An Introduction to Gödel’s Theorems - Carte Straina · An Introduction to Gödel’s Theorems In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells

An Introduction to Gödel’s Theorems

In 1931, the young Kurt Gödel published his First Incompleteness Theorem,which tells us that, for any sufficiently rich theory of arithmetic, there are somearithmetical truths the theory cannot prove. This remarkable result is amongthe most intriguing (and most misunderstood) in logic. Gödel also outlined anequally significant Second Incompleteness Theorem. How are these Theoremsestablished, and why do they matter? Peter Smith answers these questions bypresenting an unusual variety of proofs for the First Theorem, showing how toprove the Second Theorem, and exploring a family of related results (includingsome not easily available elsewhere). The formal explanations are interwovenwith discussions of the wider significance of the two Theorems. This book –extensively rewritten for its second edition – will be accessible to philosophystudents with a limited formal background. It is equally suitable for mathematicsstudents taking a first course in mathematical logic.

Peter Smith was formerly Senior Lecturer in Philosophy at the University ofCambridge. His books include Explaining Chaos (1998) and An Introduction toFormal Logic (2003), and he is also a former editor of the journal Analysis.

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Cambridge University Press978-1-107-02284-3 - An Introduction to Gödel’s Theorems: Second EditionPeter SmithFrontmatterMore information

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An Introduction to

Gödel’s TheoremsSecond edition

Peter SmithUniversity of Cambridge






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For Patsy, as ever











Contents

Preface xiii

Thanks xv

1 What Gödel’s Theorems say 1Basic arithmetic · Incompleteness · More incompleteness · Some implica-tions? · The unprovability of consistency · More implications? · What’snext?

2 Functions and enumerations 8Kinds of function · Characteristic functions · Enumerable sets · Enumer-ating pairs of numbers · An indenumerable set: Cantor’s theorem

3 Effective computability 14Effectively computable functions · Effectively decidable properties and sets· Effective enumerability · Another way of defining e.e. sets of numbers ·The Basic Theorem about e.e. sets

4 Effectively axiomatized theories 25Formalization as an ideal · Formalized languages · Formalized theories· More definitions · The effective enumerability of theorems · Negation-complete theories are decidable

5 Capturing numerical properties 36Three remarks on notation · The language LA · A quick remark abouttruth · Expressing numerical properties and functions · Capturing nu-merical properties and functions · Expressing vs. capturing: keeping thedistinction clear

6 The truths of arithmetic 46Sufficiently expressive languages · The truths of a sufficiently expressivelanguage · Unaxiomatizability · An incompleteness theorem

7 Sufficiently strong arithmetics 49The idea of a ‘sufficiently strong’ theory · An undecidability theorem ·Another incompleteness theorem

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8 Interlude: Taking stock 53Comparing incompleteness arguments · A road-map

9 Induction 56The basic principle · Another version of the induction principle · Inductionand relations · Rule, schema, or axiom?

10 Two formalized arithmetics 62BA, Baby Arithmetic · BA is negation-complete · Q, Robinson Arithmetic· Which logic? · Q is not complete · Why Q is interesting

11 What Q can prove 71Capturing less-than-or-equal-to in Q · ‘≤’ and bounded quantifiers · Q isorder-adequate · Q can correctly decide all Δ0 sentences · Σ1 and Π1 wffs· Q is Σ1-complete · Intriguing corollaries · Proving Q is order-adequate

12 IΔ0, an arithmetic with induction 83The formal Induction Schema · Introducing IΔ0 · What IΔ0 can prove ·IΔ0 is not complete · On to IΣ1

13 First-order Peano Arithmetic 90Being generous with induction · Summary overview of PA · Hoping forcompleteness · Is PA consistent?

14 Primitive recursive functions 97Introducing the primitive recursive functions · Defining the p.r. functionsmore carefully · An aside about extensionality · The p.r. functions arecomputable · Not all computable numerical functions are p.r. · Definingp.r. properties and relations · Building more p.r. functions and relations ·Further examples

15 LA can express every p.r. function 113Starting the proof · The idea of a β-function · Finishing the proof · Thep.r. functions and relations are Σ1-expressible

16 Capturing functions 119Capturing defined · ‘Weak’ capturing · ‘Strong’ capturing

17 Q is p.r. adequate 124The idea of p.r. adequacy · Starting the proof · Completing the proof ·All p.r. functions can be captured in Q by Σ1 wffs

18 Interlude: A very little about Principia 130Principia’s logicism · Gödel’s impact · Another road-map

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19 The arithmetization of syntax 136Gödel numbering · Acceptable coding schemes · Coding sequences · Term,Atom, Wff, Sent and Prf are p.r. · Some cute notation for Gödel numbers· The idea of diagonalization

20 Arithmetization in more detail 144The concatenation function · Proving that Term is p.r. · Proving thatAtom, Wff and Sent are p.r. · Towards proving Prf is p.r.

21 PA is incomplete 152Reminders · ‘G is true if and only if it is unprovable’ · PA is incomplete: thesemantic argument · ‘There is an undecidable sentence of Goldbach type’· Starting the syntactic argument for incompleteness · ω-incompleteness,ω-inconsistency · Finishing the syntactic argument · Canonical Gödel sen-tences and what they say

22 Gödel’s First Theorem 161Generalizing the semantic argument · Incompletability · Generalizing thesyntactic argument · The First Theorem

23 Interlude: About the First Theorem 167What we have proved · Some ways to argue that GT is true · What doesn’tfollow from incompleteness · What does follow from incompleteness? ·What’s next?

24 The Diagonalization Lemma 177Provability predicates · An easy theorem about provability predicates ·Proving G ↔ ¬Prov(�G�) · The Diagonalization Lemma · Incompletenessagain · ‘Gödel sentences’ again · Capturing provability?

25 Rosser’s proof 185Σ1-soundness and 1-consistency · Rosser’s construction: the basic idea ·The Gödel-Rosser Theorem · Improving the theorem

26 Broadening the scope 191Generalizing beyond p.r. axiomatized theories · True Basic Arithmeticcan’t be axiomatized · Generalizing beyond overtly arithmetic theories ·A word of warning

27 Tarski’s Theorem 197Truth-predicates, truth-theories · The undefinability of truth · Tarski’sTheorem: the inexpressibility of truth · Capturing and expressing again ·The Master Argument?

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Contents

28 Speed-up 201The length of proofs · The idea of speed-up · Long proofs, via diagonal-ization

29 Second-order arithmetics 204Second-order syntax · Second-order semantics · The Induction Axiomagain · ‘Neat’ second-order arithmetics · Introducing PA2 · Categoricity ·Incompleteness and categoricity · Another arithmetic · Speed-up again

30 Interlude: Incompleteness and Isaacson’s Thesis 219Taking stock · The unprovability-in-PA of Goodstein’s Theorem · An asideon proving the Kirby-Paris Theorem · Isaacson’s Thesis · Ever upwards ·Ancestral arithmetic

31 Gödel’s Second Theorem for PA 233Defining Con · The Formalized First Theorem in PA · The Second Theoremfor PA · On ω-incompleteness and ω-consistency again · So near, yet sofar · How should we interpret the Second Theorem?

32 On the ‘unprovability of consistency’ 239Three 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 245More notation · The Hilbert-Bernays-Löb derivability conditions · T ’signorance about what it can’t prove · The Formalized Second Theorem ·Jeroslow’s Lemma and the Second Theorem

34 Löb’s Theorem and other matters 252Theories that ‘prove’ their own inconsistency · The equivalence of fixedpoints for ¬Prov · Consistency extensions · Henkin’s Problem and Löb’sTheorem · Löb’s Theorem and the Second Theorem

35 Deriving the derivability conditions 258The second derivability condition for PA · The third derivability conditionfor PA · Generalizing to nice* theories · The Second Theorem for weakerarithmetics

36 ‘The best and most general version’ 262There are provable consistency sentences · The ‘intensionality’ of the Sec-ond Theorem · Reflection · The best version? · Another route to acceptinga Gödel sentence?

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Contents

37 Interlude: The Second Theorem, Hilbert, minds and machines 272‘Real’ vs. ‘ideal’ mathematics · A quick aside: Gödel’s caution · Relatingthe real and the ideal · Proving real-soundness? · The impact of Gödel ·Minds and computers · The rest of this book: another road-map

38 μ-Recursive functions 285Minimization and μ-recursive functions · Another definition of μ-recursive-ness · The Ackermann–Péter function · Ackermann–Péter is μ-recursivebut not p.r. · Introducing Church’s Thesis · Why can’t we diagonalize out?· Using Church’s Thesis

39 Q is recursively adequate 297Capturing a function defined by minimization · The recursive adequacytheorem · Sufficiently strong theories again · Nice theories can only captureμ-recursive functions

40 Undecidability and incompleteness 300Some more definitions · Q and PA are undecidable · The Entscheidungs-problem · Incompleteness theorems for nice′ theories · Negation-completetheories are recursively decidable · Recursively adequate theories are notrecursively decidable · Incompleteness again · True Basic Arithmetic isnot r.e.

41 Turing machines 310The basic conception · Turing computation defined more carefully · Somesimple examples · ‘Turing machines’ and their ‘states’

42 Turing machines and recursiveness 321μ-Recursiveness entails Turing computability · μ-Recursiveness entailsTuring computability: the details · Turing computability entails μ-recurs-iveness · Generalizing

43 Halting and incompleteness 328Two simple results about Turing programs · The halting problem · TheEntscheidungsproblem again · The halting problem and incompleteness ·Another incompleteness argument · Kleene’s Normal Form Theorem · Anote on partial computable functions · Kleene’s Theorem entails Gödel’sFirst Theorem

44 The Church–Turing Thesis 338Putting 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? 348

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Contents

Vagueness and the idea of computability · Formal proofs and informaldemonstrations · Squeezing arguments – the very idea · Kreisel’s squeez-ing argument · The first premiss for a squeezing argument · The otherpremisses, thanks to Kolmogorov and Uspenskii · The squeezing argumentdefended · To summarize

46 Looking back 367

Further reading 370

Bibliography 372

Index 383

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Preface

In 1931, the young Kurt Gödel published his First and Second IncompletenessTheorems; very often, these are referred to simply as ‘Gödel’s Theorems’ (eventhough he proved many other important results). These Incompleteness Theo-rems settled – or at least, seemed to settle – some of the crucial questions of theday concerning the foundations of mathematics. They remain of the greatest sig-nificance for the philosophy of mathematics, though just what that significanceis continues to be debated. It has also frequently been claimed that Gödel’s The-orems have a much wider impact on very general issues about language, truthand the mind.

This book gives proofs of the Theorems and related formal results, and touches– necessarily briefly – on some of their implications. Who is the book for?Roughly speaking, for those who want a lot more fine detail than you get inbooks for a general audience (the best of those is Franzén, 2005), but who findthe rather forbidding presentations in classic texts in mathematical logic (likeMendelson, 1997) too short on explanatory scene-setting. I assume only a mod-est amount of background in logic. So I hope philosophy students will find thebook useful, as will mathematicians who want a more accessible exposition.

But don’t be misled by the relatively relaxed style; don’t try to browse throughtoo quickly. We do cover a lot of ground in quite a bit of detail, and new ideasoften come thick and fast. Take things slowly!

I originally intended to write a shorter book, leaving many of the formal detailsto be filled in from elsewhere. But while that plan might have suited somereaders, I soon realized that it would seriously irritate others to be sent hitherand thither to consult a variety of textbooks with different terminologies anddifferent notations. So, in the end, I have given more or less full proofs of mostof the key results we cover (‘�’ serves as our end-of-proof marker, as we wantthe more usual ‘�’ for another purpose).

However, my original plan shows through in two ways. First, some proofs arestill only partially sketched in. Second, I try to signal very clearly when thedetailed proofs I do give can be skipped without much loss of understanding.With judicious skimming, you should be able to follow the main formal themesof the book even if you have limited taste for complex mathematical arguments.For those who want to fill in more details and test their understanding there areexercises on the book’s website at www.godelbook.net, where there are also othersupplementary materials.

As we go through, there is also an amount of broadly philosophical commen-tary. I follow Gödel in believing that our formal investigations and our general

xiii






Preface

reflections on foundational matters should illuminate and guide each other. Ihope that these brief philosophical discussions – relatively elementary thoughcertainly not always uncontentious – will also be reasonably widely accessible.Note however that I am more interested in patterns of ideas and argumentsthan in being historically very precise when talking e.g. about logicism or aboutHilbert’s Programme.

Writing a book like this presents many problems of organization. For example, wewill need to call upon a number of ideas from the general theory of computation– we will make use of both the notion of a ‘primitive recursive function’ andthe more general notion of a ‘μ-recursive function’. Do we explain these relatedideas all at once, up front? Or do we give the explanations many chapters apart,when the respective notions first get put to use?

I’ve mostly adopted the second policy, introducing new ideas as and whenneeded. This has its costs, but I think that there is a major compensating benefit,namely that the way the book is organized makes it clearer just what depends onwhat. It also reflects something of the historical order in which ideas emerged.

How does this second edition differ from the first? This edition is over twentypages longer, but that isn’t because there is much new material. Rather, I havemostly used the extra pages to make the original book more reader-friendly;there has been a lot of rewriting and rearrangement, particularly in the openingchapters. Perhaps the single biggest change is in using a more traditional line ofproof for the adequacy of Robinson Arithmetic (Q) for capturing all the primi-tive recursive functions. I will probably have disappointed some readers by stillresisting the suggestion that I provide a full-blown, warts-and-all, proof of theSecond Theorem, though I do say rather more than before. But after all, this issupposed to be a relatively introductory book.

Below, I acknowledge the help that I have so generously been given by so many.But here I must express thanks of a quite different order to Patsy Wilson-Smith,without whose continuing love and support neither edition of this book wouldever have been written. This book is for her.

xiv






Thanks

‘Acknowledgements’ is far too cold a word. I have acquired many intellectualdebts in the course of writing this book: with great kindness, a lot of peoplehave given me comments, suggestions and corrections. As a result, the book –whatever its remaining shortcomings – is so much better than it might havebeen.

My colleague Michael Potter has been an inspiring presence ever since I returnedto Cambridge. Many thanks are due to him and to all those who gave me ad-vice on draft extracts while I was writing the first edition, including the lateTorkel Franzén, Tim Button, Luca Incurvati, Jeffrey Ketland, Aatu Koskensilta,Christopher Leary, Mary Leng, Toby Ord, Alex Paseau, Jacob Plotkin, José F.Ruiz, Kevin Scharp, Hartley Slater, and Tim Storer. I should especially men-tion Richard Zach, whose comments were particularly extensive and particularlyhelpful.

I no longer have a list of all those who found errors in the first printing. ButArnon Avron, Peter Milne, Saeed Salehi, and Adil Sanaulla prompted the mostsignificant changes in content that made their way into corrected reprints of thefirst edition. Orlando May then spotted some still remaining technical errors,as well as a distressing number of residual typos. Jacob Plotkin, Tony Roy andAlfredo Tomasetta found further substantive mistakes. Again, I am very grateful.

It is an oddity that the books which are read the most – texts aimed atstudents – are reviewed the least in the journals: but Arnon Avron and CraigSmoryński did write friendly reviews of the first edition, and I have now tried tomeet at least some of their expressed criticisms.

When I started working on this second edition, I posted some early parts on myblog Logic Matters, and received very useful suggestions from a number of peo-ple. Encouraged by that, at a late stage in the writing I experimentally asked forvolunteers to proof-read thirty-page chunks of the book: the bribe I offered wastiny, just a mention here! But over forty more people took up the invitation, soevery page was looked at again three or four times. Many of these readers appliedthemselves to the task with quite extraordinary care and attention, telling menot just about the inevitable typos, but about ill-phrased sentences, obscurities,phrases that puzzled a non-native speaker of English, and more besides. A hand-ful of readers – I report this with mixed feelings – also found small technical er-rors still lurking in the text. The experiment, then, was a resounding success. Sowarm thanks are due to, among others, Sama Agahi, Amir Anvari, Bert Baum-gaertner, Alex Blum, Seamus Bradley, Matthew Brammall, Benjamin Briggs,

xv






Thanks

Catrin Campbell-Moore, Jordan Collins, Irena Cronin, Matthew Dentith, NeilDewar, Jan Eißfeldt, Özge Ekin, Nicolas Fillion, Stefan Daniel Firth, Joshua P.Fry, Marc Alcobé García, André Gargoura, Jaime Gaspar, Jake Goritski, ScottHendricks, Harrison Hibbert, Daniel Hoek, Anil Joshi, Abolfazl Karimi, AmirKhamseh, Baldur Arnviður Kristinsson, Jim Laird, S. P. Lam, Jim Lippard, CarlMummett, Fredrick Nelson, Robert Rynasiewicz, Noah David Schweber, AlinSoare, Shane Steinert-Threlkeld, Andrew Stephenson, Alexander auf der Straße,S. P. Suresh, Andrew Tedder, Arhat Virdi, Benjamin Wells, John Wigglesworth,and Andrew Withy. I am very grateful to them all!

Three more readers require special mention. David Auerbach’s many sugges-tions have uniformly made for significant improvements. Matthew Donald hasalso given me extremely helpful comments on the whole book. Frank Scheld hasproof-read successive versions with enthusiasm and an eagle’s eye. Again, verywarm thanks.

Finally, I must thank Hilary Gaskin at Cambridge University Press, who initiallyaccepted the book for publication and then offered me the chance to write asecond edition.

xvi






Documents

An Introduction to Gödel’s Theorems - Carte Straina · An Introduction to Gödel’s Theorems In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells