The Theory of Gödel

164 Pages·2022·2.077 MB·English

by  Carlo Cellucci

Synthese Library 470
Studies in Epistemology, Logic, Methodology,
and Philosophy of Science
Carlo Cellucci
The Theory
of Gödel
Synthese Library
Studies in Epistemology, Logic, Methodology,
and Philosophy of Science
Volume 470
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OtávioBueno,DepartmentofPhilosophy,UniversityofMiami,CoralGables,USA
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Carlo Cellucci
The Theory of Gödel
CarloCellucci
DepartmentofPhilosophy
SapienzaUniversityofRome
Rome,Italy
ISSN0166-6991 ISSN2542-8292 (electronic)
SyntheseLibrary
ISBN978-3-031-13416-6 ISBN978-3-031-13417-3 (eBook)
https://doi.org/10.1007/978-3-031-13417-3
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Preface
This book is meant to present most of the results of mathematical logic that are
relevanttothephilosophyofmathematics.
SincethehardcoreofsuchresultsconsistsofGödel’sincompletenesstheorems,
this explains the title of the book, The Theory of Gödel, which is reminiscent of
the subtitle of the very first book on the subject that Mostowski described as an
attempttopresent“thetheoryofGödel,”namely“thefamoustheoryofundecidable
sentencescreatedbyKurtGödelin1931”(Mostowski1952,v).
ThepresentationofGödel’sincompletenesstheoremsandotherlimitativeresults
in this book is in the spirit, though not in the letter of Jeroslow (1973). Other
presentationscanbefoundinBoolos(1993),EpsteinandCarnielli(2008),Felscher
(2000),Fitting(2007),Girard(1987),GoldsternandJudah(1998),Grandy(1977),
HalbeisenandKrapf(2020),Isaacson(2018),Kennedy(2022),Lindström(1997),
Murawski(1999),Robbin(2006),Smith(2013),Smullyan(1992),S´wierczkowski
(2003),Tourlakis(2003),andZach(2021).
Results are presented in the form most relevant for use in the philosophy
of mathematics. Their implications for Hilbert’s approach to the philosophy of
mathematics are discussed in the Appendix. As to their implications for the
philosophyof mathematics in general, the interested reader may refer to Cellucci
(2022).
Thebookisself-contained,allnotionsbeingexplainedinfulldetail,butofcourse
previousexposuretotheveryfirstrudimentsofmathematicallogicwillhelp.
Iamverygratefultotwoanonymousrefereesforusefulremarksandsuggestions.
I also warmly thank Elena Griniari from Springer for her invaluable help in the
editorialprocess.
Roma,Italy CarloCellucci
v
vi Preface
References
Boolos, G. (1993). The logic of provability. Cambridge: Cambridge University
Press.
Cellucci,C.(2022).Themakingofmathematics:Heuristicphilosophyofmathemat-
ics.Cham:Springer.
Epstein, R., & Carnielli, W. (2008). Computability: Computable functions, logic,
andthefoundationsofmathematics.Socorro:AdvancedReasoningForum.
Felscher,W. (2000).The logic ofarithmetic:Lectureson mathematicallogic (vol.
3).NewYork:GordonandBreach.
Fitting,M.(2007).Incompletenessinthelandofsets.London:CollegePublications.
Girard,J.-Y.(1987).Prooftheoryandlogicalcomplexity.Naples:Bibliopolis.
Goldstern,M.,&Judah,H.(1998).Theincompletenessphenomenon:Anewcourse
inmathematicallogic.Wellesley:AKPeters.
Grandy,R.(1977).Advancedlogicforapplications.Dordrecht:Reidel.
Halbeisen,L.,&Krapf,R.(2020).Gödel’stheoremsandZermelo’saxioms.Cham:
Springer.
Isaacson, D. (2018). Gödel’s incompleteness theorems. Oxford: Oxford Univer-
sity (unpublished lecture notes). https://courses-archive.maths.ox.ac.uk/node/
view_material/38136.
Jeroslow, R. (1973). Redundancies in the Hilbert-Bernays derivability conditions
forGödel’ssecondincompletenesstheorem.TheJournalofSymbolicLogic,38,
359–367.
Kennedy, J. (2022). Gödel’s incompleteness theorems. Cambridge: Cambridge
UniversityPress.
Lindström,P.(1997).Aspectsofincompleteness.Berlin:Springer.
Mostowski,A.(1952).Sentencesundecidableinformalizedarithmetic:Anexposi-
tionofthetheoryofKurtGödel.Amsterdam:North-Holland.
Murawski, R. (1999). Recursive functions and metamathematics: Problems of
completenessanddecidability,Gödel’sTheorems.Dordrecht:Springer.
Robbin,J.(2006).Mathematicallogic:Afirstcourse.Mineola:Dover.
Smith,P.(2013).AnintroductiontoGödel’sincompletenesstheorems.Cambridge:
CambridgeUniversityPress.
Smullyan,R.(1992).Gödel’sincompletenesstheorems.Oxford:OxfordUniversity
Press.
S´wierczkowski, S. (2003). Finite sets and Gödel’s incompleteness theorems. Dis-
sertationesMathematicae,422,1–58.
Tourlakis,G.(2003).Mathematicallogic:Lecturesinlogicandsettheory(vol.1).
Cambridge:CambridgeUniversityPress.
Zach, R. (2021). Incompleteness and computability: An open introduction to
Gödel’stheorems.ScottsValley:CreateSpace.
Contents
1 First-OrderLogic............................................................ 1
1.1 First-OrderLanguages ................................................ 1
1.2 FreeVariables ......................................................... 5
1.3 Substitution ............................................................ 5
1.4 ADeductiveCalculus ................................................. 8
1.5 ConnectiveRules ...................................................... 10
1.6 QuantifierRules ....................................................... 13
1.7 SubstitutivityofEquivalence ......................................... 15
1.8 Equality ................................................................ 17
2 Completeness................................................................. 21
2.1 Interpretations ......................................................... 21
2.2 SubstitutionProperties ................................................ 23
2.3 Soundness ............................................................. 28
2.4 Consistency ............................................................ 30
2.5 RichSets .............................................................. 31
2.6 CompletenessTheorem ............................................... 35
2.7 IsomorphismsofInterpretations...................................... 39
2.8 ElementaryEquivalence .............................................. 43
2.9 DefinabilityinanInterpretation ...................................... 46
3 First-OrderTheories ........................................................ 47
3.1 GeneralitiesonFirst-OrderTheory .................................. 47
3.2 ExtensionsofTheories................................................ 51
3.3 DefinitionalExtensions ............................................... 52
4 PrimitiveRecursiveArithmetic ............................................ 57
4.1 PrimitiveRecursiveFunctions ........................................ 57
4.2 TheTheoriesPRAandPA ........................................... 62
4.3 ElementaryPropertiesofPRA ....................................... 63
4.4 DevelopingArithmeticinPRA ...................................... 66
vii
viii Contents
4.5 BoundedFormulas .................................................... 74
4.6 Non-StandardModelsofPA ......................................... 77
5 Encoding...................................................................... 81
5.1 EncodingofFiniteSequences ........................................ 81
5.2 EncodingofSyntax ................................................... 83
5.3 RE-Theories ........................................................... 88
6 Incompleteness ............................................................... 91
6.1 TraditionalGödel’sFirstIncompletenessTheorem ................. 91
6.2 Gödel’sFirstIncompletenessTheorem .............................. 92
6.3 CorollariesofGödel’sFirstIncompletenessTheorem .............. 94
6.4 Gödel’sSecondIncompletenessTheorem ........................... 95
6.5 ExpressingConsistency ............................................... 97
6.6 Rosser’sIncompletenessTheorem ................................... 99
6.7 Gödel’sThirdIncompletenessTheorem ............................. 101
6.8 ReflectionPrinciple ................................................... 102
6.9 Löb’sTheorem ........................................................ 103
6.10 ExtensiontoOtherFirst-OrderTheories ............................. 104
7 OtherLimitativeResults.................................................... 107
7.1 Tarski’sUndefinabilityTheorems .................................... 107
7.2 UndecidabilityTheorem .............................................. 109
7.3 Church’sTheorem .................................................... 110
7.4 ExtensiontoOtherFirst-OrderTheories ............................. 113
7.5 DecidabilityofMonadicFirst-OrderLogic.......................... 113
8 Second-OrderLogic ......................................................... 117
8.1 Second-OrderLanguages ............................................. 117
8.2 FreeVariables ......................................................... 118
8.3 Substitution ............................................................ 119
8.4 ADeductiveCalculus ................................................. 120
8.5 QuantifierRules ....................................................... 121
8.6 SubstitutivityofEquivalence ......................................... 123
8.7 Equality ................................................................ 123
8.8 Interpretations ......................................................... 124
8.9 SubstitutionProperties ................................................ 126
8.10 Soundness ............................................................. 127
8.11 Consistency ............................................................ 129
8.12 NegativeResults ...................................................... 130
8.13 IsomorphismofInterpretations....................................... 133
9 Second-OrderArithmetic................................................... 135
9.1 Second-OrderTheories ............................................... 135
9.2 TheTheoryPA2 ....................................................... 137
9.3 PrimitiveRecursiveFunctionsinPA2 ............................... 138
9.4 LimitativeResultsforPA2 ........................................... 143
Contents ix
9.5 CategoricityofPA2 ................................................... 145
9.6 StrongIncompletenessTheoremforSecond-OrderLogic .......... 147
9.7 Non-recursiveEnumerabilityofConsequencesofPA2 ............. 147
Appendix.......................................................................... 149
A.1 Hilbert’sApproach .................................................... 149
A.2 TheConservationProgram ........................................... 150
A.3 TheConsistencyProgram ............................................ 150
A.4 EquivalenceoftheTwoPrograms .................................... 151
A.5 FalloftheConsistencyProgram ..................................... 151
A.6 FalloftheConservationProgram .................................... 152
A.7 OtherShortcomingsofHilbert’sApproach .......................... 153
References......................................................................... 155
Index............................................................................... 157