Set theory with a universal set: exploring an untyped universe
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Clarendon Press
2002
|
| Edition: | 2. ed., [reprint.] |
| Series: | Oxford logic guides
31 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 166 S. |
| ISBN: | 0198514778 9780198514770 |
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MARC
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
1
Introduction
1
1.1
Annotated definitions
4
1.1.1
Quantifier hierarchies
5
1.1.2
Mainly concerning type theory
6
1.1.3
Other definitions
9
1.1.4
Theories
10
1.2
Some motivations and axioms
11
1.2.1
Sets as predicates-in-extension
11
1.2.2
Sets as natural kinds
21
1.3
A brief survey
22
1.4
How do theories with V
€
F
avoid the paradoxes?
24
1.5
Chronology
25
2
NF and related systems
26
2.1
NF
26
2.1.1
The axiom of counting
30
2.1.2
BorTa s lemma on n-formulae, and the auto¬
morphism lemma for set abstracts
33
2.1.3
Miscellaneous combinatorics
35
2.1.4
Well-founded sets
40
2.2
Cardinal and ordinal arithmetic
44
2.2.1
Some remarks on inductive definitions
55
2.2.2
Closure properties of small sets
57
2.3
The Kaye—Specker equiconsistency lemma
58
2.3.1
NF3
65
2.3.2
NFU
67
2.3.3
Lake s model
72
2.3.4
KF
72
2.4
Subsystems, term models, and prefix classes
83
2.5
The converse consistency problem
89
3
Permutation models
92
3.1
Permutations in NF
96
3.1.1
Inner permutations in NF
97
3.1.2
Outer automorphisms in NF
119
3.2
Applications to other theories
121
PREFACE
TO THE FIRST EDITION
4
Church-Oswald models
122
4.1
Oswald s model
122
4.2
Low sets
124
4.2.1
Other definitions of low
125
4.3
^-extensions and permutation models
126
4.3.1
P-extensions
126
4.3.2
Hereditarily low sets and permutation mod¬
els
127
4.3.3
Permutation models of CO structures
129
4.4
Two applications
130
4.4.1
An elementary example
130
4.4.2
^-extending models of Zermelo to models of
NFO
132
4.5
Church s model
136
4.6
Mitchell s set theory
139
4.7
Conclusions
140
5
Open problems
143
5.1
Permutation models and quantifier hierarchies
143
5.2
Cardinals and ordinals in NF
144
5.3
KF
144
5.4
Other subsystems
145
5.5
Well-founded extensional relations
145
5.6
Term models
146
5.7
Miscellaneous
146
Bibliography
148
Index of definitions
161
Author index
163
General index
164
|
| adam_txt |
CONTENTS
1
Introduction
1
1.1
Annotated definitions
4
1.1.1
Quantifier hierarchies
5
1.1.2
Mainly concerning type theory
6
1.1.3
Other definitions
9
1.1.4
Theories
10
1.2
Some motivations and axioms
11
1.2.1
Sets as predicates-in-extension
11
1.2.2
Sets as natural kinds
21
1.3
A brief survey
22
1.4
How do theories with V
€
F
avoid the paradoxes?
24
1.5
Chronology
25
2
NF and related systems
26
2.1
NF
26
2.1.1
The axiom of counting
30
2.1.2
BorTa's lemma on n-formulae, and the auto¬
morphism lemma for set abstracts
33
2.1.3
Miscellaneous combinatorics
35
2.1.4
Well-founded sets
40
2.2
Cardinal and ordinal arithmetic
44
2.2.1
Some remarks on inductive definitions
55
2.2.2
Closure properties of small sets
57
2.3
The Kaye—Specker equiconsistency lemma
58
2.3.1
NF3
65
2.3.2
NFU
67
2.3.3
Lake's model
72
2.3.4
KF
72
2.4
Subsystems, term models, and prefix classes
83
2.5
The converse consistency problem
89
3
Permutation models
92
3.1
Permutations in NF
96
3.1.1
Inner permutations in NF
97
3.1.2
Outer automorphisms in NF
119
3.2
Applications to other theories
121
PREFACE
TO THE FIRST EDITION
4
Church-Oswald models
122
4.1
Oswald's model
122
4.2
Low sets
124
4.2.1
Other definitions of low
125
4.3
^-extensions and permutation models
126
4.3.1
'P-extensions
126
4.3.2
Hereditarily low sets and permutation mod¬
els
127
4.3.3
Permutation models of CO structures
129
4.4
Two applications
130
4.4.1
An elementary example
130
4.4.2
^-extending models of Zermelo to models of
NFO
132
4.5
Church's model
136
4.6
Mitchell's set theory
139
4.7
Conclusions
140
5
Open problems
143
5.1
Permutation models and quantifier hierarchies
143
5.2
Cardinals and ordinals in NF
144
5.3
KF
144
5.4
Other subsystems
145
5.5
Well-founded extensional relations
145
5.6
Term models
146
5.7
Miscellaneous
146
Bibliography
148
Index of definitions
161
Author index
163
General index
164 |
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| author | Forster, T. E. |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. ed., [reprint.] |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T21:39:17Z |
| indexdate | 2024-07-09T21:19:50Z |
| institution | BVB |
| isbn | 0198514778 9780198514770 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016667236 |
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| physical | X, 166 S. |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Clarendon Press |
| record_format | marc |
| series | Oxford logic guides |
| series2 | Oxford logic guides |
| spelling | Forster, T. E. Verfasser aut Set theory with a universal set exploring an untyped universe T. E. Forster 2. ed., [reprint.] Oxford [u.a.] Clarendon Press 2002 X, 166 S. txt rdacontent n rdamedia nc rdacarrier Oxford logic guides 31 Mengenlehre (DE-588)4074715-3 gnd rswk-swf Mengenlehre (DE-588)4074715-3 s DE-604 Oxford logic guides 31 (DE-604)BV000013997 31 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016667236&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Forster, T. E. Set theory with a universal set exploring an untyped universe Oxford logic guides Mengenlehre (DE-588)4074715-3 gnd |
| subject_GND | (DE-588)4074715-3 |
| title | Set theory with a universal set exploring an untyped universe |
| title_auth | Set theory with a universal set exploring an untyped universe |
| title_exact_search | Set theory with a universal set exploring an untyped universe |
| title_exact_search_txtP | Set theory with a universal set exploring an untyped universe |
| title_full | Set theory with a universal set exploring an untyped universe T. E. Forster |
| title_fullStr | Set theory with a universal set exploring an untyped universe T. E. Forster |
| title_full_unstemmed | Set theory with a universal set exploring an untyped universe T. E. Forster |
| title_short | Set theory with a universal set |
| title_sort | set theory with a universal set exploring an untyped universe |
| title_sub | exploring an untyped universe |
| topic | Mengenlehre (DE-588)4074715-3 gnd |
| topic_facet | Mengenlehre |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016667236&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000013997 |
| work_keys_str_mv | AT forsterte settheorywithauniversalsetexploringanuntypeduniverse |