An introduction to Gödel's theorems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2013
|
| Edition: | 2. ed. |
| Series: | Cambridge introductions to philosophy
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 388 S. |
| ISBN: | 9781107606753 9781107022843 |
Staff View
MARC
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| 245 | 1 | 0 | |a An introduction to Gödel's theorems |c Peter Smith |
| 250 | |a 2. ed. | ||
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| 300 | |a XVI, 388 S. | ||
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| 490 | 0 | |a Cambridge introductions to philosophy | |
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Record in the Search Index
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|---|---|
| adam_text | IMAGE 1
CONTENTS
PREFACE THANKS
1 WHAT GOEDEL S THEOREMS SAY
BASIC ARITHRNETIC . INCORNPLETENESS . MORE INCORNPLETENESS . SOME
IMPLICATIONS? . THE ILNPROVABILITY OF CONSISTENCY . MORE IMPLICATIONS? .
WHAT S
NEXT?
2 FUNCTIONS AND ENUMERATIONS
KINDS OF FLLNCTION . CHARACTERISTIC FLLNCTIONS . ENLLMERABLE SETS
ENLLMERATING PAIRS OF NLLMBERS . AN INDENLLMERABLE SET: CANTOR S THEOREM
3 EFFECTIVE COMPUTABILITY
EFFECTIVELY COMPLLTABLE FUNCTIONS . EFFECTIVELY DECIDABLE PROPERTIES AND
SETS . EFFECTIVE ENUMERABILITY . ANOTHER WAY OF DEFINING E.E. SETS OF
NUMBERS .
THE BASIC THEOREM ABOUT E.E. SETS
XIII
XV
1
8
14
4 EFFECTIVELY AXIOMATIZED THEORIES 25
}ORMALIZATION AS AN IDEAL . FORMALIZED LANGUAGES . FORMALIZED THEORIES .
MORE DEFINITIONS THE EFFECTIVE ENUMERABILITY OF THEOREMS .
NEGATIONCOMPLETE THEORIES ARE DECIDABLE
5 CAPTURING NUMERICAL PROPERTIES 36
THREE RE MARKS ON NOTATION THE LANGUAGE LA . A QUICK RE MARK ABOUT
TRUTH . EXPRESSING NUMERICAL PROPERTIES AND FUNCTIONS . CAPTURING NU-
MERICAL PROPERTICS AND FUNCTIONS . EXPRESSING VS. CAPTURING: KEEPING THE
DISTINCTION CLEAR
6 THE TRUTHS OF ARITHMETIC
SUFFICIENTLY EXPRESSIVE LANGUAGES . THE TRUTHS OF A SUFFICIENTLY
EXPRESSIVE LANGUAGE . UNAXIOMATIZABILITY . AN INCOMPLETENESS THEOREM
7 SUFFICIENTLY STRANG ARITHMETICS
THE IDEA OF A SLLFFICIENTLY STRONG THEORY . AN UNDECIDABILITY THEOREM
.
ANOTHER INCOMPLETENESS THEOREM
46
49
VII
IMAGE 2
CONTENTS
8 INTERLUDE: TAKING STOCK COMPARING INCOMPLETENESS ARGUMENTS A ROAD-MAP
9 INDUCTION
THE BASIC PRINCIPLE . ANOTHER VERSION OF THE INDUCTION PRINCIPLE .
IIHLUCTION AND RELATIONS RULE, SCHEMA, OR AXIOM?
10 TWO FORMALIZED ARITHMETICS
BA, BABY ARITHMETIC . BA IS NEGATION-COMPLETE . Q, ROBINSON ARITHMETIC .
WHICH LOGIC? . Q IS NOT COMPLETE . WHY Q IS INTERESTING
11 WH AT Q CAN PROVE
CAPTURING LESS-THAN-OR-EQUAL-TO IN Q . S AND BOUNDED QUANTIFIERS . Q
IS ORDER-ADEQUATE . Q CAN CORRECTLY DECIDE ALL 60 SENTENCES . ~L AND TI]
WFFS
. Q IS ~L-COMPLETE . INTRIGUING COROLLARIES . PROVING Q IS
ORDER-ADEQUATE
53
56
62
71
12 1.0. 0, AN ARITHMETIC WITH INDUCTION 83
THE FORMAL INDUCTION SCHEMA INTRODUCING I.TL. O . WH AT I.TL.O EAN
PROVE .
I.TL. O IS NOT COMPLETE . ON TO I~L
13 FIRST-ORDER PEANO ARITHMETIC 90
BEING GENEROUS WITH INDUCTION SUMMARY OVERVIEW OF PA . HOPING FOR
COMPLETENESS . IS PA CONSISTENT?
14 PRIMITIVE RECURSIVE FUNCTIONS 97
INTRODUCING THE PRIMITIVE RECURSIVE FUNCTIONS . DEFINING THE P.! .
FUNCTIONS MORE EAREFULLY . AN ASIDE AB OUT EXTENSIONALITY . THE P.! .
FUNCTIONS ARE
CORNPUTABLE . NOT ALL COMPUTABLE NURNERICAL FUNCTIONS ARE P.R ..
DEFINING
P.! . PROPERTIES AND RELATIONS BUILDING MORE P.! . FUNCTIONS AND
RELATIONS
FURTHER EXARNPLES
15 LA CAN EXPRESS EVERY P.R. FUNCTION 113
STARTING THE PROOF . THE IDEA OF A SS-FUNCTION FINISHING THE PROOF . THE
P.! . FUNCTIONS AND RELATIONS ARE ~L-EXPRESSIBLE
16 CAPTURING FUNCTIONS
CAPTURING DEFINED . WEAK EAPTURING . STRONG CAPTURING
17 Q IS P.R. ADEQUATE
THE IDEA OF P.! . ADEQUACY . STARTING THE PROOF . COMPLDING THE PROOF .
ALL P.! . FUNCTIONS CAN BE CAPTURED IN Q BY ~L WFFS
18 INTERLUDE: A VERY LITTLE ABOUT PRINCIPIA
PRINCIPIA S LOGICISM . GOEDEL S IRNPACT . ANOTHER ROAD-MAP
VIII
119
124
130
IMAGE 3
CONTENTS
19 THE ARITHMETIZATION OF SYNTAX 136
GOEDEL NMNBERING . ACCEPTABLE CODING SCHEMES . CODING SEQUENCES . TERM,
ATOM, WFF, SENT AMI PRJ ARE P.L . SOME CUTE NOTATION FOR GOEDEL NUMBERS .
THE IDEA OF DIAGONALIZATION
20 ARITHMETIZATION IN MORE DETAIL 144
THE CONCATENATION FUNETION . PROVING THAT TERM IS P.L . PROVING THAT
ATOM, WJF AND SENT ARE P.L . TOWARDS PROVING PR! IS P.L
21 PA IS INCOMPLETE
REMINDERS . G IS TRUE IF AND ONLY IF IT IS UNPROVABLE . PA IS
INCOMPLETE: THE SEMANTIC ARGUMENT THERE IS AN UNDECIDABLE SENTENCE OF
GOLDBACH TYPE
. STARTING THE SYNTACTIC ARGUMENT FOR INCOMPLETENESS . W-INCOMPLETENESS,
W-INCONSISTENCY . FINISHING THE SYNTACTIC ARGUMENT CANONICAL GOEDEL
SENTENCES AND WHAT THEY SAY
22 GOEDEL S FIRST THEOREM
GENERALIZING THE SEMANTIC ARGUMENT . INCOMPLETABILITY . GENERALIZING THE
SYNTAETIC ARGUMENT . THE FIRST THEOREM
23 INTERLUDE: ABOUT THE FIRST THEOREM
WH AT WE HAVE PROVED . SOME WAYS TO ARGUE THAT G T IS TRUE . WHAT
DOESN T FOLLOW FROM INCOMPLETENESS . WHAT DOES FOLLOW FROM
INCOMPLETENESS? .
WHAT S NEXT?
24 THE DIAGONALIZATION LEMMA
PROVABILITY PREDICATES . AN EASY THEOREM ABOUT PROVABILITY PREDICATES .
PROVING G +-+ ~PROV( G ) . THE DIAGONALIZATION LEMMA INCOMPLETENESS
AGAIN GOEDEL SENTENCES AGAIN CAPTURING PROVABILITY?
25 ROSSER S PROOF
~L-SOUNDNESS AND L-CONSISTENCY . ROSSER S CONSTRUCTION: THE BASIC IDEA .
THE GOEDEL-ROSSER THEOREM IMPROVING THE THEOREM
26 BROADENING THE SCOPE
GENERALIZING BEYOND P.L AXIOMATIZED THEORIES . TRUE BASIC ARITHMETIC
CAN T BE AXIORNATIZED . GENERALIZING BEYOND OVERTLY ARITHMETIC THEORIES
.
A WORD OF WARNING
27 TARSKI S THEOREM
TRUTH-PREDICATES, TRUTH-THEORIES . THE UNDEFINABILITY OF TRUTH .
TARSKI S THEOREM: THE INEXPRESSIBILITY OF TRUTH . CAPTURING AND
EXPRESSING AGAIN .
THE MASTER ARGUMENT?
152
161
167
177
185
191
197
IX
IMAGE 4
CONTENTS
28 SPEED-UP THE LENGTH OF PROOFS . THE IDEA OF SPEED-UP . LONG PROOFS,
VIA DIAGONALIZATION
29 SECOND-ORDER ARITHMETICS
SECOND-ORDER SYNTAX . SECOND-ORDER SEMANTICS . THE INDUCTION AXIOM
AGAIN NEAT SECOND-ORDER ARITHMETICS . INTRODUCING PA 2 * CATEGORICITY
.
INCOMPLETENESS AND CATEGORICITY . ANOTHER ARITHMETIC . SPEED-UP AGAIN
201
204
30 INTERLUDE: INCOMPLETENESS AND ISAACSON S THESIS 219
TAKING STOCK THE UNPROVABILITY-IN-PA OF GOODSTEIN S THEOREM AN ASIDE
ON PROVING THE KIRBY-PARIS THEOREM ISAACSON S THESIS EVER UPWARDS .
ANCESTRAL ARITHMETIC
31 GOEDEL S SECOND THEOREM FOR PA 233
DEFINING (ON THE FORMALIZED FIRST THEOREM IN PA . THE SECOND THEOREM
FOR PA . ON W-INCOMPLETENESS AND W-CONSISTENCY AGAIN SO NEAR, YET SO
FAR . HOW SHOULD WE INTERPRET THE SECOND THEOREM?
32 ON THE UNPROVABILITY OF CONSISTENCY ,
239
THREE COROLLARIES . WEAKER THEORIES CANNOT PROVE THE CONSISTENCY OF PA .
PA CANNOT PROVE THE CONSISTENCY OF STRONGER THEORIES . INTRODUCING
GENTZEN .
WHAT DO WE LEARN FROM GENTZEN S PROOF?
33 GENERALIZING THE SECOND THEOREM 245
MORE NOTATION . THE HILBERT-BERNAYS-LOEB DERIVABILITY CONDITIONS . T S
IGNORANCE ABOUT WHAT IT CAN T PROVE . THE FORMALIZED SECOND THEOREM .
JEROSLOW S LEMMA AND THE SECOND THEOREM
34 LOEB S THEOREM AND OTHER MATTERS 252
THEORIES THAT PROVE THEIR OWN INCONSISTENCY . THE EQUIVALENCE OF FIXED
POINTS FOR ,PROV . CONSISTENCY EXTENSIONS HENKIN S PROBLEM AND LOEB S
THEOREM . LOEB S THEOREM AND THE SECOND THEOREM
35 DERIVING THE DERIVABILITY CONDITIONS 258
THE SECOND DERIVABILITY CONDITION FOR PA . THE THIRD DERIVABILITY
CONDITION
FOR PA . GENERALIZING TO NICE* THEORIES . THE SECOND THEOREM FOR WEAKER
ARITHMETICS
36 THE BEST AND MOST GENERAL VERSION 262
THERE ARE PROVABLE CONSISTENCY SENTENCES . THE INTENSIONALITY OF THE
SECOND THEOREM . REFLECTION . THE BEST VERSION? . ANOTHER ROUTE TO
ACCEPTING
A GOEDEL SENTENCE?
X
IMAGE 5
37 INTERLUDE: THE SECOND THEOREM, HILBERT, MINDS AND MACHINES
CONTENTS
272
REAL VS. IDEAL MATHEMATICS . A QUICK ASIDE: GOEDEL S CAUTION .
RELATING THE REAL AND THE IDEAL . PROVING REAL-SOUNDNESS? . THE IMPACT
OF GOEDEL
MINDS AND COMPUTERS THE REST OF THIS BOOK: ANOTHER ROAD-MAP
38 IL-RECURSIVE FUNCTIONS
MINIMIZATION AND J1-RECURSIVE FUNCTIONS . ANOTHER DEFINITION OF
J1-RECURSIVENESS . THE ACKERMANN PET ER FUNCTION . ACKERMANN-PETER IS
J1-RECURSIVE
BUT NOT P.L . INTRODUCING CHURCH S THESIS WHY CAN T WE DIAGONALIZE OUT?
. USING CHURCH S THESIS
285
39 Q IS RECURSIVELY ADEQUATE 297
CAPTURING A FUNCTION DEFINED BY MINIMIZATION . THE RECURSIVE ADEQUACY
THEOREM SUFFICIENTLY STRONG THEORIES AGAIN NICE THEORIES CAN ONLY
CAPTURE
J1-RECURSIVE FUNCTIONS
40 UNDECIDABILITY AND INCOMPLETENESS 300
SOME MORE DEFINITIONS . Q AND PA ARE UNDECIDABLE . THE
ENTSCHEIDUNGSPROBLEM . INCORNPLDENESS THEOREMS FOR NICE THEORIES .
NEGATION-CORNPLETE
THEORIES ARE RECURSIVELY DECIDABLE . RECURSIVELY ADEQUATE THEORIES ARE
NOT
RECURSIVELY DECIDABLE . INCORNPLETENESS AGAIN . TRUE BASIC ARITHMETIC IS
NOT R.E.
41 TURING MACHINES 310
THE BASIC CONCEPTION . TURING COMPUTATION DEFINED MORE CAREFULLY . SOME
SIMPLE EXAMPLES . TURING MACHINES AND THEIR STATES
42 TURING MACHINES AND RECURSIVENESS 321
J1- RECURSIVENESS ENTAILS TURING COMPUTABILITY . J1- RECURSIVENESS
ENTAILS
TURING COMPUTABILITY: THE DETAILS TURING COMPUTABILITY ENTAILS
J1-RECURSIVENESS . GENERALIZING
43 HALTING AND INCOMPLETENESS 328
TWO SIMPLE RESULTS ABOUT TURING PROGRAMS . THE HALTING PROBLEM . THE
ENTSCHEIDUNGSPMBLEM AGAIN . THE HALTING PROBLEM AND INCOMPLETENESS .
ANOTHER INCOMPLETENESS ARGUMENT KLEENE S NORMAL FORM THEOREM A
NOTE ON PARTIAL COMPUTABLE FUNCTIONS . KLEENE S THEOREM ENTAILS GOEDEL S
FIRST THEOREM
44 THE CHURCH-TURING THESIS
PUTTING THINGS TOGETHER . FROM EUCLID TO HILBERT . 1936 AND ALL THAT .
WHAT THE CHURCH-TURING THESIS IS AND IS NOT . THE STATUS OF THE THESIS
45 PROVING THE THESIS?
338
348
XI
IMAGE 6
CONTENTS
VAGUENESS AND THE IDEA OF COMPUTABILITY . FORMAL PROOFS AND INFORMAL
DEMONSTRATIONS SQUEEZING ARGUMENTS - THE VERY IDEA . KREISEL S
SQUEEZING ARGUMENT . THE FIRST PREMISS FOR A SQUEEZING ARGUMENT . THE
OTHER
PREMISSES, THANKS TO KOLMOGOROV AND USPENSKII . THE SQUEEZING ARGUMENT
DEFENDED . TO SUMMARIZE
46 LOOKING BACK
FURTHER READING
BIBLIOGRAPHY
INDEX
XII
367
370
372
383
|
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| author | Smith, Peter 1944- |
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| ctrlnum | (OCoLC)840743461 (DE-599)BVBBV040898170 |
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| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.3 |
| dewey-search | 511.3 |
| dewey-sort | 3511.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Philosophie |
| edition | 2. ed. |
| format | Book |
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| genre_facet | Einführung |
| id | DE-604.BV040898170 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T00:34:52Z |
| institution | BVB |
| isbn | 9781107606753 9781107022843 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025877699 |
| oclc_num | 840743461 |
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| owner_facet | DE-384 DE-29T DE-11 DE-19 DE-BY-UBM DE-29 DE-M468 DE-473 DE-BY-UBG |
| physical | XVI, 388 S. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series2 | Cambridge introductions to philosophy |
| spelling | Smith, Peter 1944- Verfasser (DE-588)173658121 aut An introduction to Gödel's theorems Peter Smith 2. ed. Cambridge [u.a.] Cambridge Univ. Press 2013 XVI, 388 S. txt rdacontent n rdamedia nc rdacarrier Cambridge introductions to philosophy Gödel, Kurt Logic, Symbolic and mathematical Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 s DE-604 Digitalisierung UB Erlangen application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025877699&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Smith, Peter 1944- An introduction to Gödel's theorems Gödel, Kurt Logic, Symbolic and mathematical Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 gnd |
| subject_GND | (DE-588)4021417-5 (DE-588)4151278-9 |
| title | An introduction to Gödel's theorems |
| title_auth | An introduction to Gödel's theorems |
| title_exact_search | An introduction to Gödel's theorems |
| title_full | An introduction to Gödel's theorems Peter Smith |
| title_fullStr | An introduction to Gödel's theorems Peter Smith |
| title_full_unstemmed | An introduction to Gödel's theorems Peter Smith |
| title_short | An introduction to Gödel's theorems |
| title_sort | an introduction to godel s theorems |
| topic | Gödel, Kurt Logic, Symbolic and mathematical Gödelscher Unvollständigkeitssatz (DE-588)4021417-5 gnd |
| topic_facet | Gödel, Kurt Logic, Symbolic and mathematical Gödelscher Unvollständigkeitssatz Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025877699&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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