Philosophy and model theory:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Oxford
Oxford University Press
2018
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| Edition: | First edition |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvi, 517 Seiten |
| ISBN: | 9780198790402 9780198790396 |
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| 245 | 1 | 0 | |a Philosophy and model theory |c Tim Button and Sean Walsh ; with a historical appendix by Wilfrid Hodges |
| 250 | |a First edition | ||
| 264 | 1 | |a Oxford |b Oxford University Press |c 2018 | |
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Contents
Reference and realism
Logics and languages
1.1 Signatures and structures...........
1.2 First-order logic: a first look ....
1.3 The Tarskian approach to semantics . .
1.4 Semantics for variables ............
1.5 The Robinsonian approach to semantics
1.6 Straining the notion of language . . . .
1.7 The Hybrid approach to semantics . . .
1.8 Linguistic compositionality.........
1.9 Second-order logic: syntax..........
1.10 Full semantics......................
1.11 Henkin semantics....................
1.12 Consequence.........................
1.13 Definability........................
i.a First- and second-order arithmetic . . .
i.b First- and second-order set theory . . .
l.C Deductive systems.....................
Permutations and referential indeterminacy
2.1 Isomorphism and the Push-Through Construction
2.2 Benacerraf s use of Push-Through.............
2.3 Putnams use of Push-Through..................
2.4 Attempts to secure reference in mathematics . . .
2.5 Supervaluationism and indeterminacy..........
2.6 Conclusion...................................
2.A Eligibility definitions, and Completeness......
2.B Isomorphism and satisfaction...................
Ramsey sentences and Newmans objection
3.1 The o/t dichotomy ..............
3.2 Ramsey sentences................
3.3 The promise of Ramsey sentences .
3.4 A caveat on the o/t dichotomy . . .
3.5 Newmans criticism of Russell....
xii CONTENTS
3.6 The Newman-conservation-objection............................. 60
3.7 Observation vocabulary versus observable objects.............. 63
3.8 The Newman-cardinality-objection.............................. 64
3.9 Mixed-predicates again: the case of causation................... 66
3.10 Natural properties and just more theory......................... 67
3.A Newman and elementary extensions................................ 69
3. B Conservation in first-order theories............................ 72
4 Compactness, infinitesimals, and the reals 75
4.1 The Compactness Theorem ........................................ 75
4.2 Infinitesimals.................................................. 77
4.3 Notational conventions.......................................... 79
4.4 Differentials, derivatives, and the use of infinitesimals....... 79
4.5 The orders of infinite smallness................................ 81
4.6 Non-standard analysis with a valuation.......................... 84
4.7 Instrumentalism and conservation................................ 88
4.8 Historical fidelity............................................. 91
4.9 Axiomatising non-standard analysis.............................. 93
4.10 Axiomatising the reals........................................ 97
4, a Godels Completeness Theorem..................................... 99
4.B A model-theoretic proof of Compactness......................... 103
4. C The valuation function of §4.6................................. 104
5 Sameness of structure and theory 107
5.1 Definitional equivalence....................................... 107
5.2 Sameness of structure and ante rem structuralism............... 108
5.3 Interpretability................................................ no
5.4 Biinterpretability............................................. 113
5.5 From structures to theories.................................... 114
5.6 Interpretability and the transfer of truth..................... 119
5.7 Interpretability and arithmetical equivalence.................. 123
5.8 Interpretability and transfer of proof......................... 126
5.9 Conclusion..................................................... 129
5. A Arithmetisation of syntax and incompleteness................... 130
5.B Definitional equivalence in second-order logic................. 132
B Categoricity 137
6 Modelism and mathematical doxology 143
6.1 Towards modelism............................................... 143
xiii
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CONTENTS
6.2 Objects-modelism.....................................
6.3 Doxology, objectual version..........................
6.4 Concepts-modelism ...................................
6.5 Doxology, conceptual version.........................
Categoricity and the natural numbers
7.1 Moderate modelism....................................
7.2 Aspirations to Categoricity..........................
7.3 Categoricity within first-order model theory.........
7.4 Dedekind s Categoricity Theorem......................
7.5 Metatheory of full second-order logic................
7.6 Attitudes towards full second-order logic............
7.7 Moderate modelism and full second-order logic........
7.8 Clarifications.......................................
7.9 Moderation and compactness...........................
7.10 Weaker logics which deliver categoricity.............
7.11 Application to specific kinds of moderate modelism . . .
7.12 Two simple problems for modelists ...................
7. a Proof of the Löwenheim-Skolem Theorem................
Categoricity and the sets
8.1 Transitive models and inaccessibles..................
8.2 Models of first-order set theory.....................
8.3 Zermelos Quasi-Categoricity Theorem..................
8.4 . Attitudes towards full second-order logic: redux...
8. $ Axiomatising the iterative process...................
8.6 Isaacson and incomplete structure....................
8. a Zermelo Quasi-Categoricity...........................
8.b Elementary Scott-Potter foundations...................
8. c Scott-Potter Quasi-Categoricity......................
Transcendental arguments against model-theoretical scepticism
9.1 Model-theoretical scepticism.........................
9.2 Moorean versus transcendental arguments..............
9.3 The Metaresources Transcendental Argument............
9.4 The Disquotational Transcendental Argument...........
9.5 Ineffable sceptical concerns ........................
9. a Application: the (non-) absoluteness of truth........
Internal categoricity and the natural numbers
10.1 Metamathematics without semantics ...................
XIV CONTENTS
10.2 The internal categoricity of arithmetic........................ 227
10.3 Limits on what internal categoricity could show................. 229
10.4 The intolerance of arithmetic................................... 232
10.5 A canonical theory.............................................. 232
10.6 The algebraic / univocal distinction............................ 233
10.7 Situating intemalism in the landscape........................... 236
10.8 Moderate internalists ......................................... 237
10.A Connection to Parsons .......................................... 239
io.b Proofs of internal categoricity and intolerance ................ 242
10. c Predicative Comprehension........................................246
11 Internal categoricity and the sets 251
11.1 Internalising S cott-Potter set theory.......................... 251
11.2 Quasi-intolerance for pure set theory........................... 253
11.3 The status of the continuum hypothesis.......................... 255
11.4 Total internal categoricity for pure set theory................. 256
11.5 Total intolerance for pure set theory........................... 257
11.6 Intemalism and indefinite extensibility......................... 258
11. a Connection to McGee..............................................260
11.B Connection to Martin............................................ 262
11.C Internal quasi-categoricity for SP.............................. 263
11.D Total internal categoricity for CSP............................ 266
11. E Internal quasi-categoricity of ordinals......................... 268
12 Internal categoricity and truth 271
12.1 The promise of truth-intemalism................................. 271
12.2 Truth operators.............................................. 273
12.3 Intemalism about model theory and internal realism.............. 276
12.4 Truth in higher-order logic..................................... 282
12.5 Two general issues for truth-intemalism......................... 284
12. A Satisfaction in higher-order logic.............................. 285
13 Boolean-valued structures 295
13.1 Semantic-underdetermination via Push-Through.................... 295
13.2 The theory of Boolean algebras.................................. 296
13.3 Boolean-valued models........................................... 298
13.4 Semantic-underdetermination via filters......................... 301
13.5 Semanticism..................................................... 304
13.6 Bilateralism.................................................... 307
13.7 Open-ended-inferentialism........................................ 3u
13.8 Intemal-inferentialism.......................................... 314
CONTENTS XV
13.9 Suszkos Thesis.................................................. 316
13.A Boolean-valued structures with filters.......................... 321
13.B Full second-order Boolean-valued structures..................... 323
13.C Ultrafilters, ultraproducts, Loá, and compactness............... 326
13.D The Boolean-non-categoricity of CBA............................. 328
13. E Proofs concerning bilateralism ................................. 330
C Indiscemibility and classification 333
14 Types and Stone spaces 337
14.1 Types for theories.............................................. 337
14.2 An algebraic view on compactness . ............................. 338
14.3 Stones Duality Theorem.......................................... 339
14.4 Types, compactness, and stability............................... 342
14.5 Bivalence and compactness...................................... 346
14.6 A biinterpretation.............................................. 349
14.7 Propositions and possible worlds................................ 350
14. A Topological background.......................................... 354
14. B Bivalent-calculi and bivalent-universes......................... 356
15 Indiscemibility 359
15.1 Notions of indiscemibility...................................... 359
15.2 Singling out indiscernibles..................................... 366
15.3 The identity of indiscernibles.................................. 370
15.4 Two-indiscemibles in infinitary logics.......................... 376
15.5 n-indiscemibles, order, and stability........................... 380
15. A Charting the grades of discemibility............................ 384
16 Quantifiers 387
16.1 Generalised quantifiers......................................... 387
16.2 Clarifying the question of logicality........................... 389
16.3 Tarski and Sher................................................. 389
16.4 Tarski and Kleins Erlangen Programme............................ 390
16.5 The Principle of Non-Discrimination............................. 392
16.6 The Principle of Closure ....................................... 399
16.7 McGee s squeezing argument.......................................407
16.8 Mathematical content........................................... 408
16.9 Explications and pluralism...................................... 410
17 Classification and uncountable categoricity 413
xvi CONTENTS
17.1 The nature of classification................................... 413
17.2 Shelah on classification....................................... 419
17.3 Uncountable categoricity........................................426
17.4 Conclusions .................................................. 432.
17.A Proof of Proposition 17.2...................................... 432
D Historical appendix 435
18 Wilfrid Hodges
A short history of model theory 439
18.1 A new branch of metamathematics,............................... 439
18.2 Replacing the old metamathematics...............................44°
18.3 Definable relations in one structure ...........................445
18.4 Building a structure............................................449
18.5 Maps between structures........................................ 455
18.6 Equivalence and preservation....................................460
18.7 Categoricity and classification theory......................... 465
18.8 Geometric model theory..........................................469
18.9 Other languages................................................ 472
18.10 Model theory within mathematics................................ 474
18.11 Notes.......................................................... 475
18.12 Acknowledgments................................................ 475
Bibliography 477
Index 507
Index of names 513
Index of symbols and definitions 5*5
|
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| isbn | 9780198790402 9780198790396 |
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| spelling | Button, Tim Verfasser (DE-588)1036917746 aut Philosophy and model theory Tim Button and Sean Walsh ; with a historical appendix by Wilfrid Hodges First edition Oxford Oxford University Press 2018 xvi, 517 Seiten txt rdacontent n rdamedia nc rdacarrier Modelltheorie (DE-588)4114617-7 gnd rswk-swf Philosophie (DE-588)4045791-6 gnd rswk-swf Modelltheorie (DE-588)4114617-7 s Philosophie (DE-588)4045791-6 s DE-604 Walsh, Sean Verfasser aut Hodges, Wilfrid 1941- (DE-588)108085392 wat Digitalisierung BSB Muenchen - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030147712&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Button, Tim Walsh, Sean Philosophy and model theory Modelltheorie (DE-588)4114617-7 gnd Philosophie (DE-588)4045791-6 gnd |
| subject_GND | (DE-588)4114617-7 (DE-588)4045791-6 |
| title | Philosophy and model theory |
| title_auth | Philosophy and model theory |
| title_exact_search | Philosophy and model theory |
| title_full | Philosophy and model theory Tim Button and Sean Walsh ; with a historical appendix by Wilfrid Hodges |
| title_fullStr | Philosophy and model theory Tim Button and Sean Walsh ; with a historical appendix by Wilfrid Hodges |
| title_full_unstemmed | Philosophy and model theory Tim Button and Sean Walsh ; with a historical appendix by Wilfrid Hodges |
| title_short | Philosophy and model theory |
| title_sort | philosophy and model theory |
| topic | Modelltheorie (DE-588)4114617-7 gnd Philosophie (DE-588)4045791-6 gnd |
| topic_facet | Modelltheorie Philosophie |
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