers. Order types. Isomorphism types 95
5. Consistency and independence of the axiom 98
§6. Questions unanswered by the axioms 103—119
1. The generalized continuum hypothesis 103
2. The axiom of constructibility 108
3. Axioms of strong infinity 109
4. Axioms of restriction 113
§7. The role of classes in set theory 119—153
l.The axiom system VNB of von Neumann and Bernays 119
2. Metamathematical features of VNB 128
3. The axiom of choice in VNB 133
4. The approach of von Neumann 135
5. Classes taken seriously — the system of Quine and Morse 138
6. Classes not taken seriously — systems of Bernays and
Quine 146
7. The system of Ackermann 148
CHAPTER III. TYPE THEORETICAL APPROACHES 154 209
§1. The ideal calculus 154
§2. The theory of types 158
§3. Quine s new foundations 161
§4. Quine s mathematical logic 167
§5. The hierarchy of languages and the ramified class calculus 171
§6. Wang s system 2 175
§7. Lorenzen s operationist system 179
§8. The logicistic thesis 181
§9. Types, categories, and sorts 188
§10. Impredicative concept formation 193
§11. Set theories based upon non standard logics 200—209
CONTENTS vii
1. Les niewski s ontology 200
2. The systems of Chwistek and Myhill 203
3. Fitch s system 205
4. Many valued logics 207
5. Combinatory logic 209
CHAPTER IV. INTUITIONIST1C CONCEPTIONS OF MATHE¬
MATICS 210 274
§1. Historical introduction. The abyss between discreteness and
continuity 210
§2. The constructive character of mathematics. Mathematics
and language 220
§3. The principle of the excluded middle 227
§4. Mathematics and logic. Logical calculus 238
§5. The primordial intuition of integer. Choice sequences and
Brouwer s concept of set 252
§6. Mathematics as trimmed according to the intuitionistic atti¬
tude 265
CHAPTER V. METAMATHEMATICAL AND SEMANTICAL AP¬
PROACHES 275 345
§1. The Hilbert program 275
§2. Formal systems, logistic systems, and formalized theories 280
§3. Interpretations and models 288
§4. Consistency, completeness, categoricalness, and indepen¬
dence 293
§5. The Skolem—Lowenheim theorem; Skolem s paradox 302
§6. Decidability and recursiveness; arithmetization of syntax 305
§7. The limitative theorems of Godel, Tarski, Church and their
generalizations 310
§8. The metamathematics and semantics of set theory 321
§9. Philosophical remarks 331
Bibliography 346
Index of persons 391
Index of symbols 397
Subject index 399
|
| adam_txt |
CONTENTS
Preface ix
CHAPTER I. THE ANTINOMIES 1 14
§1. Historical introduction 1
§2. Logical antinomies 5—8
1. Rusell's antinomy 5
2. Cantor's antinomy 7
3. Burali Forti's antinomy 8
§3. Semantical antinomies 8—10
1. Richard's antinomy 8
2. Grelling's antinomy 9
3. The liar 9
§4. General remarks 10
§5. The three crises 12
CHAPTER II. AXIOMATIC FOUNDATIONS OF SET THEORY 15 153
§1. Introduction 15
§2. Some basic notions, equality and extensionality 22
§3. Axioms of comprehension and infinity 30—53
1. The axiom schema of comprehension 30
2. The axiom of pairing. Ordered pairs 32
3. The axioms of union and power set 33
4. The axiom schema of subsets 35
5. Relations, order, functions 41
6. The axiom of infinity 44
7. The axiom schema of replacement 49
§4. The axiom of choice 53—86
1. Formulation of the axiom. Its introduction into mathe¬
matics 53
vi CONTENTS
2. The consistency and the independence of the axiom 58
3. Special (weakened) forms of the axiom 61
4. The existential character of the axiom. Effectivity. Selec¬
tors 67
5. Some typical applications of the axiom 73
6. Mathematicians' attitude towards the axiom 80
§5. The axiom of foundation 86—102
1. Introducing the axiom 86
2. Ordinal numbers 91
3. Well founded sets 93
4. Cardinal numbers. Order types. Isomorphism types 95
5. Consistency and independence of the axiom 98
§6. Questions unanswered by the axioms 103—119
1. The generalized continuum hypothesis 103
2. The axiom of constructibility 108
3. Axioms of strong infinity 109
4. Axioms of restriction 113
§7. The role of classes in set theory 119—153
l.The axiom system VNB of von Neumann and Bernays 119
2. Metamathematical features of VNB 128
3. The axiom of choice in VNB 133
4. The approach of von Neumann 135
5. Classes taken seriously — the system of Quine and Morse 138
6. Classes not taken seriously — systems of Bernays and
Quine 146
7. The system of Ackermann 148
CHAPTER III. TYPE THEORETICAL APPROACHES 154 209
§1. The ideal calculus 154
§2. The theory of types 158
§3. Quine's new foundations 161
§4. Quine's mathematical logic 167
§5. The hierarchy of languages and the ramified class calculus 171
§6. Wang's system 2 175
§7. Lorenzen's operationist system 179
§8. The logicistic thesis 181
§9. Types, categories, and sorts 188
§10. Impredicative concept formation 193
§11. Set theories based upon non standard logics 200—209
CONTENTS vii
1. Les'niewski's ontology 200
2. The systems of Chwistek and Myhill 203
3. Fitch's system 205
4. Many valued logics 207
5. Combinatory logic 209
CHAPTER IV. INTUITIONIST1C CONCEPTIONS OF MATHE¬
MATICS 210 274
§1. Historical introduction. The abyss between discreteness and
continuity 210
§2. The constructive character of mathematics. Mathematics
and language 220
§3. The principle of the excluded middle 227
§4. Mathematics and logic. Logical calculus 238
§5. The primordial intuition of integer. Choice sequences and
Brouwer's concept of set 252
§6. Mathematics as trimmed according to the intuitionistic atti¬
tude 265
CHAPTER V. METAMATHEMATICAL AND SEMANTICAL AP¬
PROACHES 275 345
§1. The Hilbert program 275
§2. Formal systems, logistic systems, and formalized theories 280
§3. Interpretations and models 288
§4. Consistency, completeness, categoricalness, and indepen¬
dence 293
§5. The Skolem—Lowenheim theorem; Skolem's paradox 302
§6. Decidability and recursiveness; arithmetization of syntax 305
§7. The limitative theorems of Godel, Tarski, Church and their
generalizations 310
§8. The metamathematics and semantics of set theory 321
§9. Philosophical remarks 331
Bibliography 346
Index of persons 391
Index of symbols 397
Subject index 399 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Fraenkel, Abraham Adolf 1891-1965 Bar-Hillēl, Yehôšuaʿ 1915-1975 Levy, Azriel 1934- |
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| author_sort | Fraenkel, Abraham Adolf 1891-1965 |
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| dewey-full | 511/.3 |
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| discipline | Mathematik Philosophie |
| discipline_str_mv | Mathematik Philosophie |
| edition | 2. revised edition |
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| id | DE-604.BV022123589 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:16:22Z |
| indexdate | 2024-07-09T20:50:58Z |
| institution | BVB |
| isbn | 0720422701 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015338269 |
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| owner_facet | DE-706 DE-11 |
| physical | X, 404 Seiten |
| psigel | HUB-ZB011200708 |
| publishDate | 1973 |
| publishDateSearch | 1973 |
| publishDateSort | 1973 |
| publisher | North-Holland Publishing Co. |
| record_format | marc |
| series | Studies in logic and the foundations of mathematics |
| series2 | Studies in logic and the foundations of mathematics |
| spelling | Fraenkel, Abraham Adolf 1891-1965 (DE-588)118692399 aut Foundations of set theory Abraham A. Fraenkel, Yehoshua Bar-Hillel, Azriel Levy 2. revised edition Amsterdam North-Holland Publishing Co. 1973 X, 404 Seiten txt rdacontent n rdamedia nc rdacarrier Studies in logic and the foundations of mathematics 67 Mängdteori sao Set theory Axiomatik (DE-588)4004038-0 gnd rswk-swf Logik (DE-588)4036202-4 gnd rswk-swf Axiomatische Mengenlehre (DE-588)4143743-3 gnd rswk-swf Mengenlehre (DE-588)4074715-3 gnd rswk-swf Mengenlehre (DE-588)4074715-3 s Logik (DE-588)4036202-4 s 1\p DE-604 Axiomatik (DE-588)4004038-0 s 2\p DE-604 Axiomatische Mengenlehre (DE-588)4143743-3 s 3\p DE-604 Bar-Hillēl, Yehôšuaʿ 1915-1975 (DE-588)122340477 aut Levy, Azriel 1934- Verfasser (DE-588)109437543 aut Studies in logic and the foundations of mathematics 67 (DE-604)BV000893472 67 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015338269&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Fraenkel, Abraham Adolf 1891-1965 Bar-Hillēl, Yehôšuaʿ 1915-1975 Levy, Azriel 1934- Foundations of set theory Studies in logic and the foundations of mathematics Mängdteori sao Set theory Axiomatik (DE-588)4004038-0 gnd Logik (DE-588)4036202-4 gnd Axiomatische Mengenlehre (DE-588)4143743-3 gnd Mengenlehre (DE-588)4074715-3 gnd |
| subject_GND | (DE-588)4004038-0 (DE-588)4036202-4 (DE-588)4143743-3 (DE-588)4074715-3 |
| title | Foundations of set theory |
| title_auth | Foundations of set theory |
| title_exact_search | Foundations of set theory |
| title_exact_search_txtP | Foundations of set theory |
| title_full | Foundations of set theory Abraham A. Fraenkel, Yehoshua Bar-Hillel, Azriel Levy |
| title_fullStr | Foundations of set theory Abraham A. Fraenkel, Yehoshua Bar-Hillel, Azriel Levy |
| title_full_unstemmed | Foundations of set theory Abraham A. Fraenkel, Yehoshua Bar-Hillel, Azriel Levy |
| title_short | Foundations of set theory |
| title_sort | foundations of set theory |
| topic | Mängdteori sao Set theory Axiomatik (DE-588)4004038-0 gnd Logik (DE-588)4036202-4 gnd Axiomatische Mengenlehre (DE-588)4143743-3 gnd Mengenlehre (DE-588)4074715-3 gnd |
| topic_facet | Mängdteori Set theory Axiomatik Logik Axiomatische Mengenlehre Mengenlehre |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015338269&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000893472 |
| work_keys_str_mv | AT fraenkelabrahamadolf foundationsofsettheory AT barhillelyehosuaʿ foundationsofsettheory AT levyazriel foundationsofsettheory |