Godels Proof

Gödel’s Proof

In 1931 Kurt Gödel published a revolutionary paper—onethat challenged certain basic assumptions underlying muchtraditional research in mathematics and logic. Today hisexploration of terra incognita has been recognized as oneof the major contributions to modern scientific thought.Here is the first book to present a readable explanation toboth scholars and non-specialists of the main ideas, thebroad implications of Gödel’s proof. It offers any educatedperson with a taste for logic and philosophy the chance tosatisfy his intellectual curiosity about a previouslyinaccessible subject.

Gödel’s

Proof

Ernest Nagel

and

James R.Newman

London

First published 1958 Routledge is an imprint of the Taylor & Francis Group This edition published in the Taylor & Francis e-Library, 2004. © 1958 Ernest Nagel and James R.Newman

All rights reserved. No part of this book may bereprinted or reproduced or utilized in any form orby any electronic, mechanical, or other means, nowknown or hereafter invented, including photocopyingand recording, or in any information storage orretrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication DataNagel Ernest, 1901–

Godel’s proof1. Mathematics. Theories of Godel, Kurt 1906–1978.I. Title II. Newman James R. (James Roy)510'.92'4

ISBN 0-203-40661-3 Master e-book ISBN ISBN 0-203-71485-7 (Adobe eReader Format)ISBN 0-415-04040-X (Print Edition)

to

Bertrand Russell

vii

Contents

Acknowledgments ixI Introduction 3

II The Prozblem of Consistency 8III Absolute Proofs of Consistency 26IV The Systematic Codification of Formal Logic 37V An Example of a Successful Absolute Proof of

Consistency 45VI The Idea of Mapping and Its Use in Mathematics 57

VII Gödel’s Proofs 68A Gödel numbering 68B The arithmetization of meta-mathematics 76C The heart of Gödel’s argument 85

VIII Concluding Reflections 98Appendix: Notes 103Brief Bibliography 115Index 117

ix

Acknowledgments

The authors gratefully acknowledge the generousassistance they received from Professor John C.Cooleyof Columbia University. He read critically an early draftof the manuscript, and helped to clarify the structureof the argument and to improve the exposition of pointsin logic. We wish to thank Scientific American forpermission to reproduce several of the diagrams in thetext, which appeared in an article on Gödel’s Proof inthe June 1956 issue of the magazine. We are indebtedto Professor Morris Kline of New York University forhelpful suggestions regarding the manuscript.

Gödel’s Proof

3

IIntroduction

In 1931 there appeared in a German scientific periodicala relatively short paper with the forbidding title “Uberformal unentscheidbare Sätze der PrincipiaMathematica und verwandter Systeme” (“On FormallyUndecidable Propositions of Principia Mathematica andRelated Systems”). Its author was Kurt Gödel, then ayoung mathematician of 25 at the University of Viennaand since 1938 a permanent member of the Institute forAdvanced Study at Princeton. The paper is a milestonein the history of logic and mathematics. When HarvardUniversity awarded Gödel an honorary degree in 1952,the citation described the work as one of the mostimportant advances in logic in modern times.

At the time of its appearance, however, neither the titleof Gödel’s paper nor its content was intelligible to mostmathematicians. The Principia Mathematica mentionedin the title is the monumental three-volume treatise byAlfred North Whitehead and Bertrand Russell onmathematical logic and the foundations of mathematics;

4 Gödel’s Proof

and familiarity with that work is not a prerequisite tosuccessful research in most branches of mathematics.Moreover, Gödel’s paper deals with a set of questions thathas never attracted more than a comparatively small groupof students. The reasoning of the proof was so novel at thetime of its publication that only those intimatelyconversant with the technical literature of a highlyspecialized field could follow the argument with readycomprehension. Nevertheless, the conclusions Gödelestablished are now widely recognized as beingrevolutionary in their broad philosophical import. It isthe aim of the present essay to make the substance ofGödel’s findings and the general character of his proofaccessible to the non-specialist.

Gödel’s famous paper attacked a central problem inthe foundations of mathematics. It will be helpful togive a brief preliminary account of the context in whichthe problem occurs. Everyone who has been exposedto elementary geometry will doubtless recall that it istaught as a deductive discipline. It is not presented asan experimental science whose theorems are to beaccepted because they are in agreement withobservation. This notion, that a proposition may beestablished as the conclusion of an explicit logicalproof, goes back to the ancient Greeks, who discoveredwhat is known as the “axiomatic method” and used itto develop geometry in a systematic fashion. Theaxiomatic method consists in accepting without proofcertain propositions as axioms or postulates (e.g., theaxiom that through two points just one straight line

Introduction 5

can be drawn), and then deriving from the axioms allother propositions of the system as theorems. Theaxioms constitute the “foundations” of the system; thetheorems are the “superstructure,” and are obtainedfrom the axioms with the exclusive help of principlesof logic.

The axiomatic development of geometry made apowerful impression upon thinkers throughout the ages;for the relatively small number of axioms carry thewhole weight of the inexhaustibly numerouspropositions derivable from them. Moreover, if in someway the truth of the axioms can be established—and,indeed, for some two thousand years most studentsbelieved without question that they are true of space—both the truth and the mutual consistency of all thetheorems are automatically guaranteed. For thesereasons the axiomatic form of geometry appeared tomany generations of outstanding thinkers as the modelof scientific knowledge at its best. It was natural to ask,therefore, whether other branches of thought besidesgeometry can be placed upon a secure axiomaticfoundation. However, although certain parts of physicswere given an axiomatic formulation in antiquity (e.g.,by Archimedes), until modern times geometry was theonly branch of mathematics that had what most studentsconsidered a sound axiomatic basis.

But within the past two centuries the axiomaticmethod has come to be exploited with increasing powerand vigor. New as well as old branches of mathematics,including the familiar arithmetic of cardinal (or

6 Gödel’s Proof

“whole”) numbers, were supplied with what appearedto be adequate sets of axioms. A climate of opinion wasthus generated in which it was tacitly assumed thateach sector of mathematical thought can be suppliedwith a set of axioms sufficient for developingsystematically the endless totality of true propositionsabout the given area of inquiry.

Gödel’s paper showed that this assumption isuntenable. He presented mathematicians with theastounding and melancholy conclusion that theaxiomatic method has certain inherent limitations,which rule out the possibility that even the ordinaryarithmetic of the integers can ever be fully axiomatized.What is more, he proved that it is impossible to establishthe internal logical consistency of a very large class ofdeductive systems—elementary arithmetic, forexample—unless one adopts principles of reasoning socomplex that their internal consistency is as open todoubt as that of the systems themselves. In the light ofthese conclusions, no final systematization of manyimportant areas of mathematics is attainable, and noabsolutely impeccable guarantee can be given that manysignificant branches of mathematical thought areentirely free from internal contradiction.

Gödel’s findings thus undermined deeply rootedpreconceptions and demolished ancient hopes that werebeing freshly nourished by research on the foundationsof mathematics. But his paper was not altogethernegative. It introduced into the study of foundationquestions a new technique of analysis comparable in

Introduction 7

its nature and fertility with the algebraic method thatRené Descartes introduced into geometry. Thistechnique suggested and initiated new problems forlogical and mathematical investigation. It provoked areappraisal, still under way, of widely held philosophiesof mathematics, and of philosophies of knowledge ingeneral.

The details of Gödel’s proofs in his epoch-makingpaper are too difficult to follow without considerablemathematical training. But the basic structure of hisdemonstrations and the core of his conclusions can bemade intelligible to readers with very limitedmathematical and logical preparation. To achieve suchan understanding, the reader may find useful a briefaccount of certain relevant developments in the historyof mathematics and of modern formal logic. The nextfour sections of this essay are devoted to this survey.

8

IIThe Problem of Consistency

The nineteenth century witnessed a tremendousexpansion and intensification of mathematical research.Many fundamental problems that had long withstoodthe best efforts of earlier thinkers were solved; newdepartments of mathematical study were created; andin various branches of the discipline new foundationswere laid, or old ones entirely recast with the help ofmore precise techniques of analysis. To illustrate: theGreeks had proposed three problems in elementarygeometry: with compass and straight-edge to trisect anyangle, to construct a cube with a volume twice thevolume of a given cube, and to construct a square equalin area to that of a given circle. For more than 2,000years unsuccessful attempts were made to solve theseproblems; at last, in the nineteenth century it was provedthat the desired constructions are logically impossible.There was, moreover, a valuable by-product of theselabors. Since the solutions depend essentially upondetermining the kind of roots that satisfy certainequations, concern with the celebrated exercises set in

The Problem of Consistency 9

antiquity stimulated profound investigations into thenature of number and the structure of the numbercontinuum. Rigorous definitions were eventuallysupplied for negative, complex, and irrational numbers;a logical basis was constructed for the real numbersystem; and a new branch of mathematics, the theoryof infinite numbers, was founded.

But perhaps the most significant development in itslong-range effects upon subsequent mathematicalhistory was the solution of another problem that theGreeks raised without answering. One of the axiomsEuclid used in systematizing geometry has to do withparallels. The axiom he adopted is logically equivalentto (though not identical with) the assumption thatthrough a point outside a given line only one parallelto the line can be drawn. For various reasons, this axiomdid not appear “self-evident” to the ancients. Theysought, therefore, to deduce it from the other Euclideanaxioms, which they regarded as clearly self-evident.1

Can such a proof of the parallel axiom be given?Generations of mathematicians struggled with this

1 The chief reason for this alleged lack of self-evidence seems tohave been the fact that the parallel axiom makes an assertion aboutinfinitely remote regions of space. Euclid defines parallel lines asstraight lines in a plane that, “being produced indefinitely in bothdirections,” do not meet. Accordingly, to say that two lines areparallel is to make the claim that the two lines will not meet even“at infinity.” But the ancients were familiar with lines that, thoughthey do not intersect each other in any finite region of the plane,do meet “at infinity.” Such lines are said to be “asymptotic.” Thus,a hyperbola is asymptotic to its axes. It was therefore not

10 Gödel’s Proof

question, without avail. But repeated failure to constructa proof does not mean that none can be found any morethan repeated failure to find a cure for the commoncold establishes beyond doubt that man-kind willforever suffer from running noses. It was not until thenineteenth century, chiefly through the work of Gauss,Bolyai, Lobachevsky, and Riemann, that theimpossibility of deducing the parallel axiom from theothers was demonstrated. This outcome was of thegreatest intellectual importance. In the first place, itcalled attention in a most impressive way to the factthat a proof can be given of the impossibility of provingcertain propositions within a given system. As we shallsee, Gödel’s paper is a proof of the impossibility ofdemonstrating certain important propositions inarithmetic. In the second place, the resolution of theparallel axiom question forced the realization thatEuclid is not the last word on the subject of geometry,since new systems of geometry can be constructed byusing a number of axioms different from, andincompatible with, those adopted by Euclid. Inparticular, as is well known, immensely interesting andfruitful results are obtained when Euclid’s parallelaxiom is replaced by the assumption that more thanone parallel can be drawn to a given line through agiven point, or, alternatively, by the assumption that noparallels can be drawn. The traditional belief that the

intuitively evident to the ancient geometers that from a point outsidea given straight line only one straight line can be drawn that willnot meet the given line even at infinity.


axioms of geometry (or, for that matter, the axioms ofany discipline) can be established by their apparentself-evidence was thus radically undermined. Moreover,it gradually became clear that the proper business ofthe pure mathematician is to derive theorems frompostulated assumptions, and that it is not his concernas a mathematician to decide whether the axioms heassumes are actually true. And, finally, these successfulmodifications of orthodox geometry stimulated therevision and completion of the axiomatic bases for manyother mathematical systems. Axiomatic foundationswere eventually supplied for fields of inquiry that hadhitherto been cultivated only in a more or less intuitive manner. (See Appendix, no. 1.)

The over-all conclusion that emerged from thesecritical studies of the foundations of mathematics isthat the age-old conception of mathematics as “thescience of quantity” is both inadequate and misleading.For it became evident that mathematics is simply thediscipline par excellence that draws the conclusionslogically implied by any given set of axioms orpostulates. In fact, it came to be acknowledged that thevalidity of a mathematical inference in no sense dependsupon any special meaning that may be associated withthe terms or expressions contained in the postulates.Mathematics was thus recognized to be much moreabstract and formal than had been traditionallysupposed: more abstract, because mathematicalstatements can be construed in principle to be aboutanything whatsoever rather than about some inherently

12 Gödel’s Proof

circumscribed set of objects or traits of objects; and moreformal, because the validity of mathematicaldemonstrations is grounded in the structure ofstatements, rather than in the nature of a particularsubject matter. The postulates of any branch ofdemonstrative mathematics are not inherently aboutspace, quantity, apples, angles, or budgets; and anyspecial meaning that may be associated with the terms(or “descriptive predicates”) in the postulates plays noessential role in the process of deriving theorems. Werepeat that the sole question confronting the puremathematician (as distinct from the scientist whoemploys mathematics in investigating a special subjectmatter) is not whether the postulates he assumes or theconclusions he deduces from them are true, but whetherthe alleged conclusions are in fact the necessary logicalconsequences of the initial assumptions.

Take this example. Among the undefined (or“primitive”) terms employed by the influential Germanmathematician David Hilbert in his famousaxiomatization of geometry (first published in 1899)are ‘point’, ‘line’, ‘lies on’, and ‘between’. We may grantthat the customary meanings connected with theseexpressions play a role in the process of discoveringand learning theorems. Since the meanings are familiar,we feel we understand their various interrelations, andthey motivate the formulation and selection of axioms;moreover, they suggest and facilitate the formulationof the statements we hope to establish as theorems.Yet, as Hilbert plainly states, insofar as we are


concerned with the primary mathematical task ofexploring the purely logical relations of dependencebetween statements, the familiar connotations of theprimitive terms are to be ignored, and the sole“meanings” that are to be associated with them arethose assigned by the axioms into which they enter.2

This is the point of Russell’s famous epigram: puremathematics is the subject in which we do not knowwhat we are talking about, or whether what we aresaying is true.

A land of rigorous abstraction, empty of all familiarlandmarks, is certainly not easy to get around in. But itoffers compensations in the form of a new freedom ofmovement and fresh vistas. The intensifiedformalization of mathematics emancipated men’s mindsfrom the restrictions that the customary interpretationof expressions placed on the construction of novelsystems of postulates. New kinds of algebras andgeometries were developed which marked significantdepartures from the mathematics of tradition. As themeanings of certain terms became more general, theiruse became broader and the inferences that could bedrawn from them less confined. Formalization led to agreat variety of systems of considerable mathematicalinterest and value. Some of these systems, it must be

2 In more technical language, the primitive terms are “implicitly”defined by the axioms, and whatever is not covered by the implicitdefinitions is irrelevant to the demonstration of theorems.

14 Gödel’s Proof

admitted, did not lend themselves to interpretations asobviously intuitive (i.e., commonsensical) as those ofEuclidean geometry or arithmetic, but this fact causedno alarm. Intuition, for one thing, is an elastic faculty:our children will probably have no difficulty inaccepting as intuitively obvious the paradoxes ofrelativity, just as we do not boggle at ideas that wereregarded as wholly unintuitive a couple of generationsago. Moreover, as we all know, intuition is not a safeguide: it cannot properly be used as a criterion of eithertruth or fruitfulness in scientific explorations.

However, the increased abstractness of mathematicsraised a more serious problem. It turned on the questionwhether a given set of postulates serving as foundationof a system is internally consistent, so that no mutuallycontradictory theorems can be deduced from thepostulates. The problem does not seem pressing whena set of axioms is taken to be about a definite and familiardomain of objects; for then it is not only significant toask, but it may be possible to ascertain, whether theaxioms are indeed true of these objects. Since theEuclidean axioms were generally supposed to be truestatements about space (or objects in space), nomathematician prior to the nineteenth century everconsidered the question whether a pair of contradictorytheorems might some day be deduced from the axioms.The basis for this confidence in the consistency ofEuclidean geometry is the sound principle that logicallyincompatible statements cannot be simultaneously true;accordingly, if a set of statements is true (and this was


assumed of the Euclidean axioms), these statements aremutually consistent.

The non-Euclidean geometries were clearly in adifferent category. Their axioms were initially regardedas being plainly false of space, and, for that matter,doubtfully true of anything; thus the problem ofestablishing the internal consistency of non-Euclideansystems was recognized to be both formidable andcritical. In Riemannian geometry, for example, Euclid’sparallel postulate is replaced by the assumption thatthrough a given point outside a line no parallel to itcan be drawn. Now suppose the question: Is theRiemannian set of postulates consistent? Thepostulates are apparently not true of the space ofordinary experience. How, then, is their consistencyto be shown? How can one prove they will not lead tocontradictory theorems? Obviously the question is notsettled by the fact that the theorems already deduceddo not contradict each other—for the possibilityremains that the very next theorem to be deduced mayupset the apple cart. But, until the question is settled,one cannot be certain that Riemannian geometry is atrue alternative to the Euclidean system, i.e., equallyvalid mathematically. The very possibility of non-Euclidean geometries was thus contingent on theresolution of this problem.

A general method for solving it was devised. Theunderlying idea is to find a “model” (or “interpretation”)for the abstract postulates of a system, so that eachpostulate is converted into a true statement about the

16 Gödel’s Proof

model. In the case of Euclidean geometry, as we havenoted, the model was ordinary space. The method wasused to find other models, the elements of which couldserve as crutches for determining the consistency ofabstract postulates. The procedure goes something likethis. Let us understand by the word ‘class’ a collectionor aggregate of distinguishable elements, each of whichis called a member of the class. Thus, the class of primenumbers less than 10 is the collection whose membersare 2, 3, 5, and 7. Suppose the following set of postulatesconcerning two classes K and L, whose special natureis left undetermined except as “implicitly” defined bythe postulates:

1. Any two members of K are contained injust one member of L.

2. No member of K is contained in morethan two members of L.

3. The members of K are not all containedin a single member of L.

4. Any two members of L contain just onemember of K.

5. No member of L contains more than twomembers of K.

From this small set we can derive, by using customaryrules of inference, a number of theorems. For example,it can be shown that K contains just three members.But is the set consistent, so that mutually contradictorytheorems can never be derived from it? The question


can be answered readily with the help of the followingmodel:

Let K be the class of points consisting of thevertices of a triangle, and L the class of linesmade up of its sides; and let us understandthe phrase ‘a member of K is contained in amember of L’ to mean that a point which is avertex lies on a line which is a side. Each ofthe five abstract postulates is then convertedinto a true statement. For instance, the firstpostulate asserts that any two points whichare vertices of the triangle lie on just one linewhich is a side. (See Fig. 1.) In this way theset of postulates is proved to be consistent.

Fig. 1.Model for a set of postulates about two classes, K and L, is a trianglewhose vertices are the members of K and whose sides are themembers of L. The geometrical model shows that the postulates areconsistent.

The consistency of plane Riemannian geometry can

also, ostensibly, be established by a model embodying

18 Gödel’s Proof

the postulates. We may interpret the expression ‘plane’in the Riemannian axioms to signify the surface of aEuclidean sphere, the expression ‘point’ a point on thissurface, the expression ‘straight line’ an arc of a greatcircle on this surface, and so on. Each Riemannianpostulate is then converted into a theorem of Euclid.For example, on this interpretation the Riemannianparallel postulate reads: Through a point on the surfaceof a sphere, no arc of a great circle can be drawn parallelto a given arc of a great circle. (See Fig. 2.)

At first glance this proof of the consistency ofRiemannian geometry may seem conclusive. But a closerlook is disconcerting. For a sharp eye will discern thatthe problem has not been solved; it has merely beenshifted to another domain. The proof attempts to settlethe consistency of Riemannian geometry by appealingto the consistency of Euclidean geometry. What emerges,then, is only this: Riemannian geometry is consistent ifEuclidean geometry is consistent. The authority ofEuclid is thus invoked to demonstrate the consistencyof a system which challenges the exclusive validity ofEuclid. The inescapable question is: Are the axioms ofthe Euclidean system itself consistent?

An answer to this question, hallowed, as we havenoted, by a long tradition, is that the Euclidean axiomsare true and are therefore consistent. This answer isFig. 2The non-Euclidean geometry of Bernhard Riemann can berepresented by a Euclidean model. The Riemannian plane becomesthe surface of a Euclidean sphere, points on the plane become pointson this surface, straight lines in the plane become great circles. Thus,

a portion of the Riemannian plane bounded by segments of straightlines is depicted as a portion of the sphere bounded by parts of greatcircles (center). Two line segments in the Riemannian plane are twosegments of great circles on the Euclidean sphere (bottom), and these,if extended, indeed intersect, thus contradicting the parallel postulate.

20 Gödel’s Proof

no longer regarded as acceptable; we shall return to itpresently and explain why it is unsatisfactory. Anotheranswer is that the axioms jibe with our actual, thoughlimited, experience of space and that we are justified inextrapolating from the small to the universal. But,although much inductive evidence can be adduced tosupport this claim, our best proof would be logicallyincomplete. For even if all the observed facts are inagreement with the axioms, the possibility is open thata hitherto unobserved fact may contradict them and sodestroy their title to universality. Inductiveconsiderations can show no more than that the axiomsare plausible or probably true.

Hilbert tried yet another route to the top. The clue tohis way lay in Cartesian coordinate geometry. In hisinterpretation Euclid’s axioms were simply transformedinto algebraic truths. For instance, in the axioms forplane geometry, construe the expression ‘point’ tosignify a pair of numbers, the expression ‘straight line’the (linear) relation between numbers expressed by afirst degree equation with two unknowns, the expression‘circle’ the relation between numbers expressed by aquadratic equation of a certain form, and so on. Thegeometric statement that two distinct points uniquelydetermine a straight line is then transformed into thealgebraic truth that two distinct pairs of numbersuniquely determine a linear relation; the geometrictheorem that a straight line intersects a circle in at mosttwo points, into the algebraic theorem that a pair ofsimultaneous equations in two unknowns (one of which


is linear and the other quadratic of a certain type)determine at most two pairs of real numbers; and so on.In brief, the consistency of the Euclidean postulates isestablished by showing that they are satisfied by analgebraic model. This method of establishingconsistency is powerful and effective. Yet it, too, isvulnerable to the objection already set forth. For, again,a problem in one domain is resolved by transferring itto another. Hilbert’s argument for the consistency ofhis geometric postulates shows that if algebra isconsistent, so is his geometric system. The proof isclearly relative to the assumed consistency of anothersystem and is not an “absolute” proof.

In the various attempts to solve the problem ofconsistency there is one persistent source of difficulty.It lies in the fact that the axioms are interpreted bymodels composed of an infinite number of elements.This makes it impossible to encompass the models in afinite number of observations; hence the truth of theaxioms themselves is subject to doubt. In the inductiveargument for the truth of Euclidean geometry, a finitenumber of observed facts about space are presumablyin agreement with the axioms. But the conclusion thatthe argument seeks to establish involves anextrapolation from a finite to an infinite set of data.How can we justify this jump? On the other hand, thedifficulty is minimized, if not completely eliminated,where an appropriate model can be devised thatcontains only a finite number of elements. The trianglemodel used to show the consistency of the five abstract

22 Gödel’s Proof

postulates for the classes K and L is finite; and it iscomparatively simple to determine by actual inspectionwhether all the elements in the model actually satisfythe postulates, and thus whether they are true (andhence consistent). To illustrate: by examining in turnall the vertices of the model triangle, one can learnwhether any two of them lie on just one side—so thatthe first postulate is established as true. Since all theelements of the model, as well as the relevant relationsamong them, are open to direct and exhaustiveinspection, and since the likelihood of mistakesoccurring in inspecting them is practically nil, theconsistency of the postulates in this case is not a matterfor genuine doubt.

Unfortunately, most of the postulate systems thatconstitute the foundations of important branches ofmathematics cannot be mirrored in finite models.Consider the postulate in elementary arithmetic whichasserts that every integer has an immediate successordiffering from any preceding integer. It is evident thatthe model needed to test the set to which this postulatebelongs cannot be finite, but must contain an infinityof elements. It follows that the truth (and so theconsistency) of the set cannot be established by anexhaustive inspection of a limited number of elements.Apparently we have reached an impasse. Finite modelssuffice, in principle, to establish the consistency ofcertain sets of postulates; but these are of slightmathematical importance. Non-finite models, necessaryfor the interpretation of most postulate systems of


mathematical significance, can be described only ingeneral terms; and we cannot conclude as a matter ofcourse that the descriptions are free from concealedcontradictions.

It is tempting to suggest at this point that we can besure of the consistency of formulations in which non-finite models are described if the basic notionsemployed are transparently “clear” and “distinct.” Butthe history of thought has not dealt kindly with thedoctrine of clear and distinct ideas, or with the doctrineof intuitive knowledge implicit in the suggestion. Incertain areas of mathematical research in whichassumptions about infinite collections play central roles,radical contradictions have turned up, in spite of theintuitive clarity of the notions involved in theassumptions and despite the seemingly consistentcharacter of the intellectual constructions performed.Such contradictions (technically referred to as“antinomies”) have emerged in the theory of infinitenumbers, developed by Georg Cantor in the nineteenthcentury; and the occurrence of these contradictions hasmade plain that the apparent clarity of even such anelementary notion as that of class (or aggregate) doesnot guarantee the consistency of any particular systembuilt on it. Since the mathematical theory of classes,which deals with the properties and relations ofaggregates or collections of elements, is often adoptedas the foundation for other branches of mathematics,and in particular for elementary arithmetic, it ispertinent to ask whether contradictions similar to those

24 Gödel’s Proof

encountered in the theory of infinite classes infect theformulations of other parts of mathematics.

In point of fact, Bertrand Russell constructed acontradiction within the framework of elementary logicitself that is precisely analogous to the contradictionfirst developed in the Cantorian theory of infiniteclasses. Russell’s antinomy can be stated as follows.Classes seem to be of two kinds: those which do notcontain themselves as members, and those which do. Aclass will be called “normal” if, and only if, it does notcontain itself as a member; otherwise it will be called“non-normal.” An example of a normal class is the classof mathematicians, for patently the class itself is not amathematician and is therefore not a member of itself.An example of a non-normal class is the class of allthinkable things; for the class of all thinkable things isitself thinkable and is therefore a member of itself. Let‘N’ by definition stand for the class of all normal classes.We ask whether N itself is a normal class. If N is normal,it is a member of itself (for by definition N contains allnormal classes); but, in that case, N is non-normal,because by definition a class that contains itself as amember is non-normal. On the other hand, if N is non-normal, it is a member of itself (by definition of non-normal); but, in that case, N is normal, because bydefinition the members of N are normal classes. In short,N is normal if, and only if, N is non-normal. It followsthat the statement ‘N is normal’ is both true and false.This fatal contradiction results from an uncritical useof the apparently pellucid notion of class. Other


paradoxes were found later, each of them constructedby means of familiar and seemingly cogent modes ofreasoning. Mathematicians came to realize that indeveloping consistent systems familiarity and intuitiveclarity are weak reeds to lean on.

We have seen the importance of the problem ofconsistency, and we have acquainted ourselves withthe classically standard method for solving it with thehelp of models. It has been shown that in most instancesthe problem requires the use of a non-finite model, thedescription of which may itself conceal inconsistencies.We must conclude that, while the model method is aninvaluable mathematical tool, it does not supply a finalanswer to the problem it was designed to solve.

26

IIIAbsolute Proofs of Consistency

The limitations inherent in the use of models forestablishing consistency, and the growing apprehensionthat the standard formulations of many mathematicalsystems might all harbor internal contradictions, led tonew attacks upon the problem. An alternative to relativeproofs of consistency was proposed by Hilbert. Hesought to construct “absolute” proofs, by which theconsistency of systems could be established withoutassuming the consistency of some other system. Wemust briefly explain this approach as a furtherpreparation for understanding Gödel’s achievement.

The first step in the construction of an absolute proof,as Hilbert conceived the matter, is the completeformalization of a deductive system. This involvesdraining the expressions occurring within the systemof all meaning: they are to be regarded simply as emptysigns. How these signs are to be combined andmanipulated is to be set forth in a set of precisely statedrules. The purpose of this procedure is to construct a

Absolute Proofs of Consistency 27

system of signs (called a “calculus”) which concealsnothing and which has in it only that which we explicitlyput into it. The postulates and theorems of a completelyformalized system are “strings” (or finitely longsequences) of meaningless marks, constructed accordingto rules for combining the elementary signs of the systeminto larger wholes. Moreover, when a system has beencompletely formalized, the derivation of theorems frompostulates is nothing more than the transformation(pursuant to rule) of one set of such “strings” into anotherset of “strings.” In this way the danger is eliminated ofusing any unavowed principles of reasoning.Formalization is a difficult and tricky business, but itserves a valuable purpose. It reveals structure andfunction in naked clarity, as does a cut-away workingmodel of a machine. When a system has been formalized,the logical relations between mathematical propositionsare exposed to view; one is able to see the structuralpatterns of various “strings” of “meaningless” signs, howthey hang together, how they are combined, how theynest in one another, and so on.

A page covered with the “meaningless” marks of sucha formalized mathematics does not assert anything—itis simply an abstract design or a mosaic possessing adeterminate structure. Yet it is clearly possible todescribe the configurations of such a system and to makestatements about the configurations and their variousrelations to one another. One may say that a “string” ispretty, or that it resembles another “string,” or that one“string” appears to be made up of three others, and so

28 Gödel’s Proof

on. Such statements are evidently meaningful and mayconvey important information about the formal system.It must now be observed, however, that such meaningfulstatements about a meaningless (or formalized)mathematical system plainly do not themselves belongto that system. They belong to what Hilbert called “meta-mathematics,” to the language that is about mathematics.Meta-mathematical statements are statements about thesigns occurring within a formalized mathematical system(i.e., a calculus)—about the kinds and arrangements ofsuch signs when they are combined to form longer stringsof marks called “formulas,” or about the relations betweenformulas that may obtain as a consequence of the rulesof manipulation specified for them.

A few examples will help to an understanding ofHilbert’s distinction between mathematics (i.e., a systemof meaningless signs) and meta-mathematics(meaningful statements about mathematics, the signsoccurring in the calculus, their arrangement andrelations). Consider the expression:

2+3=5

This expression belongs to mathematics (arithmetic) andis constructed entirely out of elementary arithmeticalsigns. On the other hand, the statement

‘2+3=5’ is an arithmetical formula

asserts something about the displayed expression. Thestatement does not express an arithmetical fact and doesnot belong to the formal language of arithmetic; it


belongs to meta-mathematics, because it characterizesa certain string of arithmetical signs as being a formula.The following statement belongs to metamathematics:

If the sign ‘=’ is to be used in a formula ofarithmetic, the sign must be flanked both leftand right by numerical expressions.

This statement lays down a necessary condition forusing a certain arithmetical sign in arithmeticalformulas: the structure that an arithmetical formula musthave if it is to embody that sign.

Consider next the three formulas:

x=x0=00�0

Each of these belongs to mathematics (arithmetic),because each is built up entirely out of arithmeticalsigns. But the statement:

‘x’ is a variable

belongs to meta-mathematics, since it characterizes acertain arithmetical sign as belonging to a specific classof signs (i.e., to the class of variables). Again, thefollowing statement belongs to meta-mathematics:

The formula ‘0=0’ is derivable from theformula ‘x=x’ by substituting the numeral ‘0’for the variable ‘x’.

It specifies in what manner one arithmetical formula

30 Gödel’s Proof

can be obtained from another formula, and therebydescribes how the two formulas are related to each other.Similarly, the statement

‘0�0’ is not a theorem belongs to meta-mathematics, for it says of a certainformula that it is not derivable from the axioms ofarithmetic, and thus asserts that a certain relation doesnot hold between the indicated formulas of the system.Finally, the next statement belongs to meta-mathematics:

Arithmetic is consistent (i.e., it is not possible to derive from the axioms ofarithmetic two formally contradictory formulas—forexample, the formulas ‘0=0’ and ‘0�0’). This is patentlyabout arithmetic, and asserts that pairs of formulas of acertain sort do not stand in a specific relation to theformulas that constitute the axioms of arithmetic.3

3 It is worth noting that the meta-mathematical statements givenin the text do not contain as constituent parts of themselves any ofthe mathematical signs and formulas that appear in the examples.At first glance this assertion seems palpably untrue, for the signsand formulas are plainly visible. But, if the statements are examinedwith an analytic eye, it will be seen that the point is well taken. Themeta-mathematical statements contain the names of certainarithmetical expressions, but not the arithmetical expressionsthemselves. The distinction is subtle but both valid and important.It arises out of the circumstance that the rules of English grammarrequire that no sentence literally contain the objects to which the


It may be that the reader finds the word‘metamathematics’ ponderous and the conceptpuzzling. We shall not argue that the word is pretty;but the concept itself will perplex no one if we pointout that it is used in connection with a special case ofa well-known distinction, namely between a subjectmatter under study and discourse about the subjectmatter. The statement ‘among phalaropes the malesincubate the eggs’ pertains to the subject matterinvestigated by zoologists, and belongs to zoology; butif we say that this assertion about phalaropes provesthat zoology is irrational, our statement is not aboutphalaropes, but about the assertion and the discipline

expressions in the sentence may refer, but only the names of suchobjects. Obviously, when we talk about a city we do not put the cityitself into a sentence, but only the name of the city; and, similarly,if we wish to say something about a word (or other linguistic sign),it is not the word itself (or the sign) that can appear in the sentence,but only a name for the word (or sign). According to a standardconvention we construct a name for a linguistic expression byplacing single quotation marks around it. Our text adheres to thisconvention. It is correct to write:

Chicago is a populous city.But it is incorrect to write:

Chicago is tri-syllabic.To express what is intended by this latter sentence, one must write:

‘Chicago’ is tri-syllabic.Likewise, it is incorrect to write:

x=5 is an equation.We must, instead, formulate our intent by:

‘x=5’ is an equation.

32 Gödel’s Proof

in which it occurs, and is meta-zoology. If we say thatthe id is mightier than the ego, we are making noisesthat belong to orthodox psychoanalysis; but if wecriticize this statement as meaningless and unprovable,our criticism belongs to meta-psychoanalysis. And soin the case of mathematics and meta-mathematics. Theformal systems that mathematicians construct belongin the file labeled “mathematics”; the description,discussion, and theorizing about the systems belong inthe file marked “meta-mathematics.”

The importance to our subject of recognizing thedistinction between mathematics and meta-mathematics cannot be overemphasized. Failure torespect it has produced paradoxes and confusion.Recognition of its significance has made it possible toexhibit in a clear light the logical structure ofmathematical reasoning. The merit of the distinctionis that it entails a careful codification of the varioussigns that go into the making of a formal calculus, freeof concealed assumptions and irrelevant associationsof meaning. Furthermore, it requires exact definitionsof the operations and logical rules of mathematicalconstruction and deduction, many of whichmathematicians had applied without being explicitlyaware of what they were using.

Hilbert saw to the heart of the matter, and it wasupon the distinction between a formal calculus and itsdescription that he based his attempt to build “absolute”proofs of consistency. Specifically, he sought to developa method that would yield demonstrations of


consistency as much beyond genuine logical doubt asthe use of finite models for establishing the consistencyof certain sets of postulates—by an analysis of a finitenumber of structural features of expressions incompletely formalized calculi. The analysis consists innoting the various types of signs that occur in a calculus,indicating how to combine them into formulas,prescribing how formulas can be obtained from otherformulas, and determining whether formulas of a givenkind are derivable from others through explicitly statedrules of operation. Hilbert believed it might be possibleto exhibit every mathematical calculus as a sort of“geometrical” pattern of formulas, in which the formulasstand to each other in a finite number of structuralrelations. He therefore hoped to show, by exhaustivelyexamining these structural properties of expressionswithin a system, that formally contradictory formulascannot be obtained from the axioms of given calculi.An essential requirement of Hilbert’s program in itsoriginal conception was that demonstrations ofconsistency involve only such procedures as make noreference either to an infinite number of structuralproperties of formulas or to an infinite number ofoperations with formulas. Such procedures are called“finitistic”; and a proof of consistency conforming tothis requirement is called “absolute.” An “absolute”proof achieves its objectives by using a minimum ofprinciples of inference, and does not assume theconsistency of some other set of axioms. An absoluteproof of the consistency of arithmetic, if one could be

34 Gödel’s Proof

constructed, would therefore show by a finitistic meta-mathematical procedure that two contradictoryformulas, such as ‘0=0’ and its formal negation‘~(0=0)’—where the sign ‘~’ means ‘not’—cannot bothbe derived by stated rules of inference from the axioms(or initial formulas).4

It may be useful, by way of illustration, to comparemeta-mathematics as a theory of proof with the theoryof chess. Chess is played with 32 pieces of specifieddesign on a square board containing 64 square sub-divisions, where the pieces may be moved inaccordance with fixed rules. The game can obviouslybe played without assigning any “interpretation” to thepieces or to their various positions on the board,although such an interpretation could be supplied ifdesired. For example, we could stipulate that a givenpawn is to represent a certain regiment in an army, thata given square is to stand for a certain geographicalregion, and so on. But such stipulations (orinterpretations) are not customary; and neither thepieces, nor the squares, nor the positions of the pieceson the board signify anything outside the game. In thissense, the pieces and their configurations on the board

4 Hilbert did not give an altogether precise account of justwhat meta-mathematical procedures are to count as finitistic. Inthe original version of his program the requirements for anabsolute proof of consistency were more stringent than in thesubsequent explanations of the program by members of his school.


are “meaningless.” Thus the game is analogous to aformalized mathematical calculus. The pieces and thesquares of the board correspond to the elementarysigns of the calculus; the legal positions of pieces onthe board, to the formulas of the calculus; the initialpositions of pieces on the board, to the axioms orinitial formulas of the calculus; the subsequentpositions of pieces on the board, to formulas derivedfrom the axioms (i.e., to the theorems); and the rules ofthe game, to the rules of inference (or derivation) forthe calculus. The parallelism continues. Althoughconfigurations of pieces on the board, like the formulasof the calculus, are “meaningless,” statements aboutthese configurations, like meta-mathematicalstatements about formulas, are quite meaningful. A“meta-chess” statement may assert that there aretwenty possible opening moves for White, or that,given a certain configuration of pieces on the boardwith White to move, Black is mate in three moves.Moreover, general “meta-chess” theorems can beestablished whose proof involves only a finite numberof permissible configurations on the board. The “meta-chess” theorem about the number of possible openingmoves for White can be established in this way; and socan the “meta-chess” theorem that if White has onlytwo Knights and the King, and Black only his King, itis impossible for White to force a mate against Black.These and other “meta-chess” theorems can, in otherwords, be proved by finitistic methods of reasoning,that is, by examining in turn each of a finite number of

36 Gödel’s Proof

configurations that can occur under stated conditions.The aim of Hilbert’s theory of proof, similarly, was todemonstrate by such finitistic methods theimpossibility of deriving certain contradictoryformulas in a given mathematical calculus.

37

IVThe Systematic Codification of Formal Logic

There are two more bridges to cross before enteringupon Gödel’s proof itself. We must indicate how andwhy Whitehead and Russell’s Principia Mathematicacame into being; and we must give a short illustrationof the formalization of a deductive system—we shalltake a fragment of Principia—and explain how itsabsolute consistency can be established.

Ordinarily, even when mathematical proofs conformto accepted standards of professional rigor, they sufferfrom an important omission. They embody principles(or rules) of inference not explicitly formulated, of whichmathematicians are frequently unaware. Take Euclid’sproof that there is no greatest prime number (a numberis prime if it is divisible without remainder by no numberother than 1 and the number itself). The argument, castin the form of a reductio ad absurdum, runs as follows:

Suppose, in contradiction to what the proof seeks toestablish, that there is a greatest prime number. Wedesignate it by ‘x’. Then:

38 Gödel’s Proof

1. x is the greatest prime2. Form the product of all primes less than

or equal to x, and add 1 to the product. Thisyields a new number y, where y=

(2×3×5×7×…×x)+1

3. If y is itself a prime, then x is notthe greatest prime, for y is obviously greaterthan x

4. If y is composite (i.e., not a prime), thenagain x is not the greatest prime. For if y iscomposite, it must have a prime divisor z; andz must be different from each of the primenumbers 2, 3, 5, 7,…, x, smaller than or equalto x; hence z must be a prime greater than x

5. But y is either prime or composite6. Hence x is not the greatest prime7. There is no greatest prime

We have stated only the main links of the proof. It canbe shown, however, that in forging the complete chaina fairly large number of tacitly accepted rules ofinference, as well as theorems of logic, are essential.Some of these belong to the most elementary part offormal logic, others to more advanced branches; forexample, rules and theorems are incorporated thatbelong to the “theory of quantification.” This deals withrelations between statements containing such“quantifying” particles as ‘all’, ‘some’, and theirsynonyms. We shall exhibit one elementary theorem of

The Systematic Codification of Formal Logic 39

logic and one rule of inference, each of which is anecessary but silent partner in the demonstration.

Look at line 5 of the proof. Where does it come from?The answer is, from the logical theorem (or necessarytruth): ‘Either p or non-p’, where ‘p’ is called a sententialvariable. But how do we get line 5 from this theorem?The answer is, by using the rule of inference known asthe “Rule of Substitution for Sentential Variables,”according to which a statement can be derived fromanother containing such variables by substituting anystatement (in this case, ‘y is prime’) for each occurrenceof a distinct variable (in this case, the variable ‘p’). Theuse of these rules and logical theorems is, as we havesaid, frequently an all but unconscious action. And theanalysis that exposes them, even in such relativelysimple proofs as Euclid’s, depends upon advances inlogical theory made only within the past one hundredyears.5 Like Molière’s M.Jourdain, who spoke prose allhis life without knowing it, mathematicians have beenreasoning for at least two millennia without being awareof all the principles underlying what they were doing.The real nature of the tools of their craft has becomeevident only within recent times.

For almost two thousand years Aristotle’s codificationof valid forms of deduction was widely regarded ascomplete and as incapable of essential improvement.

5 For a more detailed discussion of the rules of inference andlogical theorems needed for obtaining lines 6 and 7 of the aboveproof, the reader is referred to the Appendix, no. 2.

40 Gödel’s Proof

As late as 1787, the German philosopher Immanuel Kantwas able to say that since Aristotle formal logic “hasnot been able to advance a single step, and is to allappearances a closed and completed body of doctrine.”The fact is that the traditional logic is seriouslyincomplete, and even fails to give an account of manyprinciples of inference employed in quite elementarymathematical reasoning.6 A renaissance of logicalstudies in modern times began with the publication in1847 of George Boole’s The Mathematical Analysis ofLogic. The primary concern of Boole and his immediatesuccessors was to develop an algebra of logic whichwould provide a precise notation for handling moregeneral and more varied types of deduction than werecovered by traditional logical principles. Suppose it isfound that in a certain school those who graduate withhonors are made up exactly of boys majoring inmathematics and girls not majoring in this subject. Howis the class of mathematics majors made up, in terms ofthe other classes of students mentioned? The answer isnot readily forthcoming if one uses only the apparatusof traditional logic. But with the help of Boolean algebrait can easily be shown that the class of mathematicsmajors consists exactly of boys graduating with honorsand girls not graduating with honors.

6 For example, of the principles involved in the inference: 5 isgreater than 3; therefore, the square of 5 is greater than the squareof 3.


TABLE 1

Another line of inquiry, closely related to the work ofnineteenth-century mathematicians on the founda tions

All gentlemen are polite.No bankers are polite.No gentlemen are bankers.

Symbolic logic was invented in the middle of the19th century by the English mathematician GeorgeBoole. In this illustration a syllogism is translatedinto his notation in two different ways. In theupper group of formulas, the symbol ‘�’ means“is contained in.” Thus ‘g�p’ says that the classof gentlemen is included in the class of politepersons. In the lower group of formulas two letterstogether mean the class of things having bothcharacteristics. For example, ‘bp’ means the classof individuals who are bankers and polite; andthe equation ‘bp=0’ says that this class has nomembers. A line above a letter means “not.” (‘p̄’,for example, means impolite.)

42 Gödel’s Proof

of analysis, became associated with the Booleanprogram. This new development sought to exhibit puremathematics as a chapter of formal logic; and it receivedits classical embodiment in the Principia Mathematicaof Whitehead and Russell in 1910. Mathematicians ofthe nineteenth century succeeded in “arithmetizing”algebra and what used to be called the “infinitesimalcalculus” by showing that the various notions employedin mathematical analysis are definable exclusively inarithmetical terms (i.e., in terms of the integers and thearithmetical operations upon them). For example,instead of accepting the imaginary numberas a somewhat mysterious “entity,” it came to be definedas an ordered pair of integers (0, 1) upon which certainoperations of “addition” and “multiplication” areperformed. Similarly, the irrational number wasdefined as a certain class of rational numbers—namely,the class of rationals whose square is less than 2. WhatRussell (and, before him, the German mathematicianGottlob Frege) sought to show was that all arithmeticalnotions can be defined in purely logical ideas, and thatall the axioms of arithmetic can be deduced from a smallnumber of basic propositions certifiable as purely logicaltruths.

To illustrate: the notion of class belongs to generallogic. Two classes are defined as “similar” if there is aone-to-one correspondence between their members, thenotion of such a correspondence being explicable interms of other logical ideas. A class that has a singlemember is said to be a “unit class” (e.g., the class of


satellites of the planet Earth); and the cardinal number1 can be defined as the class of all classes similar to aunit class. Analogous definitions can be given of theother cardinal numbers; and the various arithmeticaloperations, such as addition and multiplication, can bedefined in the notions of formal logic. An arithmeticalstatement, e.g., ‘1+1=2’, can then be exhibited as acondensed transcription of a statement containing onlyexpressions belonging to general logic; and such purelylogical statements can be shown to be deducible fromcertain logical axioms.

Principia Mathematica thus appeared to advance thefinal solution of the problem of consistency ofmathematical systems, and of arithmetic in particular,by reducing the problem to that of the consistency offormal logic itself. For, if the axioms of arithmetic aresimply transcriptions of theorems in logic, the questionwhether the axioms are consistent is equivalent to thequestion whether the fundamental axioms of logic areconsistent.

The Frege-Russell thesis that mathematics is only achapter of logic has, for various reasons of detail, notwon universal acceptance from mathematicians.Moreover, as we have noted, the antinomies of theCantorian theory of transfinite numbers can beduplicated within logic itself, unless special precautionsare taken to prevent this outcome. But are the measuresadopted in Principia Mathematica to outflank theantinomies adequate to exclude all forms of self-contradictory constructions? This cannot be asserted

44 Gödel’s Proof

as a matter of course. Therefore the Frege-Russellreduction of arithmetic to logic does not provide a finalanswer to the consistency problem; indeed, the problemsimply emerges in a more general form. But, irrespectiveof the validity of the Frege-Russell thesis, two featuresof Principia have proved of inestimable value for thefurther study of the consistency question. Principiaprovides a remarkably comprehensive system ofnotation, with the help of which all statements of puremathematics (and of arithmetic in particular) can becodified in a standard manner; and it makes explicitmost of the rules of formal inference used inmathematical demonstrations (eventually, these ruleswere made more precise and complete). Principia, insum, created the essential instrument for investigatingthe entire system of arithmetic as an uninterpretedcalculus—that is, as a system of meaningless marks,whose formulas (or “strings”) are combined andtransformed in accordance with stated rules ofoperation.

45

VAn Example of a Successful Absolute Proof ofConsistency

We must now attempt the second task mentioned at theoutset of the preceding section, and familiarize ourselveswith an important, though easily understandable,example of an absolute proof of consistency. Bymastering the proof, the reader will be in a betterposition to appreciate the significance of Gödel’s paperof 1931.

We shall outline how a small portion of Principia,the elementary logic of propositions, can be formalized.This entails the conversion of the fragmentary systeminto a calculus of uninterpreted signs. We shall thendevelop an absolute proof of consistency.

The formalization proceeds in four steps. First, acomplete catalogue is prepared of the signs to be usedin the calculus. These are its vocabulary. Second, the“Formation Rules” are laid down. They declare whichof the combinations of the signs in the vocabulary areacceptable as “formulas” (in effect, as sentences). The

46 Gödel’s Proof

rules may be viewed as constituting the grammar of thesystem. Third, the “Transformation Rules” are stated.They describe the precise structure of formulas fromwhich other formulas of given structure are derivable.These rules are, in effect, the rules of inference. Finally,certain formulas are selected as axioms (or as “primitiveformulas”). They serve as foundation for the entiresystem. We shall use the phrase “theorem of the system”to denote any formula that can be derived from theaxioms by successively applying the TransformationRules. By a formal “proof” (or “demonstration”) we shallmean a finite sequence of formulas, each of which eitheris an axiom or can be derived from preceding formulasin the sequence by the Transformation Rules.7

For the logic of propositions (often called thesentential calculus) the vocabulary (or list of“elementary signs”) is extremely simple. It consists ofvariables and constant signs. The variables may havesentences substituted for them and are therefore called“sentential variables.” They are the letters

‘p’, ‘q’, ‘r’, etc. The constant signs are either “sentential connectives”or signs of punctuation. The sentential connectives are:

‘~’ which is short for ‘not’ (and is called the “tilde”),

7 It immediately follows that axioms are to be counted amongthe theorems.

An Example of a Successful Absolute Proof of Consistency 47

‘V’ which is short for ‘or’,‘�’ which is short for ‘if…then…’, and‘.’ which is short for ‘and’.

The signs of punctuation are the left-and right-handround parentheses, ‘(’ and ‘)’, respectively.

The Formation Rules are so designed thatcombinations of the elementary signs, which wouldnormally have the form of sentences, are calledformulas. Also, each sentential variable counts as aformula. Moreover, if the letter ‘S’ stands for a formula,its formal negation, namely, ~(S), is also a formula.Similarly, if S1 and S2 are formulas, so are (S1)V(S2),(S1)�(S2), and (S1).(S2). Each of the following is aformula: ‘p’, ‘~(p)’, ‘(p)�(q)’, ‘((q)V(r))�(P)’. But neither‘(p)(~(q))’ nor ‘((p)�(q))V’ is a formula: not the first,because, while ‘(p)’ and ‘(~(q))’ are both formulas, nosentential connective occurs between them; and not thesecond, because the connective ‘V’ is not, as the Rulesrequire, flanked on both left and right by a formula.8

Two Transformation Rules are adopted. One of them,the Rule of Substitution (for sentential variables), saysthat from a formula containing sentential variables it isalways permissible to derive another formula byuniformly substituting formulas for the variables. It is

8 Where there is no possibility of confusion, punctuation marks(i.e., parentheses) can be dropped. Thus, instead of writing ‘~(p)’ itis sufficient to write ‘~p’; and instead of ‘(p)�(q)’, simply ‘p�q’.

48 Gödel’s Proof

understood that, when substitutions are made for avariable in a formula, the same substitution must bemade for each occurrence of the variable. For example,on the assumption that ‘p�p’ has already beenestablished, we can substitute for the variable ‘p’ theformula ‘q’ to get ‘q�q’; or we can substitute the formula‘pVq’ to get ‘(pVq)�(pVq)’. Or, if we substitute actualEnglish sentences for ‘p’, we can obtain each of thefollowing from ‘p�p’: ‘Frogs are noisy � Frogs arenoisy’; ‘(Bats are blind V Bats eat mice) � (Bats areblind V Bats eat mice)’.9 The second TransformationRule is the Rule of Detachment (or Modus Ponens). Thisrule says that from two formulas having the form S1

and S1�S2 it is always permissible to derive the formulaS2. For example, from the two formulas ‘pV~p’ and‘(pV~p)�(p�p)’, we can derive ‘pVp’.

Finally, the axioms of the calculus (essentially thoseof Principia) are the following four formulas:

1. If (either Henry VIII wasa boor or Henry VIII wasa boor) then Henry VIIIwas a boor

1. (pVp).por, in ordinary Eng-lish, if either p orp, then p

9 On the other hand, suppose the formula ‘(p�q)�(~q�~p)’ hasalready been established, and we decide to substitute ‘r’ for thevariable ‘p’ and ‘pVr’ for the variable ‘q’. We cannot, by thissubstitution, obtain the formula ‘(r�(p�r))�(~q�~r)’, because wehave failed to make the same substitution for each occurrence ofthe variable ‘q’.


2. p�(pVq) 2. If psychoanalysis is fash-that is, if p, then ionable, then (either psy-either p or q choanalys is is fashionable

or headache powders aresold cheap)

3. (pVq)�(qVp) 3. If (either Immanuel Kantthat is, if either p or was punctual or Holly-q, then either q or p wood is sinful), then

(either Hollywood is sin-ful or Immanuel Kant waspunctual)

4. (p�q)�((rVp) 4. If (if ducks waddle then�(rVq)) 5 is a prime) then (if

that is, if (if p then (either Churchill drinksq) then (if (either r brandy or ducks waddle)or p) then (either r then (either Churchillor q)) drinks brandy or 5 is a

prime))

In the left-hand column we have stated the axioms, witha translation for each. In the right-hand column we havegiven an example for each axiom. The clumsiness ofthe translations, especially in the case of the final axiom,will perhaps help the reader to realize the advantagesof using a special symbolism in formal logic. It is alsoimportant to observe that the nonsensical illustrationsused as substitution instances for the axioms and thefact that the consequents bear no meaningful relationto the antecedents in the conditional sentences in no

The correct substitution yields ‘(r�(pVr))�(~(pVr)�~r)’.

50 Gödel’s Proof

way affect the validity of the logical connectionsasserted in the examples.

Each of these axioms may seem “obvious” and trivial.Nevertheless, it is possible to derive from them withthe help of the stated Transformation Rules anindefinitely large class of theorems which are far fromobvious or trivial. For example, the formula

‘((p�q)�((r�s)�t))�((u�((r�s)�t))�((p�u)�(s�t)))’

can be derived as a theorem. We are, however, notinterested for the moment in deriving theorems fromthe axioms. Our aim is to show that this set of axiomsis not contradictory, that is, to prove “absolutely” thatit is impossible by using the Transformation Rules toderive from the axioms a formula S together with itsformal negation ~ S.

Now, it happens that ‘p�(~p�q)’ (in words: ‘if p, thenif not-p then q’) is a theorem in the calculus. (We shallaccept this as a fact, without exhibiting the derivation.)Suppose, then, that some formula S as well as itscontradictory ~ S were deducible from the axioms. Bysubstituting S for the variable ‘p’ in the theorem (aspermitted by the Rule of Substitution), and applyingthe Rule of Detachment twice, the formula ‘q’ would bededucible.10 But, if the formula consisting of the variable

10 By substituting S for ‘p’ we first obtain: S�(~S�q). From this,together with S, which is assumed to be demonstrable, we obtainby the Detachment Rule: ~S�q. Finally, since ~S is also assumed tobe demonstrable, using the Detachment Rule once more, we get: q.


‘q’ is demonstrable, it follows at once that by substitutingany formula whatsoever for ‘q’, any formula whatsoeveris deducible from the axioms. It is thus clear that, ifboth some formula S and its contradictory ~S werededucible from the axioms, every formula would bededucible. In short, if the calculus is not consistent,every formula is a theorem—which is the same as sayingthat from a contradictory set of axioms any formula canbe derived. But this has a converse: namely, if not everyformula is a theorem (i.e., if there is at least one formulathat is not derivable from the axioms), then the calculusis consistent. The task, therefore, is to show that thereis at least one formula that cannot be derived from theaxioms.

The way this is done is to employ meta-mathematicalreasoning upon the system before us. The actualprocedure is elegant. It consists in finding acharacteristic or structural property of formulas whichsatisfies the three following conditions, (1) The propertymust be common to all four axioms. (One such propertyis that of containing not more than 25 elementary signs;however, this property does not satisfy the nextcondition.) (2) The property must be “hereditary” underthe Transformation Rules—that is, if all the axioms havethe property, any formula properly derived from themby the Transformation Rules must also have it. Sinceany formula so derived is by definition a theorem, thiscondition in essence stipulates that every theorem musthave the property. (3) The property must not belong toevery formula that can be constructed in accordance

52 Gödel’s Proof

with the Formation Rules of the system—that is, wemust seek to exhibit at least one formula that doesnot have the property. If we succeed in this threefoldtask, we shall have an absolute proof of consistency.The reasoning runs something like this: the hereditaryproperty is transmitted from the axioms to alltheorems; but if an array of signs can be found thatconforms to the requirements of being a formula inthe system and that, nevertheless, does not possessthe specified hereditary property, this formula cannotbe a theorem. (To put the matter in another way, if asuspected offspring (formula) lacks an invariably in-herited trait of the forebears (axioms), it cannot infact be their descendant (theorem)). But, if a formulais discovered that is not a theorem, we haveestablished the consistency of the system; for, as wenoted a moment ago, if the system were not consistent,every formula could be derived from the axioms (i.e.,every formula would be a theorem). In short, theexhibition of a single formula without the hereditaryproperty does the trick.

Let us identify a property of the required kind. Theone we choose is the property of being a “tautology.”In common parlance, an utterance is usually said tobe tautologous if it contains a redundancy and saysthe same thing twice over in different words—e.g.,‘John is the father of Charles and Charles is a son ofJohn’. In logic, however, a tautology is defined as astatement that excludes no logical possibilities—e.g.,‘Either it is raining or it is not raining’. Another way


of putting this is to say that a tautology is “true in allpossible worlds.” No one will doubt that, irrespectiveof the actual state of the weather (i.e., regardless ofwhether the statement that it is raining is true or false),the statement ‘Either it is raining or it is not raining’is necessarily true.

We employ this notion to define a tautology in oursystem. Notice, first, that every formula is constructed ofelementary constituents ‘p’, ‘q’, ‘r’, etc. A formula is atautology if it is invariably true, regardless of whether itselementary constituents are true or false. Thus, in thefirst axiom ‘(pVp)�p’, the only elementary constituentis ‘p’; but it makes no difference whether ‘p’ is assumedto be true or is assumed to be false—the first axiom istrue in either case. This may be made more evident if wesubstitute for ‘p’ the statement ‘Mt. Rainier is 20,000 feethigh’; we then obtain as an instance of the first axiomthe statement If either Mt. Rainier is 20,000 feet high orMt. Rainier is 20,000 feet high, then Mt. Rainier is 20,000feet high’. The reader will have no difficulty inrecognizing this long statement to be true, even if heshould not happen to know whether the constituentstatement ‘Mt. Rainier is 20,000 feet high’ is true.Obviously, then, the first axiom is a tautology—“true inall possible worlds.” It can easily be shown that each ofthe other axioms is also a tautology.

Next, it is possible to prove that the property of beinga tautology is hereditary under the TransformationRules, though we shall not turn aside to give thedemonstration. (See Appendix, no. 3.) It follows that

54 Gödel’s Proof

every formula properly derived from the axioms (i.e.,every theorem) must be a tautology.

It has now been shown that the property of beingtautologous satisfies two of the three conditionsmentioned earlier, and we are ready for the third step.We must look for a formula that belongs to the system(i.e., is constructed out of the signs mentioned in thevocabulary in accordance with the Formation Rules), yet,because it does not possess the property of being atautology, cannot be a theorem (i.e., cannot be derivedfrom the axioms). We do not have to look very hard; it iseasy to exhibit such a formula. For example, ‘pVq’ fitsthe requirements. It purports to be a gosling but is in facta duckling; it does not belong to the family: it is a formula,but it is not a theorem. Clearly, it is not a tautology. Anysubstitution instance (or interpretation) shows this atonce. We can obtain by substitution for the variables in‘pVq’ the statement ‘Napoleon died of cancer or Bismarckenjoyed a cup of coffee’. This is not a truth of logic,because it would be false if both of the two clausesoccurring in it were false; and, even if it is a truestatement, it is not true irrespective of the truth or falsityof its constituent statements. (See Appendix, no. 3.)

We have achieved our goal. We have found at leastone formula that is not a theorem. Such a formula couldnot occur if the axioms were contradictory.Consequently, it is not possible to derive from theaxioms of the sentential calculus both a formula and itsnegation. In short, we have exhibited an absolute proofof the consistency of the system.11


Before leaving the sentential calculus, we mustmention a final point. Since every theorem of thiscalculus is a tautology, a truth of logic, it is natural toask whether, conversely, every logical truth expressiblein the vocabulary of the calculus (i.e., every tautology)is also a theorem (i.e., derivable from the axioms). Theanswer is yes, though the proof is too long to be statedhere. The point we are concerned with making, however,does not depend on acquaintance with the proof. Thepoint is that, in the light of this conclusion, the axiomsare sufficient for generating all tautologous formulas—all logical truths expressible in the system. Such axiomsare said to be “complete.”

Now, it is frequently of paramount interest todetermine whether an axiomatized system is complete.Indeed, a powerful motive for axiomatizing variousbranches of mathematics has been the desire to establisha set of initial assumptions from which all the true

11 The reader may find helpful the following recapitulation ofthe sequence:

1. Every axiom of the system is a tautology.2. Tautologousness is a hereditary property.3. Every formula properly derived from the axioms (i.e., every

theorem) is also a tautology.4. Hence any formula that is not a tautology is not a theorem.5. One formula has been found (e.g., ‘pVq’) that is not a tautology.6. This formula is therefore not a theorem.7. But, if the axioms were inconsistent, every formula would be

a theorem.8. Therefore the axioms are consistent.

56 Gödel’s Proof

statements in some field of inquiry are deducible. WhenEuclid axiomatized elementary geometry, he apparentlyso selected his axioms as to make it possible to derivefrom them all geometric truths; that is, those that hadalready been established, as well as any others thatmight be discovered in the future.12 Until recently itwas taken as a matter of course that a complete set ofaxioms for any given branch of mathematics can beassembled. In particular, mathematicians believed thatthe set proposed for arithmetic in the past was in factcomplete, or, at worst, could be made complete simplyby adding a finite number of axioms to the original list.The discovery that this will not work is one of Gödel’smajor achievements.

12 Euclid showed remarkable insight in treating his famous parallelaxiom as an assumption logically independent of his other axioms.For, as was subsequently proved, this axiom cannot be derived fromhis remaining assumptions, so that without it the set of axioms isincomplete.

57

VIThe Idea of Mapping and Its Use in Mathematics

The sentential calculus is an example of a mathematicalsystem for which the objectives of Hilbert’s theory ofproof are fully realized. To be sure, this calculus codifiesonly a fragment of formal logic, and its vocabulary andformal apparatus do not suffice to develop evenelementary arithmetic. Hilbert’s program, however, isnot so limited. It can be carried out successfully formore inclusive systems, which can be shown by meta-mathematical reasoning to be both consistent andcomplete. By way of example, an absolute proof ofconsistency is available for a system of arithmetic thatpermits the addition of cardinal numbers though nottheir multiplication. But is Hilbert’s finitistic methodpowerful enough to prove the consistency of a systemsuch as Principia, whose vocabulary and logicalapparatus are adequate to express the whole ofarithmetic and not merely a fragment? Repeatedattempts to construct such a proof were unsuccessful;and the publication of Gödel’s paper in 1931 showed,

58 Gödel’s Proof

finally, that all such efforts operating within the strictlimits of Hilbert’s original program must fail.

What did Gödel establish, and how did he prove hisresults? His main conclusions are twofold. In the firstplace (though this is not the order of Gödel’s actualargument), he showed that it is impossible to give ameta-mathematical proof of the consistency of a systemcomprehensive enough to contain the whole ofarithmetic—unless the proof itself employs rules ofinference in certain essential respects different fromthe Transformation Rules used in deriving theoremswithin the system. Such a proof may, to be sure, possessgreat value and importance. However, if the reasoningin it is based on rules of inference much more powerfulthan the rules of the arithmetical calculus, so that theconsistency of the assumptions in the reasoning is assubject to doubt as is the consistency of arithmetic, theproof would yield only a specious victory: one dragonslain only to create another. In any event, if the proof isnot finitistic, it does not realize the aims of Hilbert’soriginal program; and Gödel’s argument makes itunlikely that a finitistic proof of the consistency ofarithmetic can be given.

Gödel’s second main conclusion is even moresurprising and revolutionary, because it demonstratesa fundamental limitation in the power of the axiomaticmethod. Gödel showed that Principia, or any othersystem within which arithmetic can be developed, isessentially incomplete. In other words, given anyconsistent set of arithmetical axioms, there are true

The Idea of Mapping and Its Use in Mathematics 59

arithmetical statements that cannot be derived from theset. This crucial point deserves illustration. Mathematicsabounds in general statements to which no exceptionshave been found that thus far have thwarted all attemptsat proof. A classical illustration is known as “Goldbach’stheorem,” which states that every even number is thesum of two primes. No even number has ever been foundthat is not the sum of two primes, yet no one hassucceeded in finding a proof that Gold-bach’s conjectureapplies without exception to all even numbers. Here,then, is an example of an arithmetical statement thatmay be true, but may be non-derivable from the axiomsof arithmetic. Suppose, now, that Goldbach’s conjecturewere indeed universally true, though not derivable fromthe axioms. What of the suggestion that in thiseventuality the axioms could be modified or augmentedso as to make hitherto unprovable statements (such asGoldbach’s on our supposition) derivable in the enlargedsystem? Gödel’s results show that even if thesupposition were correct the suggestion would stillprovide no final cure for the difficulty. That is, even ifthe axioms of arithmetic are augmented by an indefinitenumber of other true ones, there will always be furtherarithmetical truths that are not formally derivable fromthe augmented set.13

13 Such further truths may, as we shall see, be established bysome form of meta-mathematical reasoning about an arithmeticalsystem. But this procedure does not fit the requirement that the

60 Gödel’s Proof

How did Gödel prove these conclusions? Up to a pointthe structure of his argument is modeled, as he himselfpointed out, on the reasoning involved in one of thelogical antinomies known as the “Richard Paradox,”first propounded by the French mathematician JulesRichard in 1905. We shall outline this paradox.

Consider a language (e.g., English) in which the purelyarithmetical properties of cardinal numbers can beformulated and defined. Let us examine the definitionsthat can be stated in the language. It is clear that, onpain of circularity or infinite regress, some terms re-ferring to arithmetical properties cannot be definedexplicitly—for we cannot define everything and muststart somewhere—though they can, presumably, beunderstood in some other way. For our purposes it doesnot matter which are the undefined or “primitive” terms;we may assume, for example, that we understand whatis meant by ‘an integer is divisible by another’, ‘aninteger is the product of two integers’, and so on. Theproperty of being a prime number may then be definedby: ‘not divisible by any integer other than 1 and itself;the property of being a perfect square may be definedby: ‘being the product of some integer by itself;and so on.

We can readily see that each such definition will

calculus must, so to speak, be self-contained, and that the truths inquestion must be exhibited as the formal consequences of thespecified axioms within the system. There is, then, an inherentlimitation in the axiomatic method as a way of systematizing thewhole of arithmetic.


contain only a finite number of words, and thereforeonly a finite number of letters of the alphabet. Thisbeing the case, the definitions can be placed in serialorder: a definition will precede another if the numberof letters in the first is smaller than the number of lettersin the second; and, if two definitions have the samenumber of letters, one of them will precede the otheron the basis of the alphabetical order of the letters ineach. On the basis of this order, a unique integer willcorrespond to each definition and will represent thenumber of the place that the definition occupies in theseries. For example, the definition with the smallestnumber of letters will correspond to the number 1,the next definition in the series will correspond to 2,and so on.

Since each definition is associated with a uniqueinteger, it may turn out in certain cases that an integerwill possess the very property designated by thedefinition with which the integer is correlated.14

Suppose, for instance, the defining expression ‘notdivisible by any integer other than 1 and itself’ happensto be correlated with the order number 17; obviously17 itself has the property designated by that expression.On the other hand, suppose the defining expression‘being the product of some integer by itself’ werecorrelated with the order number 15; 15 clearly does

14 This is the same sort of thing that would happen if the Englishword ‘short’ appeared in a list of words, and we characterized eachword of the list by the descriptive tags “short” or “long.” The word‘short’ would then have the tag “short” attached to it.

62 Gödel’s Proof

not have the property designated by the expression. Weshall describe the state of affairs in the second exampleby saying that the number 15 has the property of beingRichardian; and, in the first example, by saying thatthe number 17 does not have the property of beingRichardian. More generally, we define ‘x is Richardian’as a shorthand way of stating ‘x does not have theproperty designated by the defining expression withwhich x is correlated in the serially ordered set ofdefinitions’.

We come now to a curious but characteristic turn inthe statement of the Richard Paradox. The definingexpression for the property of being Richardianostensibly describes a numerical property of integers.The expression itself therefore belongs to the series ofdefinitions proposed above. It follows that theexpression is correlated with a position-fixing integeror number. Suppose this number is n. We now pose thequestion, reminiscent of Russell’s antinomy: Is nRichardian? The reader can doubtless anticipate the fatalcontradiction that now threatens. For n is Richardianif, and only if, n does not have the property designatedby the defining expression with which n is correlated(i.e., it does not have the property of being Richardian).In short, n is Richardian if, and only if, n is notRichardian; so that the statement ‘n is Richardian’ isboth true and false.

We must now point out that the contradiction is, ina sense, a hoax produced by not playing the game quitefairly. An essential but tacit assumption underlying the


serial ordering of definitions was conveniently droppedalong the way. It was agreed to consider the definitionsof the purely arithmetical properties of integers—properties that can be formulated with the help of suchnotions as arithmetical addition, multiplication, andthe like. But then, without warning, we were asked toaccept a definition in the series that involves referenceto the notation used in formulating arithmeticalproperties. More specifically, the definition of theproperty of being Richardian does not belong to theseries initially intended, because this definition involvesmeta-mathematical notions such as the number of letters(or signs) occurring in expressions. We can outflankthe Richard Paradox by distinguishing carefully betweenstatements within arithmetic (which make no referenceto any system of notation) and statements about somesystem of notation in which arithmetic is codified.

The reasoning in the construction of the RichardParadox is clearly fallacious. The constructionnevertheless suggests that it may be possible to “map”or “mirror” meta-mathematical statements about asufficiently comprehensive formal system in the systemitself. The idea of “mapping” is well known and playsa fundamental role in many branches of mathematics.It is used, of course, in the construction of ordinarymaps, where shapes on the surface of a sphere areprojected onto a plane, so that the relations betweenthe plane figures mirror the relations between the figureson the spherical surface. It is used in coordinategeometry, which translates geometry into algebra, so

64 Gödel’s Proof

that geometric relations are mapped by algebraic ones.(The reader will recall the discussion in Section II,which explained how Hilbert used algebra to establishthe consistency of his axioms for geometry. WhatHilbert did, in effect, was to map geometry ontoalgebra.) Mapping also plays a role in mathematicalphysics where, for example, relations betweenproperties of electric currents are represented in thelanguage of hydrodynamics. And mapping is involvedwhen a pilot model is constructed before proceedingwith a full-size machine, when a small wing surface isobserved for its aerodynamic properties in a windtunnel, or when a laboratory rig made up of electriccircuits is used to study the relations between large-size masses in motion. A striking visual example ispresented in Fig. 3, which illustrates a species ofmapping that occurs in a branch of mathematics knownas projective geometry.

The basic feature of mapping is that an abstractstructure of relations embodied in one domain of“objects” can be shown to hold between “objects”(usually of a sort different from the first set) in another

Fig. 3Figure 3 (a) illustrates the Theorem of Pappus: If A, B, C are any

three distinct points on a line I, and A�, B�, C� any three distinctpoints on another line II, the three points R, S, T determined by thepair of lines AB� and A�B, BC� and B�C, CA� and C�A, respectively,are collinear (i.e., lie on line III).

Figure 3 (b) illustrates the “dual” of the above theorem: If A, B,C are any three distinct lines on a point I, and A�, B�, C� any threedistinct lines on another point II, the three lines R, S, T determined

by the pair of points AB� and A�B, BC� and B�C, CA� and C�A,respectively, are copunctal (i.e., lie on point III).

The two figures have the same abstract structure, though inappearance they are markedly different. Figure 3 (a) is so related toFigure 3 (b) that points of the former correspond to lines of thelatter, while lines of the former correspond to points of the latter.In effect, (b) is a map of (a): a point in (b) represents (or is the“mirror image” of) a line in (a), while a line in (b) represents apoint in (a).

66 Gödel’s Proof

domain. It is this feature which stimulated Gödel inconstructing his proofs. If complicated meta-mathematical statements about a formalized system ofarithmetic could, as he hoped, be translated into (ormirrored by) arithmetical statements within the systemitself, an important gain would be achieved infacilitating metamathematical demonstrations. For justas it is easier to deal with the algebraic formulasrepresenting (or mirroring) intricate geometricalrelations between curves and surfaces in space thanwith the geometrical relations themselves, so it is easierto deal with the arithmetical counterparts (or “mirrorimages”) of complex logical relations than with thelogical relations themselves.

The exploitation of the notion of mapping is the keyto the argument in Gödel’s famous paper. Followingthe style of the Richard Paradox, but carefully avoidingthe fallacy involved in its construction, Gödel showedthat meta-mathematical statements about a formalizedarithmetical calculus can indeed be represented byarithmetical formulas within the calculus. As we shallexplain in greater detail in the next section, he deviseda method of representation such that neither thearithmetical formula corresponding to a certain truemeta-mathematical statement about the formula, nor thearithmetical formula corresponding to the denial of thestatement, is demonstrable within the calculus. Sinceone of these arithmetical formulas must codify anarithmetical truth, yet neither is derivable from theaxioms, the axioms are incomplete. Gödel’s method of


representation also enabled him to construct anarithmetical formula corresponding to the meta-mathematical statement ‘The calculus is consistent’ andto show that this formula is not demonstrable withinthe calculus. It follows that the meta-mathematicalstatement cannot be established unless rules of inferenceare used that cannot be represented within the calculus,so that, in proving the statement, rules must beemployed whose own consistency may be asquestionable as the consistency of arithmetic itself.Gödel established these major conclusions by using aremarkably ingenious form of mapping.

68

VIIGödel’s Proofs

Gödel’s paper is difficult. Forty-six preliminarydefinitions, together with several important preliminarytheorems, must be mastered before the main results arereached. We shall take a much easier road; nevertheless,it should afford the reader glimpses of the ascent andof the crowning structure.

A Gödel numberingGödel described a formalized calculus within whichall the customary arithmetical notations can beexpressed and familiar arithmetical relationsestablished.15 The formulas of the calculus areconstructed out of a class of elementary signs, whichconstitute the fundamental vocabulary. A set ofprimitive formulas (or axioms) are the underpinning,and the theorems of the calculus are formulas derivable

15 He used an adaptation of the system developed in PrincipiaMathematica. But any calculus within which the cardinal numbersystem can be constructed would have served his purpose.

Gödel Numbering 69

from the axioms with the help of a carefully enumeratedset of Transformation Rules (or rules of inference).

Gödel first showed that it is possible to assign aunique number to each elementary sign, each formula(or sequence of signs), and each proof (or finite sequenceof formulas). This number, which serves as a distinctivetag or label, is called the “Gödel number” of the sign,formula, or proof.16

The elementary signs belonging to the fundamentalvocabulary are of two kinds: the constant signs and thevariables. We shall assume that there are exactly tenconstant signs,17 to which the integers from 1 to 10 areattached as Gödel numbers. Most of these signs arealready known to the reader: ‘~’ (short for ‘not’); ‘V’ (shortfor ‘or’); ‘�’ (short for ‘if…then…’); ‘=’ (short for ‘equals’);‘0’ (the numeral for the number zero); and three signs ofpunctuation, namely, the left parenthesis ‘(’, the rightparenthesis ‘)’, and the comma ‘,’. In addition, two othersigns will be used: the inverted letter ‘$’, which may beread as ‘there is’ and which occurs in “existentialquantifiers”; and the lower-case ‘s’, which is attached to

16 There are many alternative ways of assigning Gödel numbers,and it is immaterial to the main argument which is adopted. Wegive a concrete example of how the numbers can be assigned tohelp the reader follow the discussion. The method of numberingused in the text was employed by Gödel in his 1931 paper.

17 The number of constant signs depends on how the formalcalculus is set up. Gödel in his paper used only seven constantsigns. The text uses ten in order to avoid certain complexities in theexposition.

70 Gödel’s Proof

numerical expressions to designate the immediatesuccessor of a number. To illustrate: the formula‘($x)(x=s0)’ may be read ‘There is an x such that x is theimmediate successor of 0’. The table below displays theten constant signs, states the Gödel number associatedwith each, and indicates the usual meanings of the signs.

TABLE 2

Besides the constant elementary signs, three kinds ofvariables appear in the fundamental vocabulary of thecalculus: the numerical variables ‘x’, ‘y’, ‘z’, etc., for whichnumerals and numerical expressions may be substituted;the sentential variables ‘p’, ‘q’, ‘r’, etc., for which formulas

Gödel Numbering 71

(sentences) may be substituted; and the predicatevariables ‘P’, ‘Q’, ‘R’, etc., for which predicates, such as‘Prime’ or ‘Greater than’, may be substituted. The variablesare assigned Gödel numbers in accordance with thefollowing rules: associate (i) with each distinct numericalvariable a distinct prime number greater than 10; (ii) witheach distinct sentential variable the square of a primenumber greater than 10; and (iii) with each distinctpredicate variable the cube of a prime greater than 10.The accompanying table illustrates the use of these rulesto specify the Gödel numbers of a few variables.

72 Gödel’s Proof

TABLE 3

Consider next a formula of the system, for example,‘($x)(x=sy)’. (Literally translated, this reads: ‘There isan x such that x is the immediate successor of y’, andsays, in effect, that every number has an immediatesuccessor.) The numbers associated with its tenconstituent elementary signs are, respectively, 8, 4, 11,9, 8, 11, 5, 7, 13, 9. We show this schematically below:

It is desirable, however, to assign a single number tothe formula rather than a set of numbers. This can bedone easily. We agree to associate with the formula theunique number that is the product of the first ten primesin order of magnitude, each prime being raised to apower equal to the Gödel number of the correspondingelementary sign. The above formula is accordinglyassociated with the number

Gödel Numbering 73

28×34×511×79×118×1311×175×197

×2313×299; let us remfer to this number as m. In a similar fashion,a unique number, the product of as many primes asthere are signs (each prime being raised to a power equalto the Gödel number of the corresponding sign), can beassigned to every finite sequence of elementary signsand, in particular, to every formula.18

Consider, finally, a sequence of formulas, such as mayoccur in some proof, e.g., the sequence:

($x)(x=sy)($x)(x=s0)

The second formula when translated reads ‘0 has an

18 Signs may occur in the calculus which do not appear in thefundamental vocabulary; these are introduced by defining them withthe help of the vocabulary signs. For example, the sign ‘.’, thesentential connective used as an abbreviation for ‘and’, can bedefined in context as follows: ‘p.q’ is short for ‘~(~pV~q)’. WhatGödel number is assigned to a defined sign? The answer is obviousif we notice that expressions containing defined signs can beeliminated in favor of their defining equivalents; and it is clear thata Gödel number can be determined for the transformed expressions.Accordingly, the Gödel number of the formula ‘p.q’ is the Gödelnumber of the formula ‘~(~pV~q)’. Similarly, the various numeralscan be introduced by definition as follows: ‘1’ as short for ‘s0’, ‘2’as short for ‘ss0’, ‘3’ as short for ‘sss0’, and so on. To obtain theGödel number for the formula ‘~(2=3)’, we eliminate the definedsigns, thus obtaining the formula ‘~(ss0=sss0)’, and determine itsGödel number pursuant to the rules stated in the text.

74 Gödel’s Proof

immediate successor’; it is derivable from the first bysubstituting the numeral ‘0’ for the numerical variable‘y’.19 We have already determined the Gödel number ofthe first formula: it is m; and suppose that n is theGödel number of the second formula. As before, it isconvenient to have a single number as a tag for thesequence. We agree therefore to associate with it thenumber which is the product of the first two primes inorder of magnitude (i.e., the primes 2 and 3), each primebeing raised to a power equal to the Gödel number ofthe corresponding formula in the sequence. If we callthis number k, we can write k=2m×3n. By applying thiscompact procedure we can obtain a number for eachsequence of formulas. In sum, every expression in thesystem, whether an elementary sign, a sequence of signs,or a sequence of sequences, can be assigned a uniqueGödel number.

What has been done so far is to establish a method forcompletely “arithmetizing” the formal calculus. Themethod is essentially a set of directions for setting up aone-to-one correspondence between the expressions inthe calculus and a certain subset of the integers.20 Once

19 The reader will recall that we defined a proof as a finitesequence of formulas, each of which either is an axiom or can bederived from preceding formulas in the sequence with the help ofthe Transformation Rules. By this definition the above sequence isnot a proof, since the first formula is not an axiom and its derivationfrom the axioms is not shown: the sequence is only a segment of aproof. It would take too long to write out a full example of a proof,and for illustiative purposes the above sequence will suffice.

Gödel Numbering 75

an expression is given, the Gödel number uniquelycorresponding to it can be calculated. But this is only halfthe story. Once a number is given, we can determinewhether it is a Gödel number, and, if: it is, the expressionit represents can be exactly analyzed or “retrieved.” If agiven number is less than or equal to 10, it is the Gödelnumber of an elementary constant sign. The sign can beidentified. If the number is greater than 10, it can bedecomposed into its prime factors in just one way (as weknow from a famous theorem of arithmetic).21 If it is aprime greater than 10, or the second or third power ofsuch a prime, it is the Gödel number of an identifiablevariable. If it is the product of successive primes, eachraised to some power, it may be the Gödel number eitherof a formula or of a sequence of formulas. In this case the

20 Not every integer is a Gödel number. Consider, for example,the number 100. 100 is greater than 10, and therefore cannot be theGödel number of an elementary constant sign; and since it is neithera prime greater than 10, nor the square nor the cube of such aprime, it cannot be the Gödel number of a variable. On decomposing100 into its prime factors, we find that it is equal to 22×52; and theprime number 3 does not appear as a factor in the decomposition,but is skipped. According to the rules laid down, however, theGödel num ber of a formula (or of a sequence of formulas) must bethe product of successive primes, each raised to some power. Thenumber 100 does not satisfy this condition. In short, 100 cannot beassigned to constant signs, variables, or formulas; hence it is not aGödel number.

21 This theorem is known as the fundamental theorem ofarithmetic. It says that if an integer is composite (i.e., not a prime)it has a unique decomposition into prime factors.

76 Gödel’s Proof

expression to which it corresponds can be exactlydetermined. Following this program, we can take anygiven number apart, as if it were a machine, discover howit is constructed and what goes into it; and since each ofits elements corresponds to an element of the expressionit represents, we can reconstitute the expression, analyzeits structure, and the like. Table 4 illustrates for a givennumber how we can ascertain whether it is a Gödelnumber and, if so, what expression it symbolizes.

TABLE 4

B The arithmetization of meta-mathematicsGödel’s next step is an ingenious application ofmapping. He showed that all meta-mathematical

The arithmetical formula ‘zero equals zero’ hasthe Gödel number 243 million. Reading downfrom A to E, the illustration shows how the numberis translated into the expression it represents;reading up, how the number for the formula isderived.

The Arithmetization of Meta-mathematics 77

statements about the structural properties ofexpressions in the calculus can be adequately mirroredwithin the calculus itself. The basic idea underlyinghis procedure is this: Since every expression in thecalculus is associated with a (Gödel) number, a meta-mathematical statement about expressions and theirrelations to one another may be construed as astatement about the corresponding (Gödel) numbersand their arithmetical relations to one another. In thisway metamathematics becomes completely“arithmetized.” To take a trivial analogue: customersin a busy supermarket are often given, when they enter,tickets on which are printed numbers whose orderdetermines the order in which the customers are to bewaited on at the meat counter. By inspecting thenumbers it is easy to tell how many persons have beenserved, how many are waiting, who precedes whom,and by how many customers, and so on. If, for example,Mrs. Smith has number 37, and Mrs. Brown number53, instead of explaining to Mrs. Brown that she hasto wait her turn after Mrs. Smith, it suffices to pointout that 37 is less than 53.

As in the supermarket, so in meta-mathematics. Eachmeta-mathematical statement is represented by a uniqueformula within arithmetic; and the relations of logicaldependence between meta-mathematical statements arefully reflected in the numerical relations of dependencebetween their corresponding arithmetical formulas.Once again mapping facilitates an inquiry into structure.The exploration of meta-mathematical questions can

78 Gödel’s Proof

be pursued by investigating the arithmetical propertiesand relations of certain integers.

We illustrate these general remarks by an elementaryexample. Consider the first axiom of the sententialcalculus, which also happens to be an axiom in theformal system under discussion: ‘(pVp)�p’. Its Gödelnumber is 28×3112×52×7112×119×133×17112, which weshall designate by the letter ‘a’. Consider also theformula: ‘(pVp)’, whose Gödel number is28×3112×52×7112×119; we shall designate it by the letter‘b’. We now assert the meta-mathematical statementthat the formula ‘(pVp)’ is an initial part of the axiom.To what arithmetical formula in the formal system doesthis statement correspond? It is evident that thesmaller formula ‘(pVp)’ can be an initial part of thelarger formula which is the axiom if, and only if, the(Gödel) number b, representing the former, is a factor ofthe (Gödel) number a, representing the latter. On theassumption that the expression ‘factor of’ is suitablydefined in the formalized arithmetical system, thearithmetical formula which uniquely corresponds tothe above meta-mathematical statement is: ‘b is a factorof a’. Moreover, if this formula is true, i.e., if b is afactor of a, then it is true that ‘(pVp)’ is an initial partof ‘(pVp)�p’.

Let us fix attention on the meta-mathematicalstatement: ‘The sequence of formulas with Gödelnumber x is a proof of the formula with Gödel numberz’. This statement is represented (mirrored) by adefinite formula in the arithmetical calculus which


expresses a purely arithmetical relation between x andz. (We can gain some notion of the complexity of thisrelation by recalling the example used above, in whichthe Gödel number k=2m×3n was assigned to the(fragment of a) proof whose conclusion has the Gödelnumber n. A little reflection shows that there is here adefinite, though by no means simple, arithmeticalrelation between k, the Gödel number of the proof, andn, the Gödel number of the conclusion.) We write thisrelamind ourselves of the meta-mathematicalstatement to tion between x and z as the formula ‘Dem(x, z)’, to rewhich it corresponds (i.e., of the meta-mathematical statement ‘The sequence of formulaswith Gödel number x is a proof (or demonstration) ofthe formula with Gödel number z’).22 We now ask thereader to observe that a meta-mathematical statementwhich says that a certain sequence of formulas is aproof for a given formula is true, if, and only if, theGödel number of the alleged proof stands to the Gödelnumber of the conclusion in the arithmetical relationhere designated by ‘Dem’. Accordingly, to establish thetruth or falsity of the meta-mathematical statementunder discussion, we need concern ourselves only

22 The reader must keep clearly in mind that, though ‘Dem (x, z)’represents the meta-mathematical statement, the formula itselfbelongs to the arithmetical calculus. The formula could be writtenin more customary notation as ‘f(x, z)=0’, where the letter ‘f’ denotesa complex set of arithmetical operations upon numbers. But thismore customary notation does not immediately suggest themetamathematical interpretation of the formula.

80 Gödel’s Proof

with the question whether the relation Dem holdsbetween two numbers. Conversely, we can establishthat the arithmetical relation holds between a pair ofnumbers by showing that the meta-mathematicalstatement mirrored by this relation between thenumbers is true. Similarly, the meta-mathematicalstatement ‘The sequence of formulas with the Gödelnumber x is not a proof for the formula with the Gödelnumber z’ is represented by a definite formula in theformalized arithmetical system. This formula is theformal contradictory of ‘Dem (x, z)’, namely, ‘~Dem (x, z)’.

One additional bit of special notation is needed forstating the crux of Gödel’s argument. Begin with anexample. The formula ‘($x)(x=sy)’ has m for its Gödelnumber (see page 73), while the variable ‘y’ has theGödel number 13. Substitute in this formula for thevariable with Gödel number 13 (i.e., for ‘y’) the numeralfor m. The result is the formula ‘($x)(x=sm)’, whichsays literally that there is a number x such that x is theimmediate successor of m. This latter formula also hasa Gödel number, which can be calculated quite easily.But instead of making the calculation, we can identifythe number by an unambiguous meta-mathematicalcharacterization: it is the Gödel number of the formulathat is obtained from the formula with Gödel numberm, by substituting for the variable with Gödel number13 the numeral for m. This metamathematicalcharacterization uniquely determines a definite numberwhich is a certain arithmetical function of the numbers


m and 13, where the function itself can be expressedwithin the formalized system.23

23 This function is quite complex. Just how complex becomesevident if we try to formulate it in greater detail. Let us attemptsuch a formulation, without carrying it to the bitter end. It wasshown on page 73 that m, the Gödel number of ‘($x)(x=sy)’, is

28×34×511×79×118×1311×175×197×2313×299.

To find the Gödel number of ‘($x)(x=sm)’ (the formula obtainedfrom the preceding one by substituting for the variable ‘y’ in thelatter the numeral for m) we proceed as follows: This formulacontains the numeral ‘m’, which is a defined sign, and, in accordancewith the content of footnote 18, m must be replaced by its definingequivalent. When this is done, we obtain the formula:

($x)(x=ssssss…s0)

where the letter ‘s’ occurs m+1 times. This formula contains onlythe elementary signs belonging to the fundamental vocabulary, sothat its Gödel number can be calculated. To do this, we first obtainthe series of Gödel numbers associated with the elementary signs ofthe formula:

8, 4, 11, 9, 8, 11, 5, 7, 7, 7,…7, 6, 9

in which the number 7 occurs m+1 times. We next take the productof the first m+10 primes in order of magnitude, each prime beingraised to a power equal to the Gödel number of the correspondingelementary sign. Let us refer to this number as r, so that

r=28×34×511×79×118×1311×175×197×237

×297×317×…×p9m+10

where pm+10 is the (m+10)th prime in order of magnitude.

Now compare the two Gödel numbers m and r. m contains aprime factor raised to the power 13; r contains all the prime factorsof m and many others besides, but none of them are raised to the

82 Gödel’s Proof

The number can therefore be designated within thecalculus. This designation will be written as ‘sub (m,13, m)’, the purpose of this form being to recall themeta-mathematical characterization which it represents,viz., ‘the Gödel number of the formula obtained fromthe formula with Gödel number m, by substituting forthe variable with the Gödel number 13 the numeral form’. We can now drop the example and generalize. Thereader will see readily that the expression ‘sub (y, 13,y)’ is the mirror image within the formalized arithmeticalcalculus of the meta-mathematical characterization: ‘theGödel number of the formula that is obtained from theformula with Gödel number y, by substituting for thevariable with Gödel number 13 the numeral for y’. Hewill also note that when a definite numeral is substitutedfor ‘y’ in ‘sub (y, 13, y)’—for example, the numeral form or the numeral for two hundred forty three million—the resulting expression designates a definite integerwhich is the Gödel number of a certain formula.24

power 13. The number r can thus be obtained from the number m,by replacing the prime factor in m which is raised to the power 13with other primes raised to some power different from 13. To stateexactly and in full detail how r is related to m is not possible withoutintroducing a great deal of additional notational apparatus; this isdone in Gödel’s original paper. But enough has been said to indicatethat the number r is a definite arithmetical function of m and 13.

24 Several questions may occur to the reader that need to beanswered. It may be asked why, in the meta-mathematical


characterization just mentioned, we say that it is “the numeralfor y” which is to be substituted for a certain variable, ratherthan “the number y.” The answer depends on the distinction,already discussed, between mathematics and metamathematics,and calls for a brief elucidation of the difference between numbersand numerals. A numeral is a sign, a linguistic expression,something which one can write down, erase, copy, and so on. Anumber, on the other hand, is something which a numeral namesor designates, and which cannot literally be written down, erased,copied, and so on. Thus, we say that 10 is the number of ourfingers, and, in making this statement, we are attributing a certain“property” to the class of our fingers; but it would evidently beabsurd to say that this property is a numeral. Again, the number10 is named by the Arabic numeral ‘10’, as well as by the Romanletter ‘X’; these names are different, though they name the samenumber. In short, when we make a substitution for a numericalvariable (which is a letter or sign) we are putting one sign inplace of another sign. We cannot literally substitute a number fora sign, because a number is a property of classes (and is sometimessaid to be a concept), not something we can put on paper. Itfollows that, in substituting for a numerical variable, we cansubstitute only a numeral (or some other numerical expression,such as ‘s0’ or ‘7+5’), and not a number. This explains why, inthe above meta-mathematical characterization, we state that weare substituting for the variable the numeral for (the number) y,rather than the number y itself.

The reader may wonder what number is designated by ‘sub (y,13, y)’ if the formula whose Gödel number is y does not happen tocontain the variable with Gödel number 13—that is, if the formuladoes not contain the variable ‘y’. Thus, sub (243,000,000, 13,243,000,000) is the Gödel number of the formula obtained from theformula with Gödel number 243,000,000 by substituting for thevariable ‘y’ the numeral ‘243,000,000’. But if the reader consultsTable 4, he will find that 243,000,000 is the Gödel number of theformula ‘0=0’, which does not contain the variable ‘y’. What, then,

84 Gödel’s Proof

is the formula that is obtained from ‘0=0’ by substituting for thevariable ‘y’ the numeral for the number 243,000,000? The simpleanswer is that, since ‘0=0’ does not contain this variable, nosubstitution can be made—or, what amounts to the same thing, thatthe formula obtained from ‘0=0’ is this very same formula.Accordingly, the number designated by ‘sub (243,000,000, 13,243,000,000)’ is 243,000,000.

The reader may also be puzzled as to whether ‘sub (y, 13, y)’ isa formula within the arithmetical system in the sense that, forexample, ‘($x)(x=sy)’, ‘0=0’, and ‘Dem (x, z)’ are formulas. The answeris no, for the following reason. The expression ‘0=0’ is called aformula, because it asserts a relation between two numbers and isthus capable of having truth or falsity significantly attributed to it.Similarly, when definite numerals are substituted for the variablesin ‘Dem (x, z)’, this expression formulates a relation between twonumbers, and so becomes a statement that is either true or false.The same holds for ‘($x)(x=sy)’. On the other hand, even when adefinite numeral is substituted for ‘y’ in ‘sub (y, 13, y)’, the resultingexpression does not assert anything and therefore cannot be true orfalse. It merely designates or names a number, by describing it as acertain function of other numbers. The difference between a formula(which is in effect a statement about numbers, and so is either trueor false) and a name-function (which is in effect a name thatidentifies a number, and so is neither true nor false) may be clarifiedby some further illustrations. ‘5=3’ is a formula which, though false,declares that the two numbers 5 and 3 are equal; ‘52=42+32’ is alsoa formula which asserts that a definite relation holds between thethree numbers 5, 4, and 3; and, more generally, ‘y=f(x)’ is a formulawhich asserts that a certain relation holds between the unspecifiednumbers x and y. On the other hand, ‘2+3’ expresses a function ofthe two numbers 2 and 3, and so names a certain number (in fact,the number 5); it is not a formula, for it clearly would be nonsensicalto ask whether ‘2+3’ is true or false. ‘(7×5)+8’ expresses an otherfunction of the three numbers 5, 7, and 8, and designates the number43. And, more generally, ‘f(x)’ expresses a function of x, and

The Heart of Gödel’s Argument 85

C The heart of Gödel’s argumentAt last we are equipped to follow in outline Gödel’smain argument. We shall begin by enumerating the stepsin a general way, so that the reader can get a bird’s-eyeview of the sequence.

Gödel showed (i) how to construct an arithmeticalformula G that represents the meta-mathematicalstatement: ‘The formula G is not demonstrable’. Thisformula G thus ostensibly says of itself that it is notdemonstrable. Up to a point, G is constructedanalogously to the Richard Paradox. In that Paradox,the expression ‘Richardian’ is associated with a certainnumber n, and the sentence ‘n is Richardian’ isconstructed. In Gödel’s argument, the formula G is alsoassociated with a certain number h, and is soconstructed that it corresponds to the statement: ‘Theformula with the associated number h is notdemonstrable’. But (ii) Gödel also showed that G isdemonstrable if, and only if, its formal negation ~ G isdemonstrable. This step in the argument is againanalogous to a step in the Richard Paradox, in which itis proved that n is Richardian if, and only if, n is not

identifies a certain number when a definite numeral is substitutedfor ‘x’ and when a definite meaning is given to the function-sign ‘f’.In short, while ‘Dem (x, z)’ is a formula because it has the form ofa statement about numbers, ‘sub (y, 13, y)’ is not a formula becauseit has only the form of a name for numbers.

86 Gödel’s Proof

Richardian. However, if a formula and its own negationare both formally demonstrable, the arithmeticalcalculus is not consistent. Accordingly, if the calculusis consistent, neither G nor ~ G is formally derivablefrom the axioms of arithmetic. Therefore, if arithmeticis consistent, G is a formally undecidable formula. Gödelthen proved (iii) that, though G is not formallydemonstrable, it nevertheless is a true arithmeticalformula. It is true in the sense that it asserts that everyinteger possesses a certain arithmetical property, whichcan be exactly defined and is exhibited by whateverinteger is examined, (iv) Since G is both true andformally undecidable, the axioms of arithmetic areincomplete. In other words, we cannot deduce allarithmetical truths from the axioms. Moreover, Gödelestablished that arithmetic is essentially incomplete:even if additional axioms were assumed so that thetrue formula G could be formally derived from theaugmented set, another true but formally undecidableformula could be constructed, (v) Next, Gödel describedhow to construct an arithmetical formula A thatrepresents the meta-mathematical statement:‘Arithmetic is consistent’; and he proved that theformula ‘A�G’ is formally demonstrable. Finally, heshowed that the formula A is not demonstrable. Fromthis it follows that the consistency of arithmetic cannotbe established by an argument that can be representedin the formal arithmetical calculus.

Now, to give the substance of the argument more fully:


(i) The formula ‘~Dem (x, z)’ has already beenidentified. It represents within formalized arithmeticthe meta-mathematical statement: ‘The sequence offormulas with the Gödel number x is not a proof for theformula with the Gödel number z’. The prefix ‘(x)’ isnow introduced into the Dem formula. This prefixperforms the same function in the formalized system asdoes the English phrase ‘For every x’. On attaching thisprefix, we have a new formula: ‘(x) ~Dem (x, z)’, whichrepresents within arithmetic the meta-mathematicalstatement: ‘For every x, the sequence of formulas withGödel number x is not a proof for the formula withGödel number z’. The new formula is therefore theformal paraphrase (strictly speaking, it is the uniquerepresentative), within the calculus, of the meta-mathematical statement: ‘The formula with Gödelnumber z is not demonstrable’—or, to put it anotherway, ‘No proof can be adduced for the formula withGödel number z’.

What Gödel showed is that a certain special case ofthis formula is not formally demonstrable. To constructthis special case, begin with the formula displayed asline (1): (1) (x)~Dem (x, sub (y, 13, y)) This formula belongs to the arithmetical calculus, butit represents a meta-mathematical statement. Thequestion is, which one? The reader should first recallthat the expression ‘sub (y, 13, y)’ designates a number.

88 Gödel’s Proof

25 It is of utmost importance to recognize that ‘sub (y, 13, y)’,though it is an expression in formalized arithmetic, is not a formulabut rather a name-function for identifying a number (see explanatoryfootnote 24). The number so identified, however, is the Gödelnumber of a formula—of the formula obtained from the formulawith Gödel number y, by substituting for the variable ‘y’ the numeralfor y.

26 This statement can be expanded still further to read: ‘Theformula [whose Gödel number is the number of the formula]obtained from the formula with Gödel number y, by substituting forthe variable with Gödel number 13 the numeral for y, is notdemonstrable’.

The reader may be puzzled by the fact that, in themetamathematical statement ‘The formula with Gödel number sub(y, 13, y) is not demonstrable’, the expression ‘sub (y, 13, y)’ doesnot appear within quotation marks, although it has been repeatedlystated in the text that ‘sub (y, 13, y)’ is an expression. The pointinvolved hinges once more on the distinction between using anexpression to talk about what the expression designates (in whichcase the expression is not placed within quotation marks) and talkingabout the expression itself (in which case we must use a name forthe expression and, in conformity with the convention forconstructing such names, must place the expression within quotationmarks). An example will help. ‘7+5’ is an expression whichdesignates a number; on the other hand, 7+5 is a number, and notan expression. Similarly, ‘sub (243,000,000, 13, 243,000,000)’ is anexpression which designates the Gödel number of a formula (seeTable 4); but sub (243,000,000, 13, 243,000,000) is the Gödel numberof a formula, and is not an expression.

This number is the Gödel number of the formulaobtained from the formula with Gödel number y, bysubstituting for the variable with Gödel number 13 thenumeral for y.25 It will then be evident that the formulaof line (1) represents the meta-mathematical statement:‘The formula with Gödel number sub (y, 13, y) is notdemonstrable’.26


But, since the formula of line (1) belongs to thearithmetical calculus, it has a Gödel number that canactually be calculated. Suppose the number to be n. Wenow substitute for the variable with Gödel number 13(i.e., for the variable ‘y’) in the formula of line (1) thenumeral for n. A new formula is then obtained, whichwe shall call ‘G’ (after Gödel) and display under thatlabel: (G) (x)~Dem (x, sub (n, 13, n)) Formula G is the special case we promised toconstruct.

Now, this formula occurs within the arithmeticalcalculus, and therefore must have a Gödel number. Whatis the number? A little reflection shows that it is sub (n,13, n). To grasp this, we must recall that sub (n, 13, n)is the Gödel number of the formula that is obtainedfrom the formula with Gödel number n by substitutingfor the variable with Gödel number 13 (i.e., for thevariable ‘y’) the numeral for n. But the formula G hasbeen obtained from the formula with Gödel number n(i.e., from the formula displayed on line (1)) bysubstituting for the variable ‘y’ occurring in it thenumeral for n. Hence the Gödel number of C is in factsub (n, 13, n).

But we must also remember that the formula G is themirror image within the arithmetical calculus of themeta-mathematical statement: ‘The formula with Gödelnumber sub (n, 13, n) is not demonstrable’. It follows

90 Gödel’s Proof

that the arithmetical formula ‘(x)~Dem (x, sub (n, 13,n))’ represents in the calculus the metamathematicalstatement: ‘The formula ‘(x)~Dem (x, sub (n, 13, n))’ isnot demonstrable’. In a sense, therefore, this arithmeticalformula G can be construed as asserting of itself that itis not demonstrable.

(ii) We come to the next step, the proof that G is notformally demonstrable. Gödel’s demonstrationresembles the development of the Richard Paradox, butstays clear of its fallacious reasoning.27 The argument isrelatively unencumbered. It proceeds by showing thatif the formula G were demonstrable then its formal

27 It may be useful to make explicit the resemblance as well asthe dissimilarity of the present argument to that used in the RichardParadox. The main point to observe is that the formula C is notidentical with the meta-mathematical statement with which it isassociated, but only represents (or mirrors) the latter within thearithmetical calculus. In the Richard Paradox (as explained on p.63 above) the number n is the number associated with a certainmeta-mathematical expression. In the Gödel construction, thenumber n is associated with a certain arithmetical formula belongingto the formal calculus, though this arithmetical formula in factrepresents a meta-mathematical statement. (The formula representsthis statement, because the meta-mathematics of arithmetic has beenmapped onto arithmetic.) In developing the Richard Paradox, thequestion is asked whether the number n possesses the meta-mathematical property of being Richardian. In the Gödelconstruction, the question asked is whether the number sub (n, 13,n) possesses a certain arithmetical property—namely, thearithmetical property expressed by the formula ‘(x)~ Dem (x, z)’.There is therefore no confusion in the Gödel construction betweenstatements within arithmetic and statements about arithmetic, suchas occurs in the Richard Paradox.


contradictory (namely, the formula ‘~(x)~Dem (x, sub(n, 13, n))’) would also be demonstrable; and, conversely,that if the formal contradictory of G were demonstrablethen G itself would also be demonstrable. Thus we have:G is demonstrable if, and only if, ~ G is demonstrable.28

But as we noted earlier, if a formula and its formalnegation can both be derived from a set of axioms, the

28 This is not what Gödel actually proved; and the statement inthe text, an adaptation of a theorem obtained by J.Barkley Rosser in1936, is used for the sake of simplicity in exposition. What Gödelactually showed is that if G is demonstrable then ~ G is demonstrable(so that arithmetic is then inconsistent); and if ~ G is demonstrablethen arithmetic is �-inconsistent. What is �-inconsistency? Let ‘P’be some arithmetical predicate. Then arithmetic would be�-inconsistent if it were possible to demonstrate both the formula‘($x)P(x)’ (i.e., ‘There is at least one number that has the propertyP’) and also each of the infinite set of formulas ‘~P(0)’, ‘~P(1)’, ‘~P(2)’,etc. (i.e., ‘0 does not have the property P’, ‘1 does not have theproperty P’, ‘2 does not have the property P’, and so on). A littlereflection shows that if a calculus is inconsistent then it is also�-inconsistent; but the converse does not necessarily hold: a systemmay be �-inconsistent without being inconsistent. For a system tobe inconsistent, both ‘($x)P(x)’ and ‘(x)~P(x)’ must be demonstrable.However, although if a system is �-inconsistent both ‘($x) P(x)’ andeach of the infinite set of formulas ‘~P(0)’, ‘~P(1)’, ‘~P(2)’, etc., aredemonstrable, the formula ‘(x)~ P(x)’ may nevertheless not bedemonstrable, so that the system is not inconsistent.

We outline the first part of Gödel’s argument that if C isdemonstrable then ~ G is demonstrable. Suppose the formula Gwere demonstrable. Then there must be a sequence of formulaswithin arithmetic that constitutes a proof for G. Let the Gödel numberof this proof be k. Accordingly, the arithmetical relation designatedby ‘Dem (x, z)’ must hold between k, the Gödel number of the

92 Gödel’s Proof

axioms are not consistent. Whence, if the axioms ofthe formalized system of arithmetic are consistent,neither the formula G nor its negation is demonstrable.In short, if the axioms are consistent, G is formallyundecidable—in the precise technical sense thatneither G nor its contradictory can be formally deducedfrom the axioms.

(iii) This conclusion may not appear at first sight tobe of capital importance. Why is it so remarkable, itmay be asked, that a formula can be constructed withinarithmetic which is undecidable? There is a surprise instore which illuminates the profound implications ofthis result. For, although the formula G is undecidableif the axioms of the system are consistent, it can

A somewhat analogous but more complicated argument isrequired to show that if ~ G is demonstrable then G is alsodemonstrable. We shall not attempt to outline it.

proof, and sub (n, 13, n), the Gödel number of G, which is to saythat ‘Dem (k, sub (n, 13, n))’ must be a true arithmetical formula.However, it can be shown that this arithmetical relation is of suchtype that, if it holds between a definite pair of numbers, the formulathat expresses this fact is demonstrable. Consequently, the formula‘Dem (k, sub (n, 13, n))’ is not only true, but also formallydemonstrable; that is, the formula is a theorem. But, with the helpof the Transformation Rules in elementary logic, we can immediatelyderive from this theorem the formula ‘~(x)~Dem (x, sub (n, 13, n))’.We have therefore shown that if the formula G is demonstrable itsformal negation is demonstrable. It follows that if the formal systemis consistent the formula G is not demonstrable.


nevertheless be shown by meta-mathematical reasoningthat G is true. That is, it can be shown that G formulatesa complex but definite numerical property whichnecessarily holds of all integers—just as the formula‘(x)~(x+3=2)’ (which, when it is interpreted in the usualway, says that no cardinal number, when added to 3,yields a sum equal to 2) expresses another, likewisenecessary (though much simpler) property of allintegers. The reasoning that validates the truth of theundecidable formula G is straightforward. First, on theassumption that arithmetic is consistent, the meta-mathematical statement ‘The formula ‘(x) ~Dem (x, sub(n, 13, n))’ is not demonstrable’ has been proven true.Second, this statement is represented within arithmeticby the very formula mentioned in the statement. Third,we recall that metamathematical statements have beenmapped onto the arithmetical formalism in such a waythat true metamathematical statements correspond totrue arithmetical formulas. (Indeed, the setting up ofsuch a correspondence is the raison d’être of themapping; as, for example, in analytic geometry where,by virtue of this process, true geometric statementsalways correspond to true algebraic statements.) Itfollows that the formula G, which corresponds to a truemeta-mathematical statement, must be true. It shouldbe noted, however, that we have established anarithmetical truth, not by deducing it formally from theaxioms of arithmetic, but by a meta-mathematicalargument.

(iv) We now remind the reader of the notion of

94 Gödel’s Proof

“completeness” introduced in the discussion of thesentential calculus. It was explained that the axiomsof a deductive system are “complete” if every truestatement that can be expressed in the system isformally deducible from the axioms. If this is not thecase, that is, if not every true statement expressible inthe system is deducible, the axioms are “incomplete.”But, since we have just established that G is a trueformula of arithmetic not formally deducible withinit, it follows that the axioms of arithmetic areincomplete—on the hypothesis, of course, that theyare consistent. Moreover, they are essentiallyincomplete: even if G were added as a further axiom,the augmented set would still not suffice to yieldformally all arithmetical truths. For, if the initialaxioms were augmented in the suggested manner,another true but undecidable arithmetical formulacould be constructed in the enlarged system; such aformula could be constructed merely by repeating inthe new system the procedure used originally forspecifying a true but undecidable formula in the initialsystem. This remarkable conclusion holds, no matterhow often the initial system is enlarged. We are thuscompelled to recognize a fundamental limitation inthe power of the axiomatic method. Against previousassumptions, the vast continent of arithmetical truthcannot be brought into systematic order by laying downonce for all a set of axioms from which every truearithmetical statement can be formally derived.

(v) We come to the coda of Gödel’s amazing


intellectual symphony. The steps have been traced bywhich he grounded the meta-mathematical statement:‘If arithmetic is consistent, it is incomplete’. But it canalso be shown that this conditional statement taken asa whole is represented by a demonstrable formulawithin formalized arithmetic.

This crucial formula can be easily constructed. Aswe explained in Section V, the meta-mathematicalstatement ‘Arithmetic is consistent’ is equivalent to thestatement ‘There is at least one formula of arithmeticthat is not demonstrable’. The latter is represented inthe formal calculus by the following formula, whichwe shall call ‘A’: (A) ($y)(x)~Dem (x, y) In words, this says: ‘There is at least one number ysuch that, for every number x, x does not stand in therelation Dem to y’. Interpreted meta-mathematically, theformula asserts: ‘There is at least one formula ofarithmetic for which no sequence of formulas constitutesa proof’. The formula A therefore represents theantecedent clause of the meta-mathematical statement‘If arithmetic is consistent, it is incomplete’. On theother hand, the consequent clause in this statement—namely, ‘It [arithmetic] is incomplete’—follows directlyfrom ‘There is a true arithmetical statement that is notformally demonstrable in arithmetic’; and the latter, asthe reader will recognize, is represented in thearithmetical calculus by an old friend, the formula G.

96 Gödel’s Proof

Accordingly, the conditional meta-mathematicalstatement ‘If arithmetic is consistent, it is incomplete’is represented by the formula:

($y)(x)~Dem (x, y)�(x)~Dem (x, sub (n, 13, n)) which, for the sake of brevity, can be symbolized by ‘A�G’.(This formula can be proved formally demonstrable, butwe shall not in these pages undertake the task.)

We now show that the formula A is not demonstrable.For suppose it were. Then, since A�G is demonstrable,by use of the Rule of Detachment the formula G wouldbe demonstrable. But, unless the calculus isinconsistent, G is formally undecidable, that is, notdemonstrable. Thus if arithmetic is consistent, theformula A is not demonstrable.

What does this signify? The formula A representsthe meta-mathematical statement ‘Arithmetic isconsistent’. If, therefore, this statement could beestablished by any argument that can be mapped ontoa sequence of formulas which constitutes a proof in thearithmetical calculus, the formula A would itself bedemonstrable. But this, as we have just seen, isimpossible, if arithmetic is consistent. The grand finalstep is before us: we must conclude that if arithmetic isconsistent its consistency cannot be established by anymeta-mathematical reasoning that can be representedwithin the formalism of arithmetic!

This imposing result of Gödel’s analysis should notbe misunderstood: it does not exclude a meta-mathematical proof of the consistency of arithmetic.


What it excludes is a proof of consistency that can bemirrored by the formal deductions of arithmetic.29

Meta-mathematical proofs of the consistency ofarithmetic have, in fact, been constructed, notably byGerhard Gentzen, a member of the Hilbert school, in1936, and by others since then.30 These proofs are ofgreat logical significance, among other reasons becausethey propose new forms of meta-mathematicalconstructions, and because they thereby help makeclear how the class of rules of inference needs to beenlarged if the consistency of arithmetic is to beestablished. But these proofs cannot be representedwithin the arithmetical calculus; and, since they arenot finitistic, they do not achieve the proclaimedobjectives of Hilbert’s original program.

29 The reader may be helped on this point by the reminder that,similarly, the proof that it is impossible to trisect an arbitrary anglewith compass and straight-edge does not mean that an angle cannotbe trisected by any means whatever. On the contrary, an arbitraryangle can be trisected if, for example, in addition to the use ofcompass and straight-edge, one is permitted to employ a fixeddistance marked on the straightedge.

30 Gentzen’s proof depends on arranging all the demonstrationsof arithmetic in a linear order according to their degree of “simplicity.”The arrangement turns out to have a pattern that is of a certain“transfinite ordinal” type. (The theory of transfinite ordinal numberswas created by the German mathematician Georg Cantor in thenineteenth century.) The proof of consistency is obtained by applyingto this linear order a rule of inference called “the principle oftransfinite induction.” Gentzen’s argument cannot be mapped ontothe formalism of arithmetic. Moreover, although most students donot question the cogency of the proof, it is not finitistic in the senseof Hilbert’s original stipulations for an absolute proof of consistency.

98

VIIIConcluding Reflections

The import of Gödel’s conclusions is far-reaching,though it has not yet been fully fathomed. Theseconclusions show that the prospect of finding for everydeductive system (and, in particular, for a system inwhich the whole of arithmetic can be expressed) anabsolute proof of consistency that satisfies the finitisticrequirements of Hilbert’s proposal, though not logicallyimpossible, is most unlikely.31 They show also that thereis an endless number of true arithmetical statementswhich cannot be formally deduced from any given setof axioms by a closed set of rules of inference. It followsthat an axiomatic approach to number theory, for

31 The possibility of constructing a finitistic absolute proof ofconsistency for arithmetic is not excluded by Gödel’s results. Gödelshowed that no such proof is possible that can be representedwithin arithmetic. His argument does not eliminate the possibilityof strictly finitistic proofs that cannot be represented withinarithmetic. But no one today appears to have a clear idea of whata finitistic proof would be like that is not capable of formulationwithin arithmetic

Concluding Reflections 99

example, cannot exhaust the domain of arithmeticaltruth. It follows, also, that what we understand by theprocess of mathematical proof does not coincide withthe exploitation of a formalized axiomatic method. Aformalized axiomatic procedure is based on an initiallydetermined and fixed set of axioms and transformationrules. As Gödel’s own arguments show, no antecedentlimits can be placed on the inventiveness ofmathematicians in devising new rules of proof.Consequently, no final account can be given of theprecise logical form of valid mathematicaldemonstrations. In the light of these circumstances,whether an all-inclusive definition of mathematical orlogical truth can be devised, and whether, as Gödelhimself appears to believe, only a thoroughgoingphilosophical “realism” of the ancient Platonic typecan supply an adequate definition, are problems stillunder debate and too difficult for further considerationhere.32

32 Platonic realism takes the view that mathematics does notcreate or invent its “objects,” but discovers them as Columbusdiscovered America. Now, if this is true, the objects must in somesense “exist” prior to their discovery. According to Platonic doctrine,the objects of mathematical study are not found in the spatio-temporal order. They are disembodied eternal Forms or Archetypes,which dwell in a distinctive realm accessible only to the intellect.On this view, the triangular or circular shapes of physical bodiesthat can be perceived by the senses are not the proper objects ofmathematics. These shapes are merely imperfect embodiments ofan indivisible “perfect” Triangle or “perfect” Circle, which isuncreated, is never fully manifested by material things, and can be

100 Gödel’s Proof

Gödel’s conclusions bear on the question whether acalculating machine can be constructed that wouldmatch the human brain in mathematical intelligence.Today’s calculating machines have a fixed set ofdirectives built into them; these directives correspondto the fixed rules of inference of formalized axiomaticprocedure. The machines thus supply answers toproblems by operating in a step-by-step manner, eachstep being controlled by the built-in directives. But, asGödel showed in his incompleteness theorem, there areinnumerable problems in elementary number theorythat fall outside the scope of a fixed axiomatic method,and that such engines are incapable of answering,however intricate and ingenious their built-inmechanisms may be and however rapid their operations.Given a definite problem, a machine of this type mightbe built for solving it; but no one such machine can bebuilt for solving every problem. The human brain may,to be sure, have built-in limitations of its own, and theremay be mathematical problems it is incapable of solving.But, even so, the brain appears to embody a structure of

grasped solely by the exploring mind of the mathematician. Gödelappears to hold a similar view when he says, “Classes and conceptsmay…be conceived as real objects…existing independently of ourdefinitions and constructions. It seems to me that the assumptionof such objects is quite as legitimate as the assumption of physicalbodies and there is quite as much reason to believe in their existence”(Kurt Gödel, “Russell’s Mathematical Logic,” in The Philosophy ofBertrand Russell (ed. Paul A.Schilpp, Evanston and Chicago, 1944),p. 137).

Concluding Reflections 101

rules of operation which is far more powerful than thestructure of currently conceived artificial machines.There is no immediate prospect of replacing the humanmind by robots.

Gödel’s proof should not be construed as an invitationto despair or as an excuse for mystery-mongering. Thediscovery that there are arithmetical truths which cannotbe demonstrated formally does not mean that there aretruths which are forever incapable of becoming known,or that a “mystic” intuition (radically different in kindand authority from what is generally operative inintellectual advances) must replace cogent proof. It doesnot mean, as a recent writer claims, that there are“ineluctable limits to human reason.” It does mean thatthe resources of the human intellect have not been, andcannot be, fully formalized, and that new principles ofdemonstration forever await invention and discovery.We have seen that mathematical propositions whichcannot be established by formal deduction from a givenset of axioms may, nevertheless, be established by“informal” meta-mathematical reasoning. It would beirresponsible to claim that these formally indemon-strable truths established by meta-mathematical argu-ments are based on nothing better than bare appeals tointuition.

Nor do the inherent limitations of calculatingmachines imply that we cannot hope to explain livingmatter and human reason in physical and chemicalterms. The possibility of such explanations is neitherprecluded nor affirmed by Gödel’s incompleteness

102 Gödel’s Proof

theorem. The theorem does indicate that the structureand power of the human mind are far more complexand subtle than any non-living machine yet envisaged.Gödel’s own work is a remarkable example of suchcomplexity and subtlety. It is an occasion, not fordejection, but for a renewed appreciation of the powersof creative reason.

103

Appendix

Notes

1. (page 11) It was not until 1899 that the arithmetic ofcardinal numbers was axiomatized, by the Italianmathematician Giuseppe Peano. His axioms are five innumber. They are formulated with the help of threeundefined terms, acquaintance with the latter beingassumed. The terms are: ‘number’, ‘zero’, and‘immediate successor of’. Peano’s axioms can be statedas follows:

1. Zero is a number.2. The immediate successor of a number is

a number.3. Zero is not the immediate successor of a

number.4. No two numbers have the same

immediate successor.5. Any property belonging to zero, and also

to the immediate successor of every numberthat has the property, belongs to all numbers.

The last axiom formulates what is often called the“principle of mathematical induction.”

104 Appendix

2. (page 39) The reader may be interested in seeing afuller account than the text provides of the logicaltheorems and rules of inference tacitly employed evenin elementary mathematical demonstrations. We shallfirst analyze the reasoning that yields line 6 in Euclid’sproof, from lines 3, 4, and 5.

We designate the letters ‘p’, ‘q’, and ‘r’ as “sententialvariables,” because sentences may be substituted forthem. Also, to economize space, we write conditionalstatements of the form ‘if p then q’ as ‘p�q’; and we callthe expression to the left of the horseshoe sign ‘�’ the“antecedent,” and the expression to the right of it the“consequent.” Similarly, we write ‘pVq’ as short for thealternative form ‘either p or q’.

There is a theorem in elementary logic which reads:

(P�r)�[(q�r)�((pVq)�r)]

It can be shown that this formulates a necessary truth.The reader will recognize that this formula states morecompactly what is conveyed by the following muchlonger statement:

If (if p then r), then [if (if q then r) then (if (either por q) then r)]

As pointed out in the text, there is a rule of inference inlogic called the Rule of Substitution for SententialVariables. According to this Rule, a sentence S2 followslogically from a sentence S1 which contains sententialvariables, if the former is obtained from the latter byuniformly substituting any sentences for the variables.If we apply this rule to the theorem just mentioned,

Notes 105

substituting ‘y is prime’ for ‘p’, ‘y is composite’ for ‘q’,and ‘x is not the greatest prime’ for ‘r’, we obtain thefollowing:

(y is prime � x is not the greatest prime)

� [(y is composite � x is not the greatest prime)� ((y is prime V y is composite) � x is not the

greatest prime)]

The reader will readily note that the conditionalsentence within the first pair of parentheses (it occurson the first line of this instance of the theorem) simplyduplicates line 3 of Euclid’s proof. Similarly, theconditional sentence within the first pair ofparentheses inside the square brackets (it occurs asthe second line of the instance of the theorem)duplicates line 4 of the proof. Also, the alternativesentence inside the square brackets duplicates line 5of the proof.

We now make use of another rule of inference knownas the Rule of Detachment (or “Modus Ponens”). Thisrule permits us to infer a sentence S2 from two othersentences, one of which is S1 and the other, S1�S2. Weapply this Rule three times: first, using line 3 of Euclid’sproof and the above instance of the logical theorem;next, the result obtained by this application and line 4of the proof; and, finally, this latest result of theapplication and line 5 of the proof. The outcome is line6 of the proof.

The derivation of line 6 from lines 3, 4, and 5 thusinvolves the tacit use of two rules of inference and atheorem of logic. The theorem and rules belong to theelementary part of logical theory, the sentential calculus.

106 Appendix

This deals with the logical relations between statementscompounded out of other statements with the help ofsentential connectives, of which ‘�’ and ‘V’ areexamples. Another such connective is the conjunction‘and’, for which the dot ‘.’ is used as a shorthand form;thus the conjunctive statement ‘p and q’ is written as‘p.q’. The sign ‘~’ represents the negative particle ‘not’;thus ‘not-p’ is written as ‘~ p’.

Let us examine the transition in Euclid’s proof fromline 6 to line 7. This step cannot be analyzed with thehelp of the sentential calculus alone. A rule ofinference is required which belongs to a more advancedpart of logical theory—namely, that which takes noteof the internal complexity of statements embodyingexpressions such as ‘all’, ‘every’, ‘some’, and theirsynonyms. These are traditionally called quantifiers,and the branch of logical theory that discusses theirrole is the theory of quantification.

It is necessary to explain some of the notationemployed in this more advanced sector of logic, as apreliminary to analyzing the transition in question. Inaddition to the sentential variables for which sentencesmay be substituted, we must consider the category of“individual variables,” such as ‘x’, ‘y’, ‘z’, etc., forwhich the names of individuals can be substituted.Using these variables, the universal statement ‘Allprimes greater than 2 are odd’ can be rendered: ‘Forevery x, if x is a prime greater than 2, then x is odd’.The expression ‘for every x’ is called the universalquantifier, and in current logical notation isabbreviated by the sign ‘(x)’. The universal statementmay therefore be written:

Notes 107

(x)(x is a prime greater than 2�x is odd) Furthermore, the “particular” (or “existential”)statement ‘Some integers are composite’ can be renderedby ‘There is at least one x such that x is an integer andx is composite’. The expression ‘there is at least one x’is called the existential quantifier, and is currentlyabbreviated by the notation ‘($x)’. The existentialstatement just mentioned can be transcribed:

($x)(x is an integer.x is composite) It is now to be observed that many statements implicitlyuse more than one quantifier, so that in exhibiting theirtrue structure several quantifiers must appear. Beforeillustrating this point, let us adopt certain abbreviationsfor what are usually called predicate expressions or,more simply, predicates. We shall use ‘Pr (x)’ as shortfor ‘x is a prime number’; and ‘Gr (x, z)’ as short for ‘xis greater than z’. Consider the statement: ‘x is thegreatest prime’. Its meaning can be made more explicitby the following locution: ‘x is a prime, and, for everyz which is a prime but different from x, x is greaterthan z’. With the help of our various abbreviations, thestatement ‘x is the greatest prime’ can be written:

Pr (x).(z) [(Pr (z).~(x=z))�Gr (x, z)]

Literally, this says: ‘x is a prime and, for every z, if z isa prime and z is not equal to x then x is greater than z’.We recognize in the symbolic sequence a formal,painfully explicit rendition of the content of line 1 inEuclid’s proof.

108 Appendix

Next, consider how to express in our notation thestatement ‘x is not the greatest prime’, which appearsas line 6 of the proof. This can be presented as:

Pr (x).($z) [Pr (z).Gr (z, x)] Literally, it says: ‘x is a prime and there is at least onez such that z is a prime and z is greater than x’.

Finally, the conclusion of Euclid’s proof, line 7, whichasserts that there is no greatest prime, is symbolicallytranscribed by:

(x) [Pr (x)�($z)(Pr (z).Gr (z, x))] which says: ‘For every x, if x is a prime, there is at leastone z such that z is a prime and z is greater than x’. Thereader will observe that Euclid’s conclusion implicitlyinvolves the use of more than one quantifier.

We are ready to discuss the step from Euclid’s line 6to line 7. There is a theorem in logic which reads:

(p.q)�(p�q)

or when translated, ‘If both p and q, then (if p thenq)’. Using the Rule of Substitution, and substituting‘Pr (x)’ for ‘p’, and ‘($z) [Pr (z).Gr (z, x)]’ for ‘q’, weobtain:

(Pr (x).($z) [Pr (z).Gr (z, x)]) �(Pr (x)�($z) [Pr (z).Gr (z, x)])

The antecedent (first line) of this instance of the theorem

Notes 109

simply duplicates line 6 of Euclid’s proof; if we applythe Rule of Detachment, we get

(Pr (x)�($z) [Pr (z).Gr (z, x)]) According to a Rule of Inference in the logical theoryof quantification, a sentence S2 having the form‘(x)(…x…)’ can always be inferred from a sentence S1

having the form ‘(…x…)’. In other words, the sentencehaving the quantifier ‘(x)’ as a prefix can be derivedfrom the sentence that does not contain the prefix butis like the former in other respects. Applying this ruleto the sentence last displayed, we have line 7 ofEuclid’s proof.

The moral of our story is that the proof of Euclid’stheorem tacitly involves the use not only of theoremsand rules of inference belonging to the sententialcalculus, but also of a rule of inference in the theory ofquantification.

3. (page 54) The careful reader may demur at this point.His reservations may run something like this. Theproperty of being a tautology has been defined in notionsof truth and falsity. Yet these notions obviously involvea reference to something outside the formal calculus.Therefore, the procedure mentioned in the text in effectoffers an interpretation of the calculus, by supplying amodel for the system. This being so, the authors havenot done what they promised, namely, to define aproperty of formulas in terms of purely structuralfeatures of the formulas themselves. It seems that thedifficulty noted in Section II of the text—that proofs ofconsistency which are based on models, and which

110 Appendix

argue from the truth of axioms to their consistency,merely shift the problem—has not, after all, beensuccessfully outflanked. Why then call the proof“absolute” rather than relative?

The objection is well taken when directed againstthe exposition in the text. But we adopted this form soas not to overwhelm the reader unaccustomed to ahighly abstract presentation resting on an intuitivelyopaque proof. Because more venturesome readers maywish to be exposed to the real thing, to see anunprettified definition that is not open to the criticismsin question, we shall supply it.

Remember that a formula of the calculus is eitherone of the letters used as sentential variables (we willcall such formulas elementary) or a compound of theseletters, of the signs employed as sentential connectives,and of the parentheses. We agree to place eachelementary formula in one of two mutually exclusiveand exhaustive classes K1 and K2. Formulas that are notelementary are placed in these classes pursuant to thefollowing conventions:

i) A formula having the form S1VS2 is placedin class K2 if both S1 and S2 are in K2;otherwise, it is placed in K1.

ii) A formula having the form S1�S2 isplaced in K2, if S1 is in K1 and S2 is in K2;otherwise, it is placed in K1.

iii) A formula having the form S1.S2 is

placed in K1, if both S1 and S2 are in K1;otherwise, it is placed in K2.

Notes 111

iv) A formula having the form ~S is placedin K2, if S is in K1; otherwise, it is placed inK1.

We then define the property of being tautologous: aformula is a tautology if, and only if, it falls in theclass K1 no matter in which of the two classes itselementary constituents are placed. It is clear that theproperty of being a tautology has now been describedwithout using any model or interpretation for thesystem. We can discover whether or not a formula is atautology simply by testing its structure by the aboveconventions.

Such an examination shows that each of the fouraxioms is a tautology. A convenient procedure is toconstruct a table that lists all the possible ways inwhich the elementary constituents of a given formulacan be placed in the two classes. From this list we candetermine, for each possibility, to which class the non-elementary component formulas of the given formulabelong, and to which class the entire formula belongs.Take the first axiom. The table for it consists of threecolumns, each headed by one of the elementary ornon-elementary component formulas of the axiom, aswell as by the axiom itself. Under each heading isindicated the class to which the particular itembelongs, for each of the possible assignments of theelementary constituents to the two classes. The tableis as follows:

p (p V p) (p V p) � p

K1 K1 K1

K2 K2 K1

112 Appendix

The first column mentions the possible ways ofclassifying the sole elementary constituent of the axiom.The second column assigns the indicated non-elementary component to a class, on the basis ofconvention (i). The last column assigns the axiom itselfto a class, on the basis of convention (ii). The finalcolumn shows that the first axiom falls in class K1,irrespective of the class in which its sole elementaryconstituent is placed. The axiom is therefore a tautology.

For the second axiom, the table is:

p q (p V q) p � (p V q)K1 K1 K1 K1

K1 K2 K1 K1

K2 K1 K1 K1

K2 K2 K2 K1

The first two columns list the four possible ways ofclassifying the two elementary constituents of the axiom.The second column assigns the non-elementarycomponent to a class, on the basis of convention (i).The last column does this for the axiom, on the basis ofconvention (ii). The final column again shows that thesecond axiom falls in class K1 for each of the fourpossible ways in which the elementary constituents canbe classified. The axiom is therefore a tautology. In asimilar way the remaining two axioms can be shown tobe tautologies.

We shall also give the proof that the property of beinga tautology is hereditary under the Rule of Detachment.(The proof that it is hereditary under the Rule ofSubstitution will be left to the reader.) Assume that any

Notes 113

two formulas S1 and S1�S2 are both tautologies; we mustshow that in this case S2 is a tautology. Suppose S2

were not a tautology. Then, for at least one classificationof its elementary constituents, S2 will fall in K2. But, byhypothesis, S1 is a tautology, so that it will fall in K1 forall classifications of its elementary constituents—and,in particular, for the classification which requires theplacing of S2 in K2. Accordingly, for this latterclassification, S1�S2 must fall in K2, because of thesecond convention. However, this contradicts thehypothesis that S1�S2 is a tautology. In consequence,S2 must be a tautology, on pain of this contradiction.The property of being a tautology is thus transmittedby the Rule of Detachment from the premises to theconclusion derivable from them by this Rule.

One final comment on the definition of a tautologygiven in the text. The two classes K1 and K2 used in thepresent account may be construed as the classes of trueand of false statements, respectively. But the account,as we have just seen, in no way depends on such aninterpretation, even if the exposition is more easilygrasped when the classes are understood in this way.

115

Brief Bibliography CARNAP, RUDOLF Logical Syntax of Language, New York,

1937.FINDLAY, J. “Goedelian sentences: a non-numerical

approach,” Mind, Vol. 51 (1942), pp.259–265.

GÖDEL, KURT “Über formal unentscheidbare Sätze derPrincipia Mathematica und verwandterSysteme I,” Monatshefte für Mathematikund Physik, Vol. 38 (1931), pp. 173–198.

KLEENE, S.C. Introduction to Metamathematics, NewYork, 1952.

LADRIÈRE, JEAN Les Limitations Internes des Formalismes,Louvain and Paris, 1957.

MOSTOWSKI, A. Sentences Undecidable in FormalizedArithmetic, Amsterdam, 1952.

QUINE, W.V.O. Methods of Logic, New York, 1950.ROSSER, BARKLEY “An informal exposition of proofs of Gödel’s

theorems and Church’s theorem,” Journalof Symbolic Logic, Vol. 4 (1939), pp.53–60.

TURING, A.M. “Computing machinery and intelligence,”Mind, Vol. 59 (1950), pp. 433–460.

WEYL, HERMANN Philosophy of Mathematics and NaturalScience, Princeton, 1949.

WILDER, R.L. Introduction to the Foundations ofMathematics, New York, 1952.

117

absolute proofs of consistency,26–33, 45–56, 109–113

antinomies, 23–24, 60–63, 85Archimedes, 5Aristotle, 39, 40arithmetic: incompleteness of, 6,

86, 94–95; consistency of, 58,95–97, 98

arithmetization: of mathematics,42; of the formal calculus ofarithmetic, 68–74; of meta-mathematics, 76–84; andmapping, 76–77

axiomatic method, 4–5; limita-tions of, 6, 58–59, 99

axioms: meaning of, 4; of thesentential calculus, 49; ofarithmetic, 103

Bolyai, Janos, 10Boole, George, 40, 41, 42

calculating machines and humanintelligence, 100

calculus and formalization,26–27, 33

Cantor, Georg, 23, 97nclass: notion of, 16; mathemati-

cal theory of, 23completeness, 55–56, 94–95consistency: meaning of, 6; pro-

blem of, 8–25; and truth, 14; ofEuclidean geometry, 14, 18; ofnon-Euclidean geometry, 15,17–19; relative proofs of, 15–23;absolute proofs of, 26–33; of thesentential calculus, 45–56,109–113; formalized definitionof, 50–51; of arithmetic, 58,95–97, 98

Index

demonstration, definition of, 46Descartes, René, 7descriptive predicates, 12

elementary signs, 46, 69essential incompleteness, 86, 94Euclid, 9, 10, 14, 15, 18, 19, 20,

21, 39, 56; his proof that thereis no largest prime number,36–38, 104–109

finite models, 21–22, 25finitistic proofs, 33, 34nformal logic: its codification,

37– 44formalization: of deductive sys-

tems, 12–13, 26–27; of thesentential calculus, 45–50,109–111; limits of, 101

formation rules, 45, 47formula in a calculus, 29, 33, 45,

47Frege, Gottlob, 42, 43, 44

Gauss, Karl F., 10Gentzen, Gerhard, 97Gödel, Kurt, 3, 4, 6, 10, 26, 37,

45, 56, 57, 58, 59, 66, 67, 68,69, 70, 71, 72, 73, 74, 75, 76,77, 78, 79, 80, 81, 82, 85, 86,87, 88, 89, 90, 91, 94, 98, 100,101, 102; his Platonic realism,99

Gödel numbering, 68–76Gödel sentence, 89Goldbach’s conjecture, 59

hereditary property, 51–52, 113Hilbert, David, 12, 13, 20, 21,26, 28, 32, 33, 34n, 36, 57,58, 64, 97, 98, 99

118 Index

implicit definition, 13nincompleteness, 56, 86; of arith-

metic, 6, 86, 94–95; essential,58–59, 86

inference, rules of, 38, 39intuitive knowledge, 14, 23, 101

Kant, Immanuel, 40

Lobachevsky, Nikolai, 10logical constants, 46

mapping: idea of, 57–67; of meta-mathematics, 76–84, 93

mathematical induction, axiomof, 103

mathematics: as science of quan-tity, 11; pure and applied, 12

meta-mathematics, 28–32; andthe theory of chess, 34–35

models: and proofs of consist-ency, 15–21, 25, 109; finiteand non-finite, 22, 25

modus ponens, 48, 105

name-forms, 85n, 88nnames of expressions, 30n, 31nnon-Euclidean geometry, consist-

ency of, 10, 15–18non-finite models, 22, 25number, definition of, 42–43numeral and number, 83numerical variable, 70

omega-consistency, 91n

Pappus, theorem of, 64–65paradoxes, 23–24, 60–63parallel axiom: Euclidean, 9–10;

non-Euclidean, 15, 18Peano, Giuseppe, 103predicate variable, 71–72primitive formula, 46, 48–49Principia Mathematica, 3, 37,

42, 43, 44, 45, 48, 68n

proof: Hilbert’s theory of, 32–36;definition of, 46, 74n

quantification theory, 38, 109quantifiers, 88, 106, 107quotation marks in names for ex-

pressions, 31n

reduction of arithmetic to logic, 42relative proofs of consistency,

15–23Richard, Jules, 60Richard Paradox, 60–63, 66, 85,

90nRiemann, Bernhard, 10, 15,

17–19Rosser, J.Barkley, 91nrule: of substitution, 39, 47–48,

50, 104; of detachment, 48,50, 105

Russell, Bertrand, 3, 13, 23, 24,37, 42, 43, 44

Russell’s Paradox, 24, 62

self-evidence: of Euclidean ax-ioms, 9–10; and intuition, 14

sentential connectives, 46sentential variable, 71statement-form, 84–85

tautology, 52–54, 109–113theorem, definition of, 46transfinite induction, principle

of, 97ntransfinite ordinals, 97ntransformation rules, 39, 47–48,

50, 104–105

undecidability, 86, 92–93use and mention, 88n, 90n

variables, 29, 39, 46; numerical,70; sentential, 71; predicate,71

Whitehead, Alfred N., 3, 37, 42

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Godels Proof