An introduction to proof theory: normalization, cut-elimination, and consistency proofs
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Oxford University Press
2021
|
| Ausgabe: | First edition |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xii, 418 Seiten Illustrationen 24 cm |
| ISBN: | 9780192895943 0192895931 9780192895936 |
Internformat
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV047434433 | ||
| 003 | DE-604 | ||
| 005 | 20220914 | ||
| 007 | t | ||
| 008 | 210823s2021 a||| |||| 00||| eng d | ||
| 020 | |a 9780192895943 |c pbk. |9 978-0-19-289594-3 | ||
| 020 | |a 0192895931 |c hbk. |9 0-19-289593-1 | ||
| 020 | |a 9780192895936 |c hbk. |9 978-0-19-289593-6 | ||
| 035 | |a (OCoLC)1277017264 | ||
| 035 | |a (DE-599)BVBBV047434433 | ||
| 040 | |a DE-604 |b ger |e rda | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-706 |a DE-11 |a DE-12 |a DE-83 | ||
| 084 | |a SK 130 |0 (DE-625)143216: |2 rvk | ||
| 100 | 1 | |a Mancosu, Paolo |d 1960- |e Verfasser |0 (DE-588)133912035 |4 aut | |
| 245 | 1 | 0 | |a An introduction to proof theory |b normalization, cut-elimination, and consistency proofs |c Paolo Mancosu, Sergio Galvan, Richard Zach |
| 250 | |a First edition | ||
| 264 | 1 | |a Oxford |b Oxford University Press |c 2021 | |
| 264 | 4 | |c © 2021 | |
| 300 | |a xii, 418 Seiten |b Illustrationen |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 0 | 7 | |a Beweistheorie |0 (DE-588)4145177-6 |2 gnd |9 rswk-swf |
| 653 | 0 | |a Proof theory | |
| 653 | 0 | |a Proof theory | |
| 689 | 0 | 0 | |a Beweistheorie |0 (DE-588)4145177-6 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Galvan, Sergio |d 1946- |e Verfasser |0 (DE-588)1048745600 |4 aut | |
| 700 | 1 | |a Zach, Richard |e Verfasser |0 (DE-588)1242279687 |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-0-19-193879-5 |
| 856 | 4 | 2 | |m Digitalisierung BSB München - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032836767&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-032836767 | ||
Datensatz im Suchindex
| _version_ | 1804182722393931776 |
|---|---|
| adam_text | Contents Preface.......................................................................................................... viii About this book......................................................................................... viii For further reading.................................................................................. x Acknowledgments .................................................................................. xii 1 Introduction............................................................................................. і 1.1 Hilbert s consistency program......................................................... і 1.2 Gentzen s proof theory................................................. 6 1.3 Proof theory after Gentzen............................................................... 10 2 Axiomatic calculi..................................................................................... 13 2.1 Propositional logic.......................................................................... 13 2.2 Reading formulas as trees............................................................... 15 2.3 Sub-formulas and main connectives .............................................. 17 2.4 Logical calculi.................................................................................. 18 2.5 Inference rules.................................................................................. 20 2.6 Derivations from assumptions and provability.............................. 24 2.7 Proofs by
induction.......................................................................... 26 2.8 The deduction theorem.................................................................... 30 2.9 Derivations as trees.......................................................................... 33 2.10 Negation.......................................................................................... 35 2.11 Independence.................................................................................. 4° 2.12 An alternative axiomatization of Jo................................................. 43 2.13 Predicate logic.................................................................................. 44 2.14 The deduction theorem for the predicate calculus ........................ 50 2.15 Intuitionistic and classical arithmetic.............................................. 53 3 Natural deduction .................................................................................. 65 3.1 Introduction..................................................................................... 65 3.2 Rules and deductions....................................................................... 67 3.3 Natural deduction for classical logic.............................................. 85 3.4 Alternative systems for classical logic ............................................ 88 3.5 Measuring deductions.................................................................... 89 3.6 Manipulating deductions, proofs about deductions ...................... 91 3.7 Equivalence of natural and axiomatic
deduction........................... 96 4 Normal deductions.................................................................................... 101 4.1 Introduction....................................................................................... 101 4.2 Double induction...............................................................................105 4.3 Normalization for Л, э, -i, V.............................................................. no 4.4 The sub-formula property................................................................. 121 4.5 The size of normal deductions............................................................130 v
VI 5 6 7 8 CONTENTS 4.6 Normalization for NJ.........................................................................132 4.7 An example...................................................................................... 143 4.8 The sub-formula property for NJ......................................................145 4.9 Normalization for NK......................................................................154 The sequent calculus...............................................................................168 5.1 The language of the sequent calculus.............................................. 168 5.2 Rules of LK......................................................................................... 170 5.3 Constructing proofs in LK................................................................ 175 5.4 The significance of cut......................................................................178 5.5 Examples of proofs........................................................................... 180 5.6 Atomic logical axioms ...................................................................... 187 5.7 Lemma on variable replacement ......................................................188 5.8 Translating NJ to LJ........................................................................... 191 5.9 Translating LJ to NJ........................................................................... 197 The cut-elimination theorem..................................................................202 6.1 Preliminary
definitions......................................................................204 6.2 Outline of the lemma.........................................................................206 6.3 Removing Mixes directly................................................................... 209 6.4 Reducing the degree of mix ............................................................212 6.5 Reducing the rank.............................................................................. 217 6.6 Reducing the rank: example..............................................................238 6.7 Reducing the degree: example ........................................................ 242 6.8 Intuitionistic sequent calculus LJ......................................................250 6.9 Why mix?............................................................................................252 6.10 Consequences of the Hauptsatz........................................................ 254 6.11 The mid-sequent theorem.................................................................257 The consistency of arithmetic..................................................................269 7.1 Introduction.............................................................. 269 7.2 Consistency of simple proofs ........................................................... 276 7.3 Preliminary details............................................................................282 7.4 Overview of the consistency proof................................................... 287 7.5 Replacing
inductions......................................................................... 288 7.6 Reducing suitable cuts...................................................................... 293 7.7 A first example ................................................................................. 295 7.8 Elimination of weakenings.................................................................298 7.9 Existence of suitable cuts....................................................................301 7.10 A simple example.............................................................................. 304 7.11 Summary............................................................................................ 308 Ordinal notations and induction.............................................................. 312 8.1 Orders, well-orders, and induction . . .............................................. 312 8.2 Lexicographical orderings................................................................. 315 8.3 Ordinal notations up to £0.................................................................320
CONTENTS VU 8.4 Operations on ordinal notations............................................................325 8.5 Ordinal notations are well-ordered......................................................330 8.6 Set-theoretic definitions of the ordinals................................................ 332 8.7 Constructing £0 from below................................................................. 336 8.8 Ordinal arithmetic ............................................................................. 337 8.9 Trees and Goodstein sequences............................................................339 9 The consistency of arithmetic, continued.................................................. 346 9.1 Assigning ordinal notations εοto proofs................................... 346 9.2 Eliminating inductions from the end-part.......................................... 354 9.3 Removing weakenings..........................................................................360 9.4 Reduction of suitable cuts.................................................................... 368 9.5 A simple example, revisited..................................................................376 A The Greek alphabet.................................................................................... 380 В Set-theoretic notation...................................................................................381 C Axioms, rules, and theorems of axiomatic calculi.................................... 383 C.i Axioms and rules of inference.............................................................. 383 C.2
Theorems and derived rules................................................................. 384 D Exercises on axiomatic derivations.............................................................. 386 D.i Hints for Problem 2.7............................................................................. 386 D.2 Hints for Problem 2.18.......................................................................... 390 D.3 Exercises with quantifiers....................................................................393 E Natural deduction ........................................................................................ 394 E.i Inference rules........................................................................................ 394 E.2 Conversions...........................................................................................395 F Sequent calculus...........................................................................................399 G Outline of the cut-elimination theorem......................................................401 Bibliography .....................................................................................................405 Index................................................................................................................... 412
|
| adam_txt |
Contents Preface. viii About this book. viii For further reading. x Acknowledgments . xii 1 Introduction. і 1.1 Hilbert's consistency program. і 1.2 Gentzen's proof theory. 6 1.3 Proof theory after Gentzen. 10 2 Axiomatic calculi. 13 2.1 Propositional logic. 13 2.2 Reading formulas as trees. 15 2.3 Sub-formulas and main connectives . 17 2.4 Logical calculi. 18 2.5 Inference rules. 20 2.6 Derivations from assumptions and provability. 24 2.7 Proofs by
induction. 26 2.8 The deduction theorem. 30 2.9 Derivations as trees. 33 2.10 Negation. 35 2.11 Independence. 4° 2.12 An alternative axiomatization of Jo. 43 2.13 Predicate logic. 44 2.14 The deduction theorem for the predicate calculus . 50 2.15 Intuitionistic and classical arithmetic. 53 3 Natural deduction . 65 3.1 Introduction. 65 3.2 Rules and deductions. 67 3.3 Natural deduction for classical logic. 85 3.4 Alternative systems for classical logic . 88 3.5 Measuring deductions. 89 3.6 Manipulating deductions, proofs about deductions . 91 3.7 Equivalence of natural and axiomatic
deduction. 96 4 Normal deductions. 101 4.1 Introduction. 101 4.2 Double induction.105 4.3 Normalization for Л, э, -i, V. no 4.4 The sub-formula property. 121 4.5 The size of normal deductions.130 v
VI 5 6 7 8 CONTENTS 4.6 Normalization for NJ.132 4.7 An example. 143 4.8 The sub-formula property for NJ.145 4.9 Normalization for NK.154 The sequent calculus.168 5.1 The language of the sequent calculus. 168 5.2 Rules of LK. 170 5.3 Constructing proofs in LK. 175 5.4 The significance of cut.178 5.5 Examples of proofs. 180 5.6 Atomic logical axioms . 187 5.7 Lemma on variable replacement .188 5.8 Translating NJ to LJ. 191 5.9 Translating LJ to NJ. 197 The cut-elimination theorem.202 6.1 Preliminary
definitions.204 6.2 Outline of the lemma.206 6.3 Removing Mixes directly. 209 6.4 Reducing the degree of mix .212 6.5 Reducing the rank. 217 6.6 Reducing the rank: example.238 6.7 Reducing the degree: example . 242 6.8 Intuitionistic sequent calculus LJ.250 6.9 Why mix?.252 6.10 Consequences of the Hauptsatz. 254 6.11 The mid-sequent theorem.257 The consistency of arithmetic.269 7.1 Introduction. 269 7.2 Consistency of simple proofs . 276 7.3 Preliminary details.282 7.4 Overview of the consistency proof. 287 7.5 Replacing
inductions. 288 7.6 Reducing suitable cuts. 293 7.7 A first example . 295 7.8 Elimination of weakenings.298 7.9 Existence of suitable cuts.301 7.10 A simple example. 304 7.11 Summary. 308 Ordinal notations and induction. 312 8.1 Orders, well-orders, and induction . . . 312 8.2 Lexicographical orderings. 315 8.3 Ordinal notations up to £0.320
CONTENTS VU 8.4 Operations on ordinal notations.325 8.5 Ordinal notations are well-ordered.330 8.6 Set-theoretic definitions of the ordinals. 332 8.7 Constructing £0 from below. 336 8.8 Ordinal arithmetic . 337 8.9 Trees and Goodstein sequences.339 9 The consistency of arithmetic, continued. 346 9.1 Assigning ordinal notations εοto proofs. 346 9.2 Eliminating inductions from the end-part. 354 9.3 Removing weakenings.360 9.4 Reduction of suitable cuts. 368 9.5 A simple example, revisited.376 A The Greek alphabet. 380 В Set-theoretic notation.381 C Axioms, rules, and theorems of axiomatic calculi. 383 C.i Axioms and rules of inference. 383 C.2
Theorems and derived rules. 384 D Exercises on axiomatic derivations. 386 D.i Hints for Problem 2.7. 386 D.2 Hints for Problem 2.18. 390 D.3 Exercises with quantifiers.393 E Natural deduction . 394 E.i Inference rules. 394 E.2 Conversions.395 F Sequent calculus.399 G Outline of the cut-elimination theorem.401 Bibliography .405 Index. 412 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Mancosu, Paolo 1960- Galvan, Sergio 1946- Zach, Richard |
| author_GND | (DE-588)133912035 (DE-588)1048745600 (DE-588)1242279687 |
| author_facet | Mancosu, Paolo 1960- Galvan, Sergio 1946- Zach, Richard |
| author_role | aut aut aut |
| author_sort | Mancosu, Paolo 1960- |
| author_variant | p m pm s g sg r z rz |
| building | Verbundindex |
| bvnumber | BV047434433 |
| classification_rvk | SK 130 |
| ctrlnum | (OCoLC)1277017264 (DE-599)BVBBV047434433 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | First edition |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01760nam a2200421 c 4500</leader><controlfield tag="001">BV047434433</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20220914 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">210823s2021 a||| |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9780192895943</subfield><subfield code="c">pbk.</subfield><subfield code="9">978-0-19-289594-3</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">0192895931</subfield><subfield code="c">hbk.</subfield><subfield code="9">0-19-289593-1</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9780192895936</subfield><subfield code="c">hbk.</subfield><subfield code="9">978-0-19-289593-6</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)1277017264</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV047434433</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rda</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-706</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-12</subfield><subfield code="a">DE-83</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 130</subfield><subfield code="0">(DE-625)143216:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Mancosu, Paolo</subfield><subfield code="d">1960-</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)133912035</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">An introduction to proof theory</subfield><subfield code="b">normalization, cut-elimination, and consistency proofs</subfield><subfield code="c">Paolo Mancosu, Sergio Galvan, Richard Zach</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">First edition</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Oxford</subfield><subfield code="b">Oxford University Press</subfield><subfield code="c">2021</subfield></datafield><datafield tag="264" ind1=" " ind2="4"><subfield code="c">© 2021</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">xii, 418 Seiten</subfield><subfield code="b">Illustrationen</subfield><subfield code="c">24 cm</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Beweistheorie</subfield><subfield code="0">(DE-588)4145177-6</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="653" ind1=" " ind2="0"><subfield code="a">Proof theory</subfield></datafield><datafield tag="653" ind1=" " ind2="0"><subfield code="a">Proof theory</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Beweistheorie</subfield><subfield code="0">(DE-588)4145177-6</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Galvan, Sergio</subfield><subfield code="d">1946-</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)1048745600</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Zach, Richard</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)1242279687</subfield><subfield code="4">aut</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Erscheint auch als</subfield><subfield code="n">Online-Ausgabe</subfield><subfield code="z">978-0-19-193879-5</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung BSB München - ADAM Catalogue Enrichment</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032836767&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-032836767</subfield></datafield></record></collection> |
| id | DE-604.BV047434433 |
| illustrated | Illustrated |
| index_date | 2024-07-03T17:59:17Z |
| indexdate | 2024-07-10T09:12:04Z |
| institution | BVB |
| isbn | 9780192895943 0192895931 9780192895936 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-032836767 |
| oclc_num | 1277017264 |
| open_access_boolean | |
| owner | DE-706 DE-11 DE-12 DE-83 |
| owner_facet | DE-706 DE-11 DE-12 DE-83 |
| physical | xii, 418 Seiten Illustrationen 24 cm |
| publishDate | 2021 |
| publishDateSearch | 2021 |
| publishDateSort | 2021 |
| publisher | Oxford University Press |
| record_format | marc |
| spelling | Mancosu, Paolo 1960- Verfasser (DE-588)133912035 aut An introduction to proof theory normalization, cut-elimination, and consistency proofs Paolo Mancosu, Sergio Galvan, Richard Zach First edition Oxford Oxford University Press 2021 © 2021 xii, 418 Seiten Illustrationen 24 cm txt rdacontent n rdamedia nc rdacarrier Beweistheorie (DE-588)4145177-6 gnd rswk-swf Proof theory Beweistheorie (DE-588)4145177-6 s DE-604 Galvan, Sergio 1946- Verfasser (DE-588)1048745600 aut Zach, Richard Verfasser (DE-588)1242279687 aut Erscheint auch als Online-Ausgabe 978-0-19-193879-5 Digitalisierung BSB München - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032836767&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mancosu, Paolo 1960- Galvan, Sergio 1946- Zach, Richard An introduction to proof theory normalization, cut-elimination, and consistency proofs Beweistheorie (DE-588)4145177-6 gnd |
| subject_GND | (DE-588)4145177-6 |
| title | An introduction to proof theory normalization, cut-elimination, and consistency proofs |
| title_auth | An introduction to proof theory normalization, cut-elimination, and consistency proofs |
| title_exact_search | An introduction to proof theory normalization, cut-elimination, and consistency proofs |
| title_exact_search_txtP | An introduction to proof theory normalization, cut-elimination, and consistency proofs |
| title_full | An introduction to proof theory normalization, cut-elimination, and consistency proofs Paolo Mancosu, Sergio Galvan, Richard Zach |
| title_fullStr | An introduction to proof theory normalization, cut-elimination, and consistency proofs Paolo Mancosu, Sergio Galvan, Richard Zach |
| title_full_unstemmed | An introduction to proof theory normalization, cut-elimination, and consistency proofs Paolo Mancosu, Sergio Galvan, Richard Zach |
| title_short | An introduction to proof theory |
| title_sort | an introduction to proof theory normalization cut elimination and consistency proofs |
| title_sub | normalization, cut-elimination, and consistency proofs |
| topic | Beweistheorie (DE-588)4145177-6 gnd |
| topic_facet | Beweistheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032836767&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT mancosupaolo anintroductiontoprooftheorynormalizationcuteliminationandconsistencyproofs AT galvansergio anintroductiontoprooftheorynormalizationcuteliminationandconsistencyproofs AT zachrichard anintroductiontoprooftheorynormalizationcuteliminationandconsistencyproofs |