Gödel's Proof 9780814759035

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Godel's Proof

In 1931 Kur t Godel published a revolutionary paper—one that challenged certai n basi c assumption s underlyin g muc h traditional research in mathematics and logic. Today his exploration of terra incognita has been recognized as one of the major contributions to modern scientific thought. Here i s the first book to present a readabl e explanatio n t o both scholars and non-specialist s of the main ideas, the broa d implications o f Godel' s proof . I t offer s an y educate d perso n with a taste for logic and philosophy the chance to gain genuine insight into a previously inaccessible subject. In this new edition, Pulitzer prize-winning author Douglas R. Hofstadter ha s reviewe d an d update d th e tex t o f thi s classic work, clarifyin g ambiguities , makin g argument s clearer , an d making the text more accessible. He has also added a new Foreword, which reveals his own unique personal connection to this seminal work and the impact it has had on his own professional life, explains the essence of Godel' s proof, and shows how and why Godel's proof remains relevant today.

Godel's Proof Revised Edition

by

Ernest Nagel and James R. Newman Edited and with a New Foreword by Douglas R. Hofstadter

n

N e w Yor k Universit y Pres s

New York and London

NEW YORK UNIVERSITY PRESS New York and London Copyright © 2001 by New York University All rights reserved Library of Congress Cataloging-in-Publication Data Nagel, Ernest, 1901Godel's proof/ b y Ernest Nagel and James R. Newman.—Rev. ed. / edite d and with a new foreword by Douglas R. Hofstadter. p. cm. Includes bibliographical references and index. ISBN 0-8147-5816-9 (acid-free paper) 1. Godel's theorem. I. Newman, James Roy, 1907-1966. II. Hofstadter, Douglas R., 1945- III. Title. QA9.65 .N34 2002 511.3—dc21 200104448 1 New York University Press books are printed on acid-free paper, and their binding materials are chosen for strength and durability. Manufactured in the United States of America 10 9 8 7 6 5 4 3 2 1

to B e r t r a n d Russel l

Content's Foreword to the New Edition by Douglas K Hofstadter ix xxiii Acknowledgments 1 I Introduction II The Problem of Consistency 7 III Absolute Proofs of Consistency 25 IV The Systematic Codification o f Formal Logic 37 V An Example of a Successful Absolute Proof of Consistency 45 VI The Idea of Mapping and Its Use in Mathematic s 57 VII Godel's Proof s 68 A Gode l numberin g 68 B Th e arithmetization o f meta-mathematic s 80 c Th e heart of Godel's argumen t 92 109 m i Concluding Reflection s 114 Appendix: Notes Brief Bibliography !25 Index 127

Vll

Foreword t o th e N e w Editio n by Douglas R. Hofstadter In Augus t 1959 , m y famil y r e t u r n e d t o Stanford , Cali fornia, afte r a yea r i n Geneva . I wa s fourteen , newl y fluent i n French , i n lov e wit h languages , e n t r a n c e d b y writing systems , symbols , a n d th e myster y o f m e a n i n g , a n d b r i m m i n g wit h curiosit y a b o u t mathematic s a n d how th e m i n d works . O n e evening , m y fathe r a n d I wen t t o a bookstor e where I c h a n c e d u p o n a littl e b o o k wit h th e enigmati c title GodeVs Proof. Flipping t h r o u g h it , I sa w man y in triguing figures a n d formulas , a n d wa s particularl y struck b y a footnot e a b o u t quotatio n marks , symbols , a n d symbol s symbolizin g o t h e r symbols . Intuitivel y sensing tha t GodeVs Proof a n d I wer e fate d fo r eac h other, I kne w I h a d t o bu y it . As w e walke d out , m y d a d r e m a r k e d tha t h e h a d taken a philosoph y cours e a t Cit y Colleg e o f Ne w Yor k from o n e o f it s authors , Ernes t Nagel , afte r whic h the y h a d b e c o m e goo d friends . Thi s coincidenc e a d d e d t o the book' s mystique , a n d o n c e h o m e , I voraciously gob bled u p it s ever y word. F r o m star t t o finish, GodeVs Proof resonated wit h m y passions ; suddenl y I foun d mysel f ix

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obsessed wit h trut h a n d falsity , paradoxe s a n d proofs , mappings a n d mirrorings , symbo l manipulatio n a n d symbolic logic , mathematic s a n d metamathematics , th e mystery o f creativ e leap s i n h u m a n thinking , a n d th e mechanization o f mentality . Soon thereafter , m y d a d informe d m e tha t b y chance h e h a d r u n int o Ernes t Nage l o n campus . Pro fessor Nagel , normall y a t Columbia , h a p p e n e d t o b e spending a yea r a t Stanford . Withi n a fe w days , o u r families go t together , a n d I wa s c h a r m e d b y al l fou r Nagels—Ernest a n d Edith , a n d thei r tw o sons , Sand y a n d Bobby , b o t h clos e t o m y age . I was thrille d t o m e e t the a u t h o r o f a b o o k I s o loved , a n d I foun d Ernes t a n d Edit h t o b e enormousl y welcomin g o f m y adoles cent enthusiasm s fo r science , philosophy , music , a n d art. All to o soon , th e Nagels ' sabbatica l yea r h a d nearl y drawn t o a close , b u t befor e the y left , the y warml y invited m e t o spen d a wee k tha t s u m m e r a t thei r cabi n in Vermont . Durin g tha t idylli c stay , Ernes t a n d Edit h came t o represen t fo r m e th e acm e o f civility , generos ity, a n d modesty ; thu s the y remai n i n m y memory , al l these year s later . T h e hig h poin t fo r m e wa s a pai r o f sunny afternoon s whe n Sand y a n d I sa t outdoor s i n a verdant meado w a n d I rea d alou d t o h i m th e entiret y of GodeVs Proof. Wha t a twist y deligh t t o rea d thi s b o o k to th e so n o f o n e o f it s authors ! By mai l ove r th e nex t fe w years , Sand y a n d I ex plored n u m b e r pattern s i n a wa y tha t h a d a p r o f o u n d impact u p o n th e res t o f m y life , a n d p e r h a p s o n hi s

Foreword x i as well . H e wen t o n — k n o w n a s A l e x — t o b e c o m e a mathematics professo r a t th e Universit y o f Wisconsin . Bobby, too , r e m a i n e d a frien d a n d toda y h e — k n o w n as Sidney—i s a physic s professo r a t th e Universit y o f Chicago, a n d w e se e eac h o t h e r wit h grea t pleasur e from tim e t o time . I wis h I coul d sa y that I h a d m e t J a m es Newman . I was given a s a high-schoo l graduatio n presen t hi s magnifi cent four-volum e set , The World of Mathematics, a n d I always a d m i r e d hi s writing styl e a n d hi s lov e fo r math ematics, b u t sa d t o say , we neve r crosse d paths . At Stanfor d I majore d i n mathematics , a n d m y lov e for th e idea s i n Nage l a n d Newman' s b o o k inspire d m e t o tak e a coupl e o f course s i n logi c a n d meta mathematics, b u t I wa s terribl y disappointe d b y thei r aridity. Shortl y thereafter , I e n t e r e d graduat e schoo l in m a t h a n d t h e sam e disillusionmen t recurred . I d r o p p e d o u t o f m a t h a n d t u r n e d t o physics , b u t afte r a few year s I foun d mysel f onc e agai n i n a quagmir e o f abstractness a n d confusion . O n e da y in 1972 , seeking som e relief , I was browsin g in th e universit y bookstor e a n d ra n acros s A Profile of Mathematical Logic b y Howar d D e L o n g — a b o o k tha t h a d nearl y th e sam e electrifyin g effec t o n m e a s GodeVs Proof di d i n 1959 . Thi s luci d treatis e rekindle d i n m e the long-dorman t ember s o f m y lov e fo r logic , meta mathematics, a n d tha t wondrou s tangl e o f issue s I h a d c o n n e c t e d wit h Godel' s t h e o r e m a n d it s proof. Havin g long sinc e los t m y origina l cop y o f Nage l a n d New -

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m a n ' s magica l booklet , I b o u g h t a n o t h e r one—luckily , it wa s stil l i n p r i n t ! — a n d rerea d i t wit h renewe d fasci nation. T h a t summer , takin g a brea k fro m graduat e schoo l a n d drivin g acros s th e continent , I c a m p e d o u t a n d religiously rea d abou t Godel' s work , th e n a t u r e o f rea soning, a n d th e d r e a m o f mechanizin g t h o u g h t a n d consciousness. Withou t p l a n n i n g it , I w o u nd u p i n Ne w York City , a n d th e first peopl e I contacte d wer e m y ol d friends Ernes t a n d Edit h Nagel , wh o serve d a s intellec tual a n d emotiona l m e n t o r s fo r m e . Ove r th e nex t several m o n t h s , I spen t countles s evening s i n thei r apartment, a n d w e ardentl y discusse d man y topics , in cluding, o f course , Godel' s proo f a n d it s ramifications . T h e yea r 197 2 m a r k e d th e beginnin g o f m y ow n intense persona l involvemen t wit h Godel' s t h e o r e m a n d th e ric h spher e o f idea s s u r r o u n d i n g it . Ove r th e next fe w years , I develope d a n idiosyncrati c se t o f ex plorations o n thi s nexu s o f ideas , a n d w o u n d u p callin g it Godel, Escher, Bach: an Eternal Golden Braid. T h e r e i s n o d o u b t tha t th e parent s o f m y sprawlin g volume wer e Nagel a n d Newman' s book , o n th e o n e h a n d , a n d Howard DeLong' s book , o n th e other . What i s Godel' s wor k about ? Kur t Godel , a n Austria n logician b o r n i n 1906 , was steepe d i n th e mathematica l a t m o s p h e r e o f hi s time , whic h wa s characterize d b y a relentless driv e towar d formalization . Peopl e wer e con vinced tha t mathematica l thinkin g coul d b e capture d by law s o f p u r e symbo l manipulation . Fro m a fixed se t

Foreword xii i of axiom s a n d a fixed se t o f typographica l rules , o n e could s h u n t symbol s a r o u n d a n d p r o d u c e ne w string s of symbols , calle d "theorems. " T h e pinnacl e o f thi s m o v e m e n t wa s a m o n u m e n t a l three-volum e wor k b y Bertrand Russel l a n d Alfre d N o r t h Whitehea d calle d Principia Mathematica, whic h cam e o u t i n th e year s 1 9 1 0 - 1 9 1 3 . Russel l a n d Whitehea d believe d tha t the y h a d g r o u n d e d al l o f mathematic s i n p u r e logic , a n d that thei r wor k woul d for m th e soli d foundatio n fo r al l of mathematic s forevermore . A coupl e o f decade s later , Gode l bega n t o d o u b t thi s noble vision , a n d o n e day , while studyin g th e extremel y austere pattern s o f symbol s i n thes e volumes , h e h a d a flash tha t thos e pattern s wer e s o m u c h lik e n u m b e r patterns tha t h e coul d i n fac t replac e eac h symbo l b y a n u m b e r a n d reperceiv e al l o f Principia Mathematica n o t as symbo l shuntin g b u t a s n u m b e r c r u n c h i n g (t o bor row a m o d e r n t e r m ) . Thi s ne w wa y o f lookin g a t thing s h a d a n astoundin g w r a p a r o u n d effect : sinc e th e subjec t matter o f Principia Mathematica wa s n u m b e r s , a n d sinc e Godel h a d t u r n e d th e m e d i u m o f th e volume s als o int o n u m b e r s , thi s showe d tha t Principia Mathematica wa s it s own subjec t matter , o r i n o t h e r words , tha t th e pat t e r n e d formula s o f Russel l a n d Whitehead' s syste m could b e see n a s sayin g thing s a b o u t eac h other , o r possibly eve n abou t themselves . This w r a p a r o u n d wa s a trul y u n e x p e c t e d tur n o f events, fo r i t inevitabl y b r o u g h t ancien t paradoxe s o f self-reference t o Godel' s m i n d — a b o v e all , "Thi s state m e n t i s false. " Usin g thi s typ e o f p a r a d o x a s hi s guide ,

xiv Douglas K Hofstadter Godel realize d that , i n principle , h e coul d writ e dow n a formul a o f Principia Mathematica tha t perversel y sai d about itself , "Thi s formul a i s unprovabl e b y th e rule s of Principia Mathematical T h e ver y existenc e o f suc h a twisted formul a wa s a h u g e threa t t o th e edific e o f Russell a n d Whitehead , fo r the y h a d m a d e th e absolut e elimination o f "viciou s circularity " a sacre d goal , a n d h a d b e e n convince d the y h a d wo n th e battle . Bu t no w it seeme d tha t viciou s circle s h a d e n t e r e d thei r pristin e world t h r o u g h th e bac k door , a n d P a n d o r a ' s b o x wa s wide o p e n . T h e self-underminin g Godelia n formul a h a d t o b e dealt with , a n d Gode l di d s o mos t astutely , showin g that althoug h i t resemble d a paradox , i t differe d subtl y from o n e . I n particular , i t wa s reveale d t o b e a tru e statement tha t coul d n o t b e prove n usin g th e rule s o f the system—indeed , a tru e statemen t whos e unprov ability resulte d precisel y fro m it s truth . In thi s shockingl y bol d m a n n e r , Gode l storme d th e fortress o f Principia Mathematica a n d b r o u g h t i t tum bling dow n i n ruins . H e als o showe d tha t hi s m e t h o d applied t o an y syste m whatsoeve r tha t trie d t o accom plish th e goal s o f Principia Mathematica. I n effect , then , Godel destroye d th e h o p e s o f thos e wh o believe d tha t mathematical thinkin g i s capturabl e b y th e rigidit y o f axiomatic systems , a n d h e thereb y force d mathemati cians, logicians , a n d philosopher s t o explor e th e mys terious newl y foun d chas m irrevocabl y separatin g prov ability from truth . Ever sinc e Godel , i t ha s b e e n realize d ho w subtl e

Foreword x v a n d d e e p th e ar t o f mathematica l thinkin g is , a n d th e once-bright h o p e o f mechanizin g h u m a n mathematica l t h o u g h t start s t o see m shaky , i f n o t utterl y quixotic . What, then , afte r Godel , i s mathematica l thinkin g be lieved t o be ? What , afte r Godel , i s mathematica l truth ? I n d e e d , wha t i s truth a t all ? These ar e th e centra l issue s that stil l li e unresolved , sevent y year s afte r Godel' s ep och-making p a p e r appeared . My book , despit e owin g a larg e d e b t t o Nage l a n d Newman, doe s n o t agre e wit h al l o f thei r philosophica l conclusions, a n d h e r e I would lik e t o p o i n t o u t o n e ke y difference. I n thei r "Concludin g Reflections, " Nage l a n d Newma n argu e tha t fro m Godel' s discoverie s i t follows tha t computers—"calculatin g machines, " a s they cal l t h e m — a r e i n principl e incapabl e o f reason ing a s flexibly a s w e h u m a n s reason , a resul t tha t sup posedly ensue s fro m th e fac t tha t computer s follo w " a fixed se t o f directives " (i.e. , a p r o g r a m ) . T o Nage l a n d Newman, thi s notio n correspond s t o a fixed se t o f axi oms a n d rule s o f i n f e r e n c e — a n d th e c o m p u t e r ' s be havior, a s i t execute s it s p r o g r a m , a m o u n t s t o tha t o f a m a c h i n e systematicall y c h u r n i n g o u t proof s o f theo rems i n a forma l system . Thi s m a p p i n g o f c o m p u t e r o n t o forma l syste m take s th e ter m "calculatin g ma chine" ver y literally—tha t is , a m a c h i n e buil t t o dea l with n u m b e r s a n d arithmetica l fact s alone . T h e ide a that suc h machine s b y thei r ver y n a t u r e shoul d c h u r n o u t set s o f tru e statement s a b o u t mathematic s i s seduc tive a n d certainl y ha s a grai n o f trut h t o it , b u t i t i s fa r

xvi Douglas R. Hofstadter from th e ful l visio n o f th e powe r a n d versatilit y o f computers. Although computers , a s thei r n a m e implies , ar e buil t of rigidl y arithmetic-respectin g hardware , n o t h i n g i n their desig n link s t h e m inseparabl y t o mathematica l truth. I t i s n o h a r d e r t o ge t a c o m p u t e r t o p r i n t o u t scads o f fals e calculation s (" 2 + 2 = 5 ; ° / °= 43> " etc.) tha n t o prin t o u t t h e o r e m s i n a forma l system . A subtler challeng e woul d b e t o devis e " a fixed se t o f directives" b y whic h a c o m p u t e r migh t explor e th e world o f mathematica l idea s (no t just string s o f mathe matical symbols) , guide d b y visua l imagery , th e associ ative pattern s linkin g concepts , a n d th e intuitiv e pro cesses o f guesswork , analogy , a n d estheti c choic e tha t every mathematicia n uses . W h e n Nage l a n d Newma n wer e composin g GodeVs Proof, th e goa l o f gettin g computer s t o thin k lik e p e o p l e — i n o t h e r words , artificia l intelligence—wa s very ne w a n d it s potential wa s unclear. T h e mai n thrus t in thos e earl y day s use d computer s a s mechanica l in stantiations o f axiomati c systems , a n d a s such , the y di d n o t h i n g b u t c h u r n o u t proof s o f theorems . No w admit tedly, i f thi s a p p r o a c h represente d th e ful l scop e o f how computer s migh t eve r i n principl e b e use d t o m o d e l cognition , then , indeed , Nage l a n d Newma n would b e wholl y justified i n arguing , base d o n Godel' s discoveries, tha t computers , n o matte r ho w rapi d thei r calculations o r ho w capaciou s thei r memories , ar e nec essarily les s flexibl e a n d insightfu l tha n th e h u m a n mind.

Foreword xvi i But theorem-provin g i s a m o n g th e leas t subtl e o f ways o f tryin g t o ge t computer s t o think . Conside r th e p r o g r a m "AM, " writte n i n th e mid-1970 s b y Dougla s Lenat. Instea d o f mathematica l statements , A M deal t with concepts ; it s goa l wa s t o see k "interesting " ones , using a rudimentar y m o d e l o f esthetic s a n d simplicity . Starting fro m scratch , A M discovere d man y concept s o f n u m b e r theory . Rathe r tha n logicall y provin g theo rems, A M w a n d e r e d a r o u n d th e worl d o f n u m b e r s , following it s primitiv e estheti c nose , sniffin g o u t pat terns, a n d makin g guesse s a b o u t them . A s with a brigh t h u m a n , mos t o f AM' s guesse s wer e right , som e wer e wrong, and , fo r a few , th e jury i s stil l out . For a n o t h e r wa y o f modelin g menta l processe s com putationally, tak e neura l n e t s — a s fa r fro m th e theo rem-proving paradig m a s o n e coul d imagine . Sinc e th e cells o f th e brai n ar e wire d togethe r i n certai n patterns , a n d sinc e o ne ca n imitat e an y such patter n i n software — that is , i n a "fixe d se t o f directives"— a calculatin g en gine's powe r ca n b e harnesse d t o imitat e microscopi c brain circuitr y a n d it s behavior . Suc h model s b e e n studied no w fo r man y year s b y cognitiv e scientists , wh o have foun d tha t man y pattern s o f h u m a n learning , in cluding erro r makin g a s a n automati c by-product , ar e faithfully replicated . T h e p o i n t o f thes e tw o example s (an d I coul d giv e many more ) i s tha t h u m a n thinkin g i n al l it s flexible a n d fallibl e glor y ca n i n principl e b e m o d e l e d b y a "fixed se t o f directives, " provide d o n e i s liberate d fro m the preconceptio n tha t computers , buil t o n arithmeti -

xviii Douglas K Hofstadter cal operations , ca n d o n o t h i n g b u t slavishl y p r o d u c e truth, th e whol e truth , a n d n o t h i n g b u t th e truth . T h a t idea, admittedly , lie s a t th e cor e o f forma l axiomati c reasoning systems , b u t toda y n o o n e take s suc h system s seriously a s a m o d e l o f wha t th e h u m a n m i n d does , even whe n i t i s a t it s mos t logical . W e no w u n d e r s t a n d that th e h u m a n m i n d i s fundamentall y n o t a logi c en gine b u t a n analog y engine , a learnin g engine , a guess ing engine , a n esthetics-drive n engine , a self-correctin g engine. A n d havin g profoundl y u n d e r s t o o d thi s lesson , we ar e perfectl y abl e t o mak e "fixe d set s o f directives " that hav e som e o f thes e qualities . T o b e sure , we hav e n o t ye t c o m e clos e t o p r o d u c i n g a c o m p u t e r p r o g r a m tha t ha s anythin g remotel y resem bling th e flexibility o f th e h u m a n mind , a n d i n thi s sense Ernes t Nage l a n d J a m es Newma n wer e exactl y o n the mar k i n declaring , i n thei r poeti c fashion , tha t Godel's t h e o r e m "i s an occasion , n o t fo r dejection , b u t for a renewe d appreciatio n o f th e power s o f creativ e reason." I t coul d n o t b e sai d better . T h e r e is , however , a n iron y t o Nage l a n d Newman' s interpretation o f Godel' s result . Godel' s grea t strok e o f genius—as reader s o f Nage l a n d Newma n wil l s e e — was t o realiz e tha t n u m b e r s ar e a universa l m e d i u m fo r the e m b e d d i n g o f pattern s o f an y sort , a n d tha t fo r that reason , statement s seemingl y a b o u t n u m b e r s alone ca n i n fac t e n c o d e statement s a b o u t o t h e r uni verses o f discourse . I n o t h e r words , Gode l sa w beyon d the surfac e leve l o f n u m b e r theory , realizin g tha t n u m bers coul d represen t an y kin d o f structure . T h e analo -

Foreword xi x gous Godelia n lea p wit h respec t t o computer s woul d be t o se e tha t becaus e computer s a t bas e manipulat e n u m b e r s , a n d becaus e n u m b e r s ar e a universa l me d i u m fo r th e e m b e d d i n g o f pattern s o f an y sort , com puters ca n dea l wit h arbitrar y patterns , w h e t h e r the y are logica l o r illogical , consisten t o r inconsistent . I n short, whe n o n e step s bac k fa r e n o u g h fro m myriad s of interrelate d n u m b e r patterns , o n e ca n m a k e o u t patterns fro m o t h e r domains , just a s th e ey e lookin g a t a scree n o f pixel s see s a familia r fac e a n d nar y a 1 o r o. Thi s Godelia n vie w o f computer s ha s explode d o n the m o d e r n worl d t o suc h a n exten t tha t toda y th e numerical substrat e o f computer s i s al l b u t invisible , except t o specialists . Ordinar y peopl e routinel y us e computers fo r wor d processing , g a m e playing , com munication, animation , designing , drawing , a n d o n a n d on , al l without eve r thinkin g a b o u t th e basi c arith metical operation s goin g o n d e e p dow n i n th e hard ware. Cognitiv e scientists , relyin g o n th e arithmetica l hardware o f thei r computer s t o b e error-fre e a n d un creative, giv e thei r computer s "fixe d set s o f directives " to m o d e l h u m a n error-makin g a n d creativity . T h e r e i s n o reaso n t o thin k tha t th e processe s o f creativ e math ematical thinkin g cannot , a t leas t i n principle , b e mod eled usin g computers . Bu t bac k i n th e 1950s , suc h visions o f th e potentia l o f computer s wer e h a r d t o see . Still, i t i s ironi c tha t i n a boo k devote d t o celebratin g Godel's insigh t tha t n u m b e r s engul f th e worl d o f pat terns a t large , th e primar y philosophica l conclusio n would b e base d o n n o t h e e d i n g tha t insight , a n d woul d

xx Douglas

R. Hofstadter

thereby fai l t o se e tha t calculatin g machine s ca n repli cate pattern s o f an y imaginabl e sort—eve n thos e o f the creativ e h u m a n mind . I shal l clos e wit h a fe w word s abou t wh y I hav e take n the libert y o f makin g som e technica l e m e n d a t i o n s t o this classi c text . Althoug h th e boo k receive d mostl y warm accolade s fro m reviewers , ther e wer e som e critic s who fel t tha t i n spot s i t was n o t sufficientl y precis e a n d that i t riske d misleadin g it s readers . T h e first tim e t h r o u g h , I mysel f wa s unaware o f an y suc h deficiencies , b u t man y year s later , whe n readin g GodeVs Proof with an ey e t o explainin g thes e sam e idea s mysel f a s pre cisely a n d clearl y a s possible , I stumble d ove r certai n passages i n Chapte r VI I a n d realized , afte r a while, tha t the stumblin g wa s n o t entirel y m y ow n fault . I t m a d e m e sa d t o realiz e tha t thi s belove d b o o k h a d a fe w blemishes, b u t ther e wa s obviousl y n o t h i n g I coul d d o about it . Oddl y e n o u g h , t h o u g h , i n th e margin s o f m y copy I carefull y a n n o t a t e d al l th e glitche s tha t I uncov ered, indicatin g ho w the y migh t b e corrected—almos t as i f I h a d foresee n tha t o n e da y I woul d receiv e a n email ou t o f th e blu e fro m Ne w Yor k Universit y Pres s asking m e i f I woul d conside r writin g a forewor d t o a new editio n o f th e book . I mus t certainl y b e a m o n g th e reader s mos t pro foundly affecte d b y th e littl e opu s b y Ernes t Nage l a n d J a m e s Newman , a n d fo r tha t reason , havin g b e e n give n the chance , I ow e i t t o t h e m t o polis h thei r ge m a n d t o

Foreword xx i give i t a ne w luste r fo r th e ne w millennium . I woul d like t o believ e tha t i n s o doing , I a m n o t betrayin g m y respected m e n t o r s b u t a m instea d payin g t h e m h o m age, a s a n a r d e n t a n d faithfu l disciple . Center fo r Researc h o n Concept s a n d Cognitio n Indiana University, Bloomington

Acknowledgments T h e author s gratefull y acknowledg e th e generou s assis tance the y receive d fro m Professo r J o h n C . Coole y o f Columbia University . H e rea d criticall y a n earl y draf t of th e manuscript , a n d h e l p e d t o clarif y th e structur e of th e a r g u m e n t a n d t o improv e th e expositio n o f points i n logic . W e wis h t o than k Scientific American fo r permission t o r e p r o d u c e severa l o f th e diagram s i n th e text, whic h a p p e a r e d i n a n articl e o n Godel' s Proo f i n the J u ne 195 6 issu e o f th e magazine . W e ar e i n d e b t e d to Professo r Morri s Klin e o f Ne w Yor k Universit y fo r helpful suggestion s regardin g th e manuscript .

xxni

Godel's Proof

I

Introduction In 193 1 ther e a p p e a r e d i n a G e r m a n scientifi c period ical a relativel y shor t p a p e r wit h th e forbiddin g titl e "Uber forma l u n e n t s c h e i d b a r e Satz e de r Principi a Mathematica u n d verwandte r Systeme " ("O n Formall y Undecidable Proposition s o f Principi a Mathematic a a n d Relate d Systems") . It s a u t h o r wa s Kur t Godel , t h e n a youn g mathematicia n o f 2 5 a t th e Universit y o f Vi e n n a a n d sinc e 193 8 a p e r m a n e n t m e m b e r o f th e In stitute fo r Advance d Stud y a t Princeton . T h e p a p e r i s a milestone i n th e histor y o f logi c a n d mathematics . W h e n Harvar d Universit y awarde d Gode l a n honorar y degree i n 1952 , th e citatio n describe d th e wor k a s o n e of th e mos t i m p o r t a n t advance s i n logi c i n m o d e r n times. At th e tim e o f it s appearance , however , n e i t h e r th e title o f Godel' s p a p e r n o r it s c o n t e n t wa s intelligibl e t o most mathematicians . T h e Principia Mathematica men tioned i n th e titl e i s th e m o n u m e n t a l three-volum e treatise b y Alfre d N o r t h Whitehea d a n d Bertran d Rus sell o n mathematica l logi c a n d th e foundation s o f 1

2 GodeVs

Proof

mathematics; a n d familiarit y wit h tha t wor k i s n o t a prerequisite t o successfu l researc h i n mos t branche s o f mathematics. Moreover , Godel' s p a p e r deal s wit h a se t of question s tha t ha s neve r attracte d m o r e tha n a com paratively smal l g r o u p o f students . T h e reasonin g o f the proo f wa s s o nove l a t th e tim e o f it s publicatio n that onl y thos e intimatel y conversan t wit h th e technica l literature o f a highl y specialize d field coul d follo w th e a r g u m e n t wit h read y comprehension . Nevertheless , th e conclusions Gode l establishe d ar e no w widel y recog nized a s bein g revolutionar y i n thei r b r o a d philosophi cal import . I t i s th e ai m o f th e presen t essa y t o mak e the substanc e o f Godel' s findings a n d th e genera l char acter o f hi s proo f accessibl e t o th e nonspecialist . Godel's famou s p a p e r attacke d a centra l p r o b l e m i n the foundation s o f mathematics . I t wil l b e helpfu l t o give a brie f preliminar y accoun t o f th e contex t i n which th e p r o b l e m occurs . Everyon e wh o ha s b e e n exposed t o elementar y geometr y wil l doubtles s recal l that i t i s taugh t a s a deductive discipline . I t i s n o t pre sented a s a n experimenta l scienc e whos e t h e o r e m s ar e to b e accepte d becaus e the y ar e i n a g r e e m e n t wit h observation. Thi s notion , tha t a propositio n ma y b e established a s th e conclusio n o f a n explici t logical proof, goes bac k t o th e ancien t Greeks , wh o discovere d wha t is know n a s th e "axiomati c m e t h o d " a n d use d i t t o develop geometr y i n a systemati c fashion . T h e axio matic m e t h o d consist s i n acceptin g without proo f cer tain proposition s a s axiom s o r postulate s (e.g. , th e ax iom tha t t h r o u g h tw o point s just o n e straigh t lin e ca n

Introduction 3 be drawn) , a n d t h e n derivin g fro m th e axiom s al l o t h e r propositions o f th e syste m a s theorems . T h e axiom s constitute th e "foundations " o f th e system ; th e theo rems ar e th e "superstructure, " a n d ar e obtaine d fro m the axiom s wit h th e exclusiv e hel p o f principle s o f logic. T h e axiomati c developmen t o f geometr y m a d e a powerful impressio n u p o n thinker s t h r o u g h o u t th e ages; fo r th e relativel y smal l n u m b e r o f axiom s carr y the whol e weigh t o f th e inexhaustibl y n u m e r o u s p r o p ositions derivabl e fro m them . Moreover , i f in som e wa y the trut h o f th e axiom s ca n b e established—and , in deed, fo r som e tw o t h o u s a n d year s mos t student s be lieved withou t questio n tha t the y ar e tru e o f s p a c e — both th e trut h a n d th e mutua l consistenc y o f al l th e theorems ar e automaticall y guaranteed . Fo r thes e rea sons th e axiomati c for m o f geometr y a p p e a r e d t o man y generations o f outstandin g thinker s a s th e m o d e l o f scientific knowledg e a t it s best . I t wa s natura l t o ask , therefore, w h e t h e r o t h e r branche s o f t h o u g h t beside s geometry ca n b e place d u p o n a secur e axiomati c foun dation. However , althoug h certai n part s o f physic s wer e given a n axiomati c formulatio n i n antiquit y (e.g. , b y Archimedes), unti l m o d e r n time s geometr y wa s th e only b r a n c h o f mathematic s tha t h a d wha t mos t stu dents considere d a soun d axiomati c basis . But withi n th e pas t tw o centurie s th e axiomati c m e t h o d ha s com e t o b e exploite d wit h increasin g power a n d vigor . Ne w a s wel l a s ol d branche s o f math ematics, includin g th e stud y o f th e propertie s o f th e

4 GodeVs

Proof

familiar cardina l (o r "whole" ) n u m b e r s , * were supplie d with wha t a p p e a r e d t o b e adequat e set s o f axioms . A climate o f opinio n wa s thu s g e n e r a t e d i n whic h i t wa s tacitly assume d tha t eac h secto r o f mathematica l t h o u g h t ca n b e supplie d wit h a se t o f axiom s sufficien t for developin g systematicall y th e endles s totalit y o f tru e propositions a b o u t th e give n are a o f inquiry . Godel's p a p e r showe d tha t thi s assumptio n i s u n t e n able. H e presente d mathematician s wit h th e astound ing a n d melanchol y conclusio n tha t th e axiomati c m e t h o d ha s certai n i n h e r e n t limitations , whic h rul e o u t th e possibilit y tha t eve n th e propertie s o f th e n o n negative integer s ca n eve r b e full y axiomatized . Wha t * Number theor y is the study, going back to the ancient Greeks, of th e propertie s o f th e natura l number s o , 1 , 2 , 3 , . . . —also sometimes calle d th e "cardina l numbers " o r "non-negativ e inte gers." Suc h propertie s include: ho w man y factor s a numbe r has ; how many different way s a number can be represented a s a sum of smaller numbers; whether o r not there is a biggest number having some specifie d property ; whethe r o r no t certai n equation s hav e solutions tha t ar e whol e numbers ; an d s o on . Althoug h numbe r theory i s inexhaustibly ric h an d ful l o f surprises , it s vocabulary is tiny—an alphabe t o f just a doze n symbol s allow s an y number theoretical statemen t t o b e expresse d (althoug h ofte n cumber somely). In this book, we shall occasionally use the term "arithmetic " as a synonym for "numbe r theory, " but of course what this term entails is the full, ric h univers e of properties of the natural numbers, and not merel y th e mechanic s o f addition, subtraction , multiplication , and lon g divisio n a s taught i n elementar y schools , and a s mechanized in cash registers and adding machines. [—Ed.]

Introduction 5 is m o r e , h e prove d tha t i t i s impossibl e t o establis h th e internal logica l consistenc y o f a ver y larg e clas s o f de ductive systems—numbe r theory , fo r e x a m p l e — u n l e s s o n e adopt s principle s o f reasonin g s o comple x tha t their interna l consistenc y i s as o p e n t o d o u b t a s tha t o f the system s themselves . I n th e ligh t o f thes e conclu sions, n o final systematizatio n o f man y i m p o r t a n t area s of mathematic s i s attainable , a n d n o absolutel y impec cable guarante e ca n b e give n tha t man y significan t branches o f mathematica l t h o u g h t ar e entirel y fre e from interna l contradiction . Godel's findings thu s u n d e r m i n e d deepl y r o o t e d preconceptions a n d demolishe d ancien t h o p e s tha t were bein g freshl y n o u r i s h e d b y researc h o n th e foun dations o f mathematics . Bu t hi s p a p e r wa s n o t alto gether negative . I t i n t r o d u c e d int o th e stud y o f founda tion question s a ne w techniqu e o f analysi s comparabl e in it s n a t u r e a n d fertilit y wit h th e algebrai c m e t h o d that Ren e Descarte s i n t r o d u c e d int o geometry . Thi s technique suggeste d a n d initiate d ne w problem s fo r logical a n d mathematica l investigation . I t provoke d a reappraisal, stil l u n d e r way , o f widely hel d philosophie s of mathematics , a n d o f philosophie s o f knowledg e i n general. T h e detail s o f Godel' s proof s i n hi s epoch-makin p a p e r ar e to o difficul t t o follo w withou t considerabl mathematical training . Bu t th e basi c structur e o f hi demonstrations a n d th e cor e o f hi s conclusion s ca n b m a d e intelligibl e t o reader s wit h ver y limite d mathe

g e s e -

6 GodeVs

Proof

matical a n d logica l preparation . T o achiev e suc h a n understanding, th e reade r ma y find usefu l a brie f ac c o u n t o f certai n relevan t development s i n th e histor y of mathematic s a n d o f m o d e r n forma l logic . T h e nex t four section s o f thi s essa y ar e devote d t o thi s survey .

I I

The Proble m o f Consistenc y T h e n i n e t e e n t h centur y witnesse d a t r e m e n d o u s ex pansion a n d intensificatio n o f mathematica l research . Many fundamenta l problem s tha t h a d lon g withstoo d the bes t effort s o f earlie r thinker s wer e solved ; ne w areas o f mathematica l stud y wer e created ; a n d i n vari ous branche s o f th e disciplin e ne w foundation s wer e laid, o r ol d one s entirel y recas t wit h th e h e l p o f m o r e precise technique s o f analysis . T o illustrate : th e Greek s h a d p r o p o s e d thre e problem s i n elementar y geometry : with compas s a n d straight-edg e t o trisec t an y angle , t o construct a cub e wit h a volum e twic e th e volum e o f a given cube , a n d t o construc t a squar e equa l i n are a t o that o f a give n circle . Fo r m o r e tha n 2,00 0 year s unsuc cessful attempt s wer e m a d e t o solv e thes e problems ; a t last, i n th e n i n e t e e n t h centur y i t wa s prove d tha t th e desired construction s ar e logicall y impossible . T h e r e was, moreover , a valuabl e by-produc t o f thes e labors . Since th e solution s d e p e n d essentiall y u p o n determin ing th e kin d o f root s tha t satisf y certai n equations , concern wit h th e celebrate d exercise s se t i n antiquit y 7

8 Godel's

Proof

stimulated p r o f o u n d investigation s int o th e n a t u r e o f n u m b e r a n d th e structur e o f th e n u m b e r c o n t i n u u m . Rigorous definition s wer e eventuall y supplie d fo r neg ative, complex , a n d irrationa l n u m b e r s ; a logica l basi s was constructe d fo r th e rea l n u m b e r system ; a n d a ne w b r a n c h o f mathematics , th e theor y o f infinit e n u m b e r s , was founded . But p e r h a p s th e mos t significan t developmen t i n it s long-range effect s u p o n subsequen t mathematica l his tory wa s th e solutio n o f a n o t h e r p r o b l e m tha t th e Greeks raise d withou t answering . O n e o f th e axiom s Euclid use d i n systematizin g geometr y ha s t o d o wit h parallels. T h e axio m h e a d o p t e d i s logicall y equivalen t to ( t h o u g h n o t identica l with ) th e assumptio n tha t t h r o u g h a p o i n t outsid e a give n lin e onl y o n e paralle l to th e lin e ca n b e drawn . Fo r variou s reasons , thi s axiom di d n o t a p p e a r "self-evident " t o th e ancients . They sought , therefore , t o d e d u c e i t fro m th e o t h e r Euclidean axioms , whic h the y regarde d a s clearl y sel f evident. 1 Ca n suc h a proo f o f th e paralle l axio m b e

1

The chief reason for thi s alleged lack of self-evidence seem s to have been the fact that the parallel axiom makes an assertion about infinitely remote region s o f space . Eucli d define s paralle l line s a s straight lines in a plane that , "bein g produced indefinitel y i n bot h directions," d o no t meet . Accordingly , t o sa y tha t tw o line s ar e parallel i s to make th e clai m tha t th e tw o lines will not meet eve n "at infinity." But the ancient s were familiar wit h lines that, thoug h they do not intersec t eac h othe r i n an y finite region o f the plane , do meet "a t infinity." Suc h lines are said to be "asymptotic. " Thus, a hyperbola is asymptotic to its axes. It was therefore no t intuitively

The Problem of Consistency 9 given? Generation s o f mathematician s struggle d wit h this question , withou t avail . But r e p e a t e d failur e t o con struct a proo f doe s n o t m e a n tha t n o n e ca n b e foun d any m o r e tha n r e p e a t e d failur e t o fin d a cur e fo r th e c o m m o n col d establishe s beyon d d o u b t tha t humanit y will foreve r suffe r fro m r u n n i n g noses . I t wa s n o t unti l the n i n e t e e n t h century , chiefl y t h r o u g h th e wor k o f Gauss, Bolyai , Lobachevsky , a n d Riemann , tha t th e impossibility o f d e d u c i n g th e paralle l axio m fro m th e oth ers was demonstrated . Thi s o u t c o m e wa s of th e greates t intellectual importance . I n th e firs t place , i t calle d at tention i n a mos t impressiv e wa y t o th e fac t tha t a proof can b e give n o f th e impossibility of proving certai n p r o p ositions withi n a give n system . A s w e shal l see , Godel' s p a p e r i s a proo f o f th e impossibilit y o f formall y dem onstrating certai n i m p o r t a n t proposition s i n n u m b e r theory. I n th e secon d place , th e resolutio n o f th e par allel axio m questio n force d th e realizatio n tha t Eucli d is n o t th e las t wor d o n th e subjec t o f geometry , sinc e new system s o f geometr y ca n b e constructe d b y using a n u m b e r o f axiom s differen t from , a n d incompatibl e with, thos e a d o p t e d b y Euclid . I n particular , a s i s wel l known, immensel y interestin g a n d fruitfu l result s ar e obtained whe n Euclid' s paralle l axio m i s replace d b y the assumptio n tha t m o r e tha n o n e paralle l ca n b e drawn t o a give n lin e t h r o u g h a give n point , or , alter evident to the ancient geometers that from a point outside a given straight line only one straight line can be drawn tha t will not meet the given line even at infinity.

i o GodeUs

Proof

natively, b y th e assumptio n tha t n o parallel s ca n b e drawn. T h e traditiona l belie f tha t th e axiom s o f geom etry (or , fo r tha t matter , th e axiom s o f an y discipline ) can b e establishe d b y thei r a p p a r e n t self-evidenc e wa s thus radicall y u n d e r m i n e d . Moreover , i t graduall y be came clea r tha t th e p r o p e r busines s o f p u r e mathema ticians i s t o derive theorems from postulated assumptions, a n d tha t i t i s n o t thei r concer n w h e t h e r th e axiom s assumed ar e actuall y true . And , finally, thes e successfu l modifications o f o r t h o d o x geometr y stimulate d th e re vision a n d completio n o f th e axiomati c base s fo r man y o t h e r mathematica l systems . Axiomati c foundation s were eventuall y supplie d fo r fields o f inquir y tha t h a d hitherto b e e n cultivate d onl y i n a m o r e o r les s intuitiv e m a n n e r . (Se e Appendix , n o . 1. ) T h e over-al l conclusio n tha t e m e r g e d fro m thes e critical studie s o f th e foundation s o f mathematic s i s that th e age-ol d conceptio n o f mathematic s a s "th e science o f quantity " i s b o th inadequat e a n d misleading . For i t b e c a m e eviden t tha t mathematic s i s simpl y th e discipline par excellence that draw s th e conclusion s logi cally implie d b y an y give n se t o f axiom s o r postulates . In fact , i t cam e t o b e acknowledge d tha t th e validit y o f a mathematica l inferenc e i n n o sens e d e p e n d s u p o n any specia l m e a n i n g tha t ma y b e associate d wit h th e terms o r expression s containe d i n th e postulates . Math ematics wa s thu s recognize d t o b e m u c h m o r e abstrac t a n d forma l t h a n h a d b e e n traditionall y supposed : m o r e abstract , becaus e mathematica l statement s ca n be construe d i n principl e t o b e abou t anythin g what -

The Problem of Consistency 1 1 soever rathe r t h a n a b o u t som e inherentl y circum scribed se t o f object s o r trait s o f objects ; a n d m o r e formal, becaus e th e validit y o f mathematica l d e m o n strations i s g r o u n d e d i n th e structur e o f statements , rather tha n i n th e n a t u r e o f a particula r subjec t matter . T h e postulate s o f an y b r a n c h o f demonstrativ e mathe matics ar e n o t inherentl y a b o u t space , quantity , apples , angles, o r budgets ; a n d an y specia l m e a n i n g tha t ma y be associate d wit h th e term s (o r "descriptiv e predi cates") i n th e postulate s play s n o essentia l rol e i n th e process o f derivin g theorems . W e repea t tha t th e sol e question confrontin g th e p u r e mathematicia n (a s dis tinct fro m th e scientis t wh o employ s mathematic s i n investigating a specia l subjec t matter ) i s n o t w h e t h e r the postulate s assume d o r th e conclusion s d e d u c e d from t h e m ar e true , b u t w h e t h e r th e allege d conclu sions ar e i n fac t th e necessary logical consequences of th e initial assumptions . Take thi s example . A m o n g th e undefine d (o r "prim itive") term s employe d b y th e influentia l G e r m a n mathematician Davi d Hilber t i n hi s famou s axiomati zation o f geometr y (firs t publishe d i n 1899 ) ar e 'point', 'line' , 'lie s o n ' , a n d 'between' . W e ma y g r a n t that th e customar y meaning s c o n n e c t e d wit h thes e ex pressions pla y a rol e i n th e proces s o f discoverin g a n d learning theorems . Sinc e th e meaning s ar e familiar , w e feel w e u n d e r s t a n d thei r variou s interrelations , a n d they motivat e th e formulatio n a n d selectio n o f axioms ; moreover, the y sugges t a n d facilitat e th e formulatio n of th e statement s w e h o p e t o establis h a s theorems .

12 GodeVs

Proof

Yet, a s Hilber t plainl y states , insofa r a s w e ar e con c e r n e d wit h th e primar y mathematica l tas k o f explor ing th e purel y logica l relation s o f d e p e n d e n c e betwee n statements, th e familia r connotation s o f th e primitiv e terms ar e t o b e ignored , a n d th e sol e "meanings " tha t are t o b e associate d wit h t h e m ar e thos e assigne d b y the axiom s int o whic h the y enter. 2 Thi s i s th e p o i n t o f Russell's famou s epigram : p u r e mathematic s i s the sub j e c t i n whic h w e d o n o t kno w wha t w e ar e talkin g about, o r w h e t h e r wha t w e ar e sayin g i s true . A lan d o f rigorou s abstraction , empt y o f al l familia r landmarks, i s certainl y n o t eas y t o ge t a r o u n d in . Bu t i t offers compensation s i n th e for m o f a ne w freedo m o f m o v e m e n t a n d fres h vistas . T h e intensifie d formaliza tion o f mathematic s emancipate d people' s mind s fro m the restriction s tha t th e customar y interpretatio n o f expressions place d o n th e constructio n o f nove l sys tems o f postulates . Ne w kind s o f algebra s a n d geome tries wer e develope d whic h m a r k e d significan t depar tures fro m th e mathematic s o f tradition . A s th e meanings o f certai n term s b e c a m e m o r e general , thei r use b e c a m e b r o a d e r a n d th e inference s tha t coul d b e drawn fro m t h e m les s confined . Formalizatio n le d t o a great variet y o f system s o f considerabl e mathematica l interest a n d value . Som e o f thes e systems , i t mus t b e 2

In more technical language, the primitive terms are "implicitly" denned b y the axioms, and whatever is not covered by the implicit definitions is irrelevant to the demonstration o f theorems.

The Problem of Consistency 1 3 admitted, di d n o t len d themselve s t o interpretation s a s obviously intuitiv e (i.e. , commonsensical ) a s thos e o f Euclidean geometr y o r arithmetic , b u t thi s fac t cause d n o alarm . Intuition , fo r o n e thing , i s a n elasti c faculty : o u r childre n wil l probabl y hav e n o difficult y i n accept ing a s intuitively obviou s th e paradoxe s o f relativity, j u st as w e d o n o t boggl e a t idea s tha t wer e r e g a r d e d a s wholly unintuitiv e a coupl e o f generation s ago . More over, a s w e al l know , intuitio n i s n o t a saf e guide : i t c a n n o t properl y b e use d a s a criterio n o f eithe r trut h or fruitfulnes s i n scientifi c explorations . However, th e increase d abstractnes s o f mathematic s raised a m o r e seriou s problem . I t t u r n e d o n th e ques tion whethe r a give n se t o f postulate s servin g a s foun dation o f a syste m i s internall y consistent , s o tha t n o mutually contradictor y t h e o r e m s ca n b e d e d u c e d fro m the postulates . T h e p r o b l e m doe s n o t see m pressin g when a se t o f axiom s i s take n t o b e abou t a definit e a n d familia r d o m a i n o f objects ; fo r t h e n i t i s n o t onl y significant t o ask , b u t i t ma y b e possibl e t o ascertain , w h e t h e r th e axiom s ar e i n d e e d tru e o f thes e objects . Since th e Euclidea n axiom s wer e generall y suppose d to b e tru e statement s a b o u t spac e (o r object s i n space) , n o mathematicia n prio r t o th e n i n e t e e n t h centur y eve r considered th e questio n w h e t h e r a pai r o f contradic tory t h e o r e m s migh t som e da y b e d e d u c e d fro m th e axioms. T he basi s for thi s confidenc e i n th e consistenc y of Euclidea n geometr y i s th e soun d principl e tha t logi cally incompatibl e statement s c a n n o t b e simultane -

14 GodeVs

Proof

ously true ; accordingly , i f a se t o f statement s i s tru e (and thi s wa s assume d o f th e Euclidea n axioms) , thes e statements ar e mutuall y consistent . T h e non-Euclidea n geometrie s wer e clearl y i n a dif ferent category . Thei r axiom s wer e initiall y regarde d a s being plainl y fals e o f space , and , fo r tha t matter , doubt fully tru e o f anything ; thu s th e p r o b l e m o f establishin g the interna l consistenc y o f non-Euclidea n system s wa s recognized t o b e b o t h formidabl e a n d critical . I n ellip tic geometry , fo r example , Euclid' s paralle l postulat e i s replaced b y th e assumptio n tha t t h r o u g h a give n p o i n t outside a lin e no paralle l t o i t ca n b e drawn . No w consider th e question : I s th e ellipti c se t o f postulate s consistent? T h e postulate s ar e apparentl y n o t tru e o f the spac e o f ordinar y experience . How , then , i s thei r consistency t o b e shown ? Ho w ca n o n e prov e the y wil l n o t lea d t o contradictor y theorems ? Obviousl y th e question i s n o t settle d b y th e fac t tha t th e t h e o r e m s already d e d u c e d d o n o t contradic t eac h o t h e r — f o r th e possibility remain s tha t th e ver y nex t t h e o r e m t o b e d e d u c e d ma y upse t th e appl e cart . But , unti l th e ques tion i s settled , o n e c a n n o t b e certai n tha t ellipti c ge ometry i s a tru e alternativ e t o th e Euclidea n system , i.e., equall y vali d mathematically . T h e ver y possibilit y of non-Euclidea n geometrie s wa s thu s contingen t o n the resolutio n o f thi s p r o b l e m . A genera l m e t h o d fo r solvin g i t wa s devised . T h e underlying ide a i s t o find a "model " (o r "interpreta tion") fo r th e abstrac t postulate s o f a system , s o tha t

The Problem of Consistency 1 5 each postulat e i s converte d int o a tru e statemen t abou t the model . I n th e cas e o f Euclidea n geometry , a s w e have noted , th e m o d e l wa s ordinary space . T h e m e t h o d was use d t o find o t h e r models , th e element s o f whic h could serv e a s crutche s fo r d e t e r m i n i n g th e consistenc y of abstrac t postulates . T h e p r o c e d u r e goe s somethin g like this . Le t u s u n d e r s t a n d b y th e wor d 'class ' a collec tion o r aggregat e o f distinguishabl e elements , eac h o f which i s calle d a ' m e m b e r ' o f th e class . Thus , th e clas s of p r i m e n u m b e r s les s tha n 1 o i s th e collectio n whos e m e m b e r s ar e 2 , 3 , 5 , a n d 7 . Suppos e th e followin g se t of postulate s c o n c e r n i n g tw o classe s K a n d L , whos e special n a t u r e i s lef t u n d e t e r m i n e d excep t a s "implic itly" define d b y th e postulates : 1. An y tw o m e m b e r s o f K ar e containe d i n jus t one membe r o f L . 2. N o m e m b e r o f K i s containe d i n m o r e tha n two m e m b e r s o f L . 3. T h e m e m b e r s o f K ar e n o t al l containe d i n a singl e m e m b e r o f L . 4. An y tw o m e m b e r s o f L contai n jus t o n e member o f K . 5. N o m e m b e r o f L contain s m o r e t h a n tw o members o f K . From thi s smal l se t w e ca n derive , b y usin g custom ary rule s o f inference , a n u m b e r o f theorems . Fo r example, i t ca n b e show n tha t K contain s jus t thre e m e m b e r s . Bu t i s th e se t consistent , s o tha t mutuall y

16 GodeVs

Proof

contradictory t h e o r e m s ca n neve r b e derive d fro m it ? T h e questio n ca n b e answere d readil y wit h th e h e l p o f the followin g model : Let K be th e clas s o f point s consistin g o f th e ver tices o f a triangle , an d L th e clas s o f line s mad e u p of its sides; and le t us understand th e phras e ' a mem ber o f K i s containe d i n a membe r o f L ' t o mea n that a poin t whic h i s a verte x lie s o n a lin e whic h i s a side . Eac h o f th e five abstrac t postulate s i s the n converted int o a tru e statement . Fo r instance , th e first postulat e assert s tha t an y tw o point s whic h ar e vertices o f th e triangl e li e o n just on e lin e whic h i s a side. (Se e Fig . 1. ) I n thi s way th e se t o f postulate s i s proved t o b e consistent . K

L

L

K

L

K

Fig. i Model for a set of postulates about two classes, K and L, is a triangle whose vertice s ar e th e member s o f K an d whos e side s ar e th e members o f L . The geometrica l mode l show s tha t th e postulate s are consistent. T h e consistenc y o f plan e ellipti c geometr y ca n also , ostensibly, b e establishe d b y a m o d e l embodyin g th e

The Problem of Consistency 1 7 postulates. W e ma y interpre t th e expressio n ' p l a n e ' i n the axiom s o f ellipti c geometr y a s th e surfac e o f a Euclidean sphere , th e expressio n 'point ' a s a pai r o f antipodal point s o n thi s surface , th e expressio n 'straight line ' a s a grea t circl e o n thi s surface , a n d s o on. Eac h ellipti c postulat e i s t h e n converte d int o a t h e o r e m o f Euclid . Fo r example , o n thi s interpretatio n the ellipti c paralle l postulat e reads : T h r o u g h a p o i n t o n th e surfac e o f a sphere , n o grea t circl e ca n b e draw n parallel t o a give n grea t circle . (Se e Fig . 2. ) At firs t glanc e thi s proo f o f th e consistenc y o f ellipti c geometry ma y see m conclusive . Bu t a close r loo k i s disconcerting. Fo r a shar p ey e wil l discer n tha t th e p r o b l e m ha s n o t b e e n solved ; i t ha s merel y b e e n shifted t o a n o t h e r d o m a i n . T h e proo f attempt s t o settl e the consistenc y o f ellipti c geometr y b y appealin g t o th e consistency o f Euclidea n geometry . Wha t emerges , then, i s onl y this : ellipti c geometr y i s consisten t i f Eu clidean geometr y i s consistent . T h e authorit y o f Eucli d is thu s invoke d t o demonstrat e th e consistenc y o f a system whic h challenge s th e exclusiv e validit y of Euclid . T h e inescapabl e questio n is : Ar e th e axiom s o f th e Euclidean syste m itsel f consistent ? An answe r t o thi s question , hallowed , a s w e hav e noted, b y a lon g tradition , i s tha t th e Euclidea n axiom s are tru e a n d ar e therefor e consistent . Thi s answe r i s n o longer r e g a r d e d a s acceptable ; w e shal l r e t u r n t o i t presently a n d explai n wh y i t i s unsatisfactory . A n o t h e r answer i s tha t th e axiom s j i be wit h o u r actual , t h o u g h limited, experienc e o f spac e a n d tha t w e ar e justifie d

Fig. 2 The non-Euclidea n geometr y o f th e "ellipti c plane" can b e repre sented b y a Euclidean model . The ellipti c plane become s th e sur face o f a Euclidea n sphere , point s o n th e plan e becom e pair s of antipodal points on this surface, straight lines in the plane become great circles . Thus , a portio n o f th e ellipti c plan e bounde d b y

The Problem of Consistency 1 9 in extrapolatin g fro m th e smal l t o th e universal . But , although m u c h inductiv e evidenc e ca n b e a d d u c e d t o support thi s claim , o u r bes t proo f woul d b e logicall y incomplete. Fo r eve n i f al l th e observe d fact s ar e i n a g r e e m e n t wit h th e axioms , th e possibilit y i s o p e n tha t a h i t h e r t o unobserve d fac t ma y contradic t t h e m a n d s o destroy thei r titl e t o universality . Inductiv e considera tions ca n sho w n o m o r e tha n tha t th e axiom s ar e plau sible o r probabl y true . Hilbert trie d ye t a n o t h e r r o u t e t o th e top . T h e clu e to hi s wa y la y i n Cartesia n coordinat e geometry . I n hi s interpretation Euclid' s axiom s wer e simpl y trans formed int o algebrai c truths . Fo r instance , i n th e axi oms fo r plan e geometry , constru e th e expressio n 'point' t o signif y a pai r o f n u m b e r s , th e expressio n 'straight line ' th e (linear ) relatio n betwee n n u m b e r s expressed b y a first degre e equatio n wit h tw o un knowns, th e expressio n 'circle ' th e relatio n betwee n n u m b e r s expresse d b y a quadrati c equatio n o f a certai n form, a n d s o on . T h e geometri c statemen t tha t tw o distinct point s uniquel y d e t e r m i n e a straigh t lin e i s t h e n transforme d int o th e algebrai c trut h tha t tw o dis tinct pair s o f n u m b e r s uniquel y d e t e r m i n e a linea r re lation; th e geometri c t h e o r e m tha t a straigh t lin e inter sects a circl e i n a t mos t tw o points , int o th e algebrai c segments o f straigh t line s i s depicte d a s a portio n o f th e spher e bounded b y parts of great circles {center). Two line segments in the elliptic plan e ar e tw o segment s o f grea t circle s o n th e Euclidea n sphere (bottom), an d these, if extended, indeed intersect , thus contradicting the Euclidean parallel postulate.

20 GodeVs

Proof

t h e o r e m tha t a pai r o f simultaneou s equation s i n tw o unknowns (on e o f whic h i s linea r a n d th e o t h e r quad ratic o f a certai n type ) d e t e r m i n e a t mos t tw o pair s o f real n u m b e r s ; a n d s o on . I n brief , t h e consistenc y o f the Euclidea n postulate s i s established b y showing tha t they ar e satisfie d b y a n algebrai c model . Thi s m e t h o d of establishin g consistenc y i s powerful a n d effective . Ye t it, too , i s vulnerable t o th e objectio n alread y se t forth . For, again , a p r o b l e m i n o n e d o m a i n i s resolve d b y transferring i t t o another . Hilbert' s a r g u m e n t fo r t h e consistency o f hi s geometri c postulate s show s tha t i f algebra i s consistent , s o i s hi s geometri c system . T h e proof i s clearl y relativ e t o t h e assume d consistenc y o f a n o t h e r syste m a n d i s n ot a n "absolute " proof . In th e variou s attempt s t o solv e th e p r o b l e m o f con sistency ther e i s o n e persisten t sourc e o f difficulty . I t lies i n th e fac t tha t th e axiom s ar e interprete d b y mod els compose d o f a n infinit e n u m b e r o f elements . Thi s makes i t impossible t o encompass th e models i n a finite n u m b e r o f observations ; h e n c e th e trut h o f th e axiom s themselves i s subjec t t o doubt . I n th e inductiv e argu m e n t fo r th e trut h o f Euclidea n geometry , a finite n u m b e r o f observe d fact s a b o u t spac e ar e presumabl y in a g r e e m e n t wit h th e axioms . Bu t th e conclusio n tha t the a r g u m e n t seek s t o establis h involve s a n extrapola tion fro m a finite t o a n infinit e se t of data . Ho w can we justify thi s j u m p? O n th e o t h e r h a n d , th e difficult y i s minimized, i f n o t completel y eliminated , wher e a n ap propriate m o d e l ca n b e devise d tha t contain s onl y a finite n u m b e r o f elements . T h e triangl e m o d e l use d t o

The Problem of Consistency 2 1 show th e consistenc y o f th e five abstrac t postulate s fo r the classe s K a n d L i s finite; a n d i t i s comparativel y simple t o d e t e r m i n e b y actua l inspectio n w h e t h e r al l the element s i n th e m o d e l actuall y satisf y th e postu lates, a n d thu s w h e t h e r the y ar e tru e (an d h e n c e con sistent) . To illustrate : b y examinin g i n t u r n al l th e ver tices o f th e m o d e l triangle , o n e ca n lear n w h e t h e r an y two o f t h e m li e o n jus t o n e side—s o tha t th e first postulate i s establishe d a s true . Sinc e al l th e element s of th e model , a s wel l a s th e relevan t relation s a m o n g t h e m , ar e o p e n t o direc t a n d exhaustiv e inspection , a n d sinc e th e likelihoo d o f mistake s occurrin g i n in specting t h e m i s practicall y nil , th e consistenc y o f t h e postulates i n thi s cas e i s n o t a matte r fo r g e n u i n e doubt. Unfortunately, mos t o f th e postulat e system s tha t constitute th e foundation s o f i m p o r t a n t b r a n c h e s o f mathematics c a n n o t b e m i r r o r e d i n finite models . Con sider th e postulat e i n elementar y arithmeti c whic h as serts tha t ever y intege r ha s a n immediat e successo r differing fro m an y p r e c e d i n g integer . I t i s eviden t tha t the m o d e l n e e d e d t o tes t th e se t t o which thi s postulat e belongs c a n n o t b e finite, b u t mus t contai n a n infinit y of elements . I t follow s tha t th e trut h (an d s o th e consis tency) o f th e se t c a n n o t b e establishe d b y an exhaustiv e inspection o f a limite d n u m b e r o f elements . Appar ently w e hav e r e a c h e d a n impasse . Finit e model s suf fice, i n principle , t o establis h th e consistenc y o f certai n sets o f postulates ; b u t thes e ar e o f sligh t mathematica l importance. Non-finit e models , necessar y fo r th e inter -

22 GodeVs

Proof

pretation o f mos t postulat e system s o f mathematica l significance, ca n b e describe d onl y i n genera l terms ; a n d w e c a n n o t conclud e a s a matte r o f cours e tha t th e descriptions ar e fre e fro m conceale d contradictions . It i s temptin g t o sugges t a t thi s p o i n t tha t w e ca n b e sure o f th e consistenc y o f formulation s i n whic h n o n finite model s ar e describe d i f th e basi c notion s em ployed ar e transparentl y "clear " a n d "distinct. " Bu t th e history o f t h o u g h t ha s n o t deal t kindl y wit h th e doc trine o f clea r a n d distinc t ideas , o r wit h th e doctrin e o f intuitive knowledg e implici t i n th e suggestion . I n cer tain area s o f mathematica l researc h i n whic h assump tions abou t infinit e collection s pla y centra l roles , radi cal contradiction s hav e t u r n e d u p , i n spit e o f th e intuitive clarit y o f t h e notion s involve d i n th e assump tions a n d despit e th e seemingl y consisten t characte r o f the intellectua l construction s performed . Suc h contra dictions (technicall y referre d t o a s "antinomies" ) hav e e m e r g e d i n th e theor y o f infinit e n u m b e r s , develope d by Geor g Canto r i n th e n i n e t e e n t h century ; a n d th e occurrence o f thes e contradiction s ha s m a d e plai n tha t the a p p a r e n t clarit y o f eve n suc h a n elementar y notio n as tha t o f class* (o r aggregate) doe s n o t guarante e th e

* In this book, we use the term "class" to mean what most people today ten d t o cal l "sets. " I n th e decade s whe n Whitehea d an d Russell developed Principia Mathematica and Godel made his discoveries, th e mor e commo n ter m wa s "class, " however, an d accord ingly, we shall use that word, since it is more reflective of the epoch of which we are writing. [—Ed.]

The Problem of Consistency 2 3 consistency o f an y particula r syste m buil t o n it . Sinc e the mathematica l theor y o f classes , whic h deal s wit h the propertie s a n d relation s o f aggregate s o r collec tions o f elements , i s ofte n a d o p t e d a s th e foundatio n for o t h e r branche s o f mathematics , a n d i n particula r for n u m b e r theory , i t i s p e r t i n e n t t o as k w h e t h e r con tradictions simila r t o thos e e n c o u n t e r e d i n t h e theor y of infinit e classe s infec t th e formulation s o f o t h e r part s of mathematics . In p o i n t o f fact , Bertran d Russel l constructe d a con tradiction withi n th e framewor k o f elementar y logi c itself tha t i s precisel y analogou s t o th e contradictio n first develope d i n t h e Cantoria n theor y o f infinit e clas ses. Russell' s antinom y ca n b e state d a s follows. Classe s seem t o b e o f tw o kinds : thos e whic h d o n o t contai n themselves a s m e m b e r s , a n d thos e whic h d o . A clas s will b e calle d "normal " if , a n d onl y if , i t doe s n o t con tain itsel f a s a m e m b e r; otherwis e i t will be calle d "non normal." A n exampl e o f a n o r m a l clas s i s t h e clas s o f mathematicians, fo r patentl y th e clas s itsel f i s n o t a mathematician a n d i s therefore n o t a m e m b e r o f itself . An exampl e o f a non-norma l clas s i s th e clas s o f al l thinkable things ; fo r t h e clas s o f al l thinkabl e thing s is itsel f thinkabl e a n d i s therefor e a m e m b e r o f itself . Let ' N ' b y definitio n stan d fo r t h e clas s o f all n o r m a l classes. W e as k w h e t h er N itsel f i s a n o r m a l class . I f N is normal , i t i s a m e m b e r o f itsel f (fo r b y definitio n N contains al l n o r m al classes) ; but , in tha t case , N i s n o nnormal, becaus e b y definition a class that contain s itsel f as a m e m b e r i s non-normal. O n t h e o t h e r h a n d , i f N i s

24 GodeVs

Proof

non-normal, i t i s a m e m b e r o f itsel f (b y definitio n o f "non-normal"); but , i n tha t case , N i s normal , becaus e by definitio n th e m e m b e r s o f N ar e n o r m a l classes . I n short, N i s n o r m a l if , a n d onl y if , N i s non-normal . I t follows tha t th e statemen t ' N i s n o r m a l ' i s b o t h tru e a n d false . Thi s fata l contradictio n result s fro m a n un critical us e o f th e apparentl y pelluci d n o t i o n o f "class. " O t h e r paradoxe s wer e foun d later , eac h o f t h e m con structed b y m e a n s o f familia r a n d seemingl y cogen t m o d e s o f reasoning . Mathematician s cam e t o realiz e that i n developin g consisten t system s familiarit y a n d intuitive clarit y ar e wea k reed s t o lea n on . We hav e see n th e importanc e o f th e p r o b l e m o f consistency, a n d w e hav e acquainte d ourselve s wit h th e classically standar d m e t h o d fo r solvin g i t wit h th e h e l p of models . I t ha s b e e n show n tha t i n mos t instance s the p r o b l e m require s th e us e o f a non-finit e model , the descriptio n o f which ma y itsel f concea l inconsisten cies. W e mus t conclud e that , whil e th e m o d e l m e t h o d is a n invaluabl e mathematica l tool , i t doe s n o t suppl y a final answe r t o th e p r o b l e m i t was designe d t o solve .

I

l

l

Absolute Proof s o f Consistenc y T h e limitation s i n h e r e n t i n th e us e o f model s fo r estab lishing consistency , a n d th e growin g a p p r e h e n s i o n tha t the standar d formulation s o f man y mathematica l sys tems migh t al l h a r b o r interna l contradictions , le d t o new attack s u p o n th e p r o b l e m . A n alternativ e t o rela tive proof s o f consistenc y wa s p r o p o s e d b y Hilbert . H e sought t o construc t "absolute " proofs , b y whic h th e consistency o f system s coul d b e establishe d withou t as suming th e consistenc y o f som e o t h e r system . W e mus t briefly explai n thi s a p p r o a c h a s a furthe r preparatio n for u n d e r s t a n d i n g Godel' s achievement . T h e firs t ste p i n th e constructio n o f a n absolut e proof, a s Hilber t conceive d th e matter , i s th e complete formalization o f a deductiv e system . Thi s involve s drain ing th e expression s occurrin g withi n t h e syste m o f al l meaning: the y ar e t o b e r e g a r d e d simpl y a s empt y signs. Ho w thes e sign s ar e t o b e c o m b i n e d a n d manip ulated i s to b e se t forth i n a se t o f precisel y state d rules . T h e purpos e o f thi s p r o c e d u r e i s t o construc t a syste m of sign s (calle d a "calculus" ) whic h conceal s n o t h i n g 25

26 GodeVs

Proof

a n d whic h ha s i n i t onl y tha t whic h w e explicitl y p u t into it . T h e postulate s a n d t h e o r e m s o f a completel y formalized syste m ar e "strings " (o r finitely lon g se quences) o f meaningless marks, constructe d accordin g to rule s fo r combinin g th e elementar y sign s o f th e system int o large r wholes . Moreover , whe n a syste m has b e e n completel y formalized , th e derivatio n o f t h e o r e m s fro m postulate s i s n o t h i n g m o r e tha n th e transformation (pursuan t t o rule ) o f o n e se t o f suc h "strings" int o a n o t h e r se t o f "strings. " I n thi s wa y th e d a n g e r i s eliminate d o f usin g an y unavowe d princi ples o f reasoning . Formalizatio n i s a difficul t a n d tricky business , b u t i t serve s a valuabl e purpose . I t re veals structur e a n d functio n i n nake d clarity , a s doe s a cutawa y workin g m o d e l o f a m a c h i n e . W h e n a sys tem ha s b e e n formalized , th e logica l relation s be tween mathematica l proposition s ar e expose d t o view ; o n e i s abl e t o se e th e structura l pattern s o f variou s "strings" o f "meaningless " signs , ho w the y h a n g to gether, ho w the y ar e combined , ho w the y nes t i n o n e another, a n d s o on . A pag e covere d wit h th e "meaningless " mark s o f such a formalize d mathematic s doe s n o t assert anything — i t i s simpl y a n abstrac t desig n o r a mosai c possessin g a d e t e r m i n a t e structure. * Ye t i t i s clearl y possibl e t o

* A more accurat e way to describe such a formalized calculu s is to say that it s symbols may appear to hav e meanings (an d it s rules may well push its symbols around in such a way that they act exacdy as symbols with the desired meanings would act), but their behavior

Absolute Proofs of Consistency 2 7 describe th e configuration s o f suc h a syste m a n d t o make statement s a b o u t th e configuration s a n d thei r various relation s t o o n e another . O n e ma y sa y tha t a "string" i s palindromi c (i.e. , i s th e sam e w h e n rea d backwards a s forwards) o r tha t it has two symbols in com m o n wit h a n o t h e r "string, " o r tha t o n e "string " i s m a d e u p o f thre e others , a n d s o on . Suc h statement s ar e evi dently meaningfu l a n d ma y conve y i m p o r t a n t informa tion abou t th e forma l system . I t mus t no w b e observed , however, tha t suc h meaningfu l statement s about a mean ingless (o r formalized ) mathematica l syste m plainl y d o n o t themselve s b e l o n g t o tha t system . The y belon g to wha t Hilber t calle d "meta-mathematics, " t o th e lan guage tha t i s about mathematics . Meta-mathematica l is not a consequence of their meanings ; indeed, quit e th e revers e is the case. To the extent that formal symbols appear meaningful, thi s appearance devolve s entirely from thei r behavior, which i n tur n is wholly determined b y the system' s rules an d initia l formula s (axi oms) . It is therefore no t entirely unjustified o r unreasonable to see "meaningless strings" as having a type of meaningfulness, a s long as one bear s i n min d tha t an y suc h meanin g i s passive rathe r tha n active. To explain thi s notion metaphorically , th e strings and thei r constituent symbols couldn't care less about any putative meanings that anybod y migh t wis h t o rea d int o them—al l tha t matter s t o them is how the rules manipulate them . To giv e a n analogy , yo u migh t giv e your ca r a pe t nam e an d even thin k o f i t a s animate , bu t th e ca r wil l function exactl y th e same with o r without it s pet nam e an d it s putative "soul" ; all tha t matters i s the machiner y tha t make s i t work. Th e nam e an d sou l have no effect o n that machinery, although they may make it easier for yo u t o relate t o your car . As with favorit e cars , so with forma l calculi. [—Ed.]

28 GodeVs

Proof

statements ar e statement s abou t th e sign s occurrin g within a formalize d mathematica l syste m (i.e. , a cal c u l u s ) — a b o u t th e kind s a n d arrangement s o f suc h signs when the y ar e c o m b i n e d t o for m longe r string s o f marks calle d "formulas, " o r abou t th e relation s betwee n formulas tha t ma y obtai n a s a consequenc e o f th e rule s of manipulatio n specifie d fo r them . A few example s wil l help t o conve y a n u n d e r s t a n d i n g of Hilbert' s distinctio n betwee n mathematic s (i.e. , a system o f meaningles s signs ) a n d meta-mathematic s (meaningful statement s a b o u t mathematics , th e sign s occurring i n th e calculus , thei r a r r a n g e m e n t a n d rela tions). Conside r th e expression : 2+ 3 = 5 This expressio n belong s t o mathematic s (arithmetic ) a n d i s constructe d entirel y o u t o f elementar y arithmet ical signs . O n th e o t h e r h a n d , th e statement 6

2 + 3 = 5 ' i s a n arithmetica l formul a

asserts somethin g abou t th e displaye d expression . T h e statement doe s n o t expres s a n arithmetica l fac t a n d does n o t belon g t o th e forma l languag e o f arithmetic ; it belon g t o meta-mathematics , becaus e i t characterize s a certai n strin g o f arithmetica l sign s a s being a formula . T h e followin g statemen t belong s t o meta-mathematics : If th e sig n ' =' i s t o b e use d i n a formul a o f arith metic, th e sig n mus t b e flanked bot h lef t an d righ t by numerical expressions .

Absolute Proofs of Consistency 2 9

This statemen t lay s down a necessar y conditio n fo r us ing a certain arithmetica l sig n in arithmetica l formulas : the structur e tha t a n arithmetica l formul a mus t hav e if it is to embod y tha t sign . Consider nex t th e thre e formulas : X= X

0= 0 0*0 Each o f thes e belong s t o mathematics , becaus e eac h i s built u p entirel y ou t o f mathematica l signs . Bu t th e statement: 'x' i s a variabl e belongs t o meta-mathematics , sinc e i t characterize s a certain mathematica l sig n a s belongin g t o a specifi c class of sign s (i.e. , t o th e clas s o f variables). Again, th e following statemen t belong s t o meta-mathematics : The formul a ' 0 = 0 ' i s derivabl e fro m th e formul a l x = x' b y substituting th e numera l '0 ' fo r th e vari able V . It specifie s i n wha t manne r on e mathematica l formul a can b e obtaine d fro m anothe r formula , an d thereb y describes ho w th e tw o formula s ar e relate d t o eac h other. Similarly , th e statemen t '0 i 2 0 ' i s not a theorem o f formal syste m X belongs t o meta-mathematics , fo r i t say s o f a certai n formula tha t i t is not derivabl e fro m th e axiom s o f th e

30 GodeVs

Proof

specific forma l calculu s m e n t i o n e d , a n d thu s assert s that a certai n relatio n doe s n o t hol d betwee n th e indi cated formula s o f th e system . Finally , th e nex t state m e n t belong s t o meta-mathematics : Formal syste m X i s consisten t (i.e., i t i s n o t possibl e t o deriv e fro m th e axiom s o f system X tw o formall y contradictor y formulas—fo r ex ample, th e formula s ' 0 = 0 ' a n d ' 0 ¥" 0 ' ) . Thi s i s patently abou t a forma l calculus , a n d assert s tha t pair s of formula s o f a certai n sor t d o n o t stan d i n a specifi c relation t o th e formula s tha t constitut e th e axiom s o f that calculus. 3

3

It is worth noting that the meta-mathematical statements given in the text do not contain a s constituent parts of themselves any of the mathematical signs and formulas that appear i n th e examples . At first glanc e thi s assertion seem s palpably untrue, for th e sign s and formulas ar e plainl y visible . But , i f th e statement s ar e examine d with an analytic eye, it will be seen that the point is well taken. The meta-mathematical statement s contai n th e names o f certai n arith metical expressions , bu t no t th e arithmetica l expression s them selves. Th e distinctio n i s subtl e bu t bot h vali d an d important . I t arises ou t o f th e circumstanc e tha t th e rule s o f Englis h gramma r require tha t no sentenc e literall y contain th e object s t o which th e expressions i n th e sentenc e ma y refer, bu t onl y the names of such objects. Obviously, when we talk about a city we do not put the city itself into a sentence, but only the name of the city; and, similarly, if we wish to say something about a word (o r other linguistic sign), it i s not th e wor d itsel f (o r th e sign ) tha t ca n appea r i n th e sen tence, bu t onl y a nam e fo r th e wor d (o r sign) . Accordin g t o a

Absolute Proofs of Consistency 3 1 It ma y b e tha t th e r e a d e r finds th e wor d 'meta mathematics' p o n d e r o u s a n d th e concep t puzzling . W e shall n o t argu e tha t th e wor d i s pretty ; b u t th e concep t itself wil l perple x n o o n e i f we p o i n t o u t tha t i t i s use d in connectio n wit h a specia l cas e o f a well-know n dis tinction, namel y betwee n a subjec t matte r u n d e r stud y a n d discours e abou t th e subjec t matter . T h e statemen t ' a m o n g phalarope s th e male s incubat e th e eggs ' per tains t o th e subjec t matte r investigate d b y zoologists , a n d belong s t o zoology ; b u t i f we sa y tha t thi s assertio n about phalarope s prove s tha t zoolog y i s irrational , o u r statement i s n o t a b o u t phalaropes , b u t abou t th e asser tion a n d th e disciplin e i n whic h i t occurs , a n d i s meta zoology. I f w e sa y tha t t h e id i s mightie r t h a n t h e ego, we ar e makin g noise s tha t b e l o n g t o o r t h o d o x psycho analysis; b u t i f w e criticiz e thi s statemen t a s meaning standard convention we construct a name for a linguistic expression by placing singl e quotatio n mark s aroun d it . Ou r tex t adhere s t o this convention. 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 ou r intent by: 'x = 5 ' is an equation .

32 GodeVs

Proof

less a n d unprovable , o u r criticis m belong s t o meta psychoanalysis. A n d s o i n th e cas e o f mathematic s a n d meta-mathematics . T h e forma l system s tha t math ematicians construc t belon g i n th e file labele d "math ematics"; th e description , discussion , a n d theorizin g about th e system s belon g i n th e fil e m a r k e d "meta mathematics." T h e i m p o r t a n c e t o o u r subjec t o f recognizin g th e distinction betwee n mathematic s a n d meta-mathemat ics c a n n o t b e overemphasized . Failur e t o respec t i t ha s p r o d u c e d paradoxe s a n d confusion . Recognitio n o f it s significance ha s m a d e i t possibl e t o exhibi t i n a clea r light th e logica l structur e o f mathematica l reasoning . T h e meri t o f th e distinctio n i s tha t i t entail s a carefu l codification o f th e variou s sign s tha t g o int o th e mak ing o f a forma l calculus , fre e o f conceale d assumption s a n d possibl y misleadin g association s o f meaning . Fur t h e r m o r e , i t require s exac t definition s o f th e opera tions a n d logica l rule s o f mathematica l constructio n a n d deduction , man y o f whic h mathematician s h a d applied withou t bein g explicitl y awar e o f wha t the y were using . Hilbert sa w t o th e h e a r t o f th e matter , a n d i t wa s u p o n th e distinctio n betwee n a forma l calculu s a n d it s description tha t h e base d hi s attemp t t o buil d "abso lute" proof s o f consistency . Specifically , h e sough t t o develop a m e t h o d tha t woul d yiel d demonstration s o f consistency a s m u c h beyon d g e n u i n e logica l d o u b t a s the us e o f finite model s fo r establishin g th e consistenc y of certai n set s o f postulates—b y a n analysi s o f a finite

Absolute Proofs of Consistency 3 3 n u m b e r o f structura l feature s o f expression s i n com pletely formalize d calculi . T h e analysi s consist s i n not ing th e variou s type s o f sign s tha t occu r i n a calculus , indicating ho w t o combin e t h e m int o formulas , pre scribing ho w formula s ca n b e obtaine d fro m o t h e r for mulas, a n d d e t e r m i n i n g w h e t h e r formula s o f a give n kind ar e derivabl e fro m other s t h r o u g h explicitl y state d rules o f operation . Hilber t believe d i t might b e possibl e to exhibi t ever y mathematica l calculu s a s a sor t o f "ge ometrical" patter n o f formulas , i n whic h th e formula s stand t o eac h o t h e r i n a finite n u m b e r o f structura l relations. H e therefor e h o p e d t o show , b y exhaustivel y examining thes e structura l propertie s o f expression s within a system , tha t formall y contradictor y formula s c a n n o t b e obtaine d fro m th e axiom s o f give n calculi . An essentia l r e q u i r e m e n t o f Hilbert' s p r o g r a m i n it s original conceptio n wa s tha t demonstration s o f consis tency involv e onl y suc h p r o c e d u r e s a s mak e n o refer ence eithe r t o a n infinit e n u m b e r o f structura l proper ties o f formula s o r t o a n infinit e n u m b e r o f operation s with formulas . Suc h p r o c e d u r e s ar e calle d "finitistic" ; a n d a proo f o f consistenc y conformin g t o thi s require m e n t i s calle d "absolute. " An "absolute " proo f achieve s its objective s b y using a m i n i m u m o f principle s o f infer ence, a n d doe s n o t assum e th e consistenc y o f som e o t h e r se t o f axioms . A n absolut e proo f o f th e consis tency o f a formalize d versio n o f n u m b e r theory , i f suc h a proo f coul d b e constructed , woul d therefor e sho w b y a finitistic meta-mathematica l p r o c e d u r e tha t tw o con tradictory formulas , suc h a s ' 0 = 0 ' a n d it s forma l

34 Godel's

Proof

negation ' ~ ( 0 = 0 ) ' — w h e r e th e sig n ' ~ ' i s use d i n a rule-bound wa y s o a s t o mimic , formally , o u r intuitiv e concept o f " n e g a t i o n " — c a n n o t b o t h b e derive d b y stated rule s o f inferenc e fro m th e axiom s (o r initia l formulas) . 4 It ma y b e useful , b y wa y o f illustration , t o c o m p a r e meta-mathematics a s a theor y o f proo f wit h th e theor y of chess . Ches s i s playe d wit h 3 2 piece s o f specifie d design o n a squar e b o a r d containin g 6 4 squar e subdi visions, wher e th e piece s ma y b e move d i n accordanc e with fixed rules . T h e gam e ca n obviousl y b e playe d without assignin g an y "interpretation " t o th e piece s o r to thei r variou s position s o n th e board , althoug h suc h an interpretatio n coul d b e supplie d i f desired . Fo r ex ample, w e coul d stipulat e tha t a give n paw n i s t o rep resent a certai n r e g i m e n t i n a n army , tha t a give n square i s t o stan d fo r a certai n geographica l region , a n d s o on . Bu t suc h stipulation s (o r interpretations ) are n o t customary ; a n d neithe r th e pieces , n o r th e squares, n o r th e position s o f th e piece s o n th e b o a r d signify anythin g outside th e game . I n thi s sense , th e pieces a n d thei r configuration s o n th e b o a r d ar e "meaningless." T h u s th e gam e i s analogou s t o a for -

4

Hilbert did not give an altogether precise account of just what meta-mathematical procedures are to count as finitistic. In the original version of his program th e requirements for an absolute proof of consistenc y were mor e stringen t tha n i n th e subsequen t expla nations of the program by members of his school.

Absolute Proofs of Consistency 3 5 malized mathematica l calculus . T h e piece s a n d th e squares o f th e b o a r d correspon d t o th e elementar y signs o f th e calculus ; th e lega l position s o f piece s o n the board , t o th e formula s o f th e calculus ; th e initia l positions o f piece s o n th e board , t o th e axiom s o r initial formula s o f th e calculus ; th e subsequen t posi tions o f piece s o n th e board , t o formula s derive d fro m the axiom s (i.e. , t o th e t h e o r e m s ) ; a n d th e rule s o f th e game, t o th e rule s o f inferenc e (o r derivation ) fo r the calculus . T h e parallelis m continues . Althoug h configurations o f piece s o n th e board , lik e th e formu las o f th e calculus , ar e "meaningless, " statement s abou t these configurations , lik e meta-mathematica l state ments abou t formulas , ar e quit e meaningful . A "meta chess" statemen t ma y asser t tha t ther e ar e twent y pos sible o p e n i n g move s fo r White , o r that , give n a certai n configuration o f piece s o n th e b o a r d wit h Whit e t o move, Blac k i s mat e i n thre e moves . Moreover , genera l "meta-chess" t h e o r e m s ca n b e establishe d whos e proo f involves onl y a finite n u m b e r o f permissibl e configura tions o n th e board . T h e "meta-chess " t h e o r e m abou t the n u m b e r o f possibl e o p e n i n g move s fo r Whit e ca n be establishe d i n thi s way ; a n d s o ca n th e "meta-chess " t h e o r e m tha t i f Whit e ha s onl y tw o Knight s a n d th e King, a n d Blac k ha s onl y a King , i t i s impossibl e fo r White t o forc e a mat e agains t Black . Thes e a n d o t h e r "meta-chess" t h e o r e m s can , i n o t h e r words , b e prove d by finitistic m e t h o d s o f reasoning , tha t is , by examinin g in tur n eac h o f a finite n u m b e r o f configuration s tha t

36 GodeVs

Proof

can occu r u n d e r state d conditions . T h e ai m o f Hu bert's theor y o f proof , similarly , wa s t o demonstrat e b y such finitistic m e t h o d s th e impossibilit y o f derivin g cer tain formall y contradictor y formula s i n a give n mathe matical calculus .

I

V

The Systemati c Codificatio n o f Forma l Logi c T h e r e ar e tw o m o r e bridge s t o cros s befor e enterin g u p o n Godel' s proo f itself . W e mus t indicat e ho w a n d why Whitehea d a n d Russell' s Principia Mathematica came int o being ; a n d w e mus t giv e a shor t illustratio n of th e formalizatio n o f a deductiv e system—w e shal l take a fragmen t o f Principia —and explai n ho w it s ab solute consistenc y ca n b e established . Ordinarily, eve n w h e n mathematica l proof s confor m to accepte d standard s o f professiona l rigor , the y suffe r from a n i m p o r t a n t omission . The y embod y principle s (or rules ) o f inferenc e n o t explicitl y formulated , o f which mathematician s ar e frequentl y unaware . Tak e Euclid's proo f tha t ther e i s n o greates t prim e n u m b e r (a n u m b e r i s p r i m e i f i t i s divisibl e withou t r e m a i n d e r by n o n u m b e r o t h e r t h a n 1 a n d th e n u m b e r itself) . T h e argument , cas t i n th e for m o f a reductio ad absurdum, run s a s follows : Suppose, i n contradictio n t o wha t th e proo f seek s t o establish, tha t ther e i s a greates t p r i m e n u m b e r . W e designate i t b y V . T h e n : 37

38 Godel's

Proof

1. x i s th e greates t prim e 2. For m th e p r o d u c t o f al l prime s les s tha n o r equal t o x, a n d a d d l t o th e product . Thi s yield s a new n u m b e r y, wher e y = ( 2 X 3 X 5 X 7 X . . . X x )+ i 3. I f y is itself a prime , t h e n x is n ot th e greates t prime, fo r y i s obviousl y greate r tha n x 4. I f y i s composit e (i.e. , n o t a p r i m e ) , t h e n again x i s n o t th e greates t prime . Fo r i f y i s com posite, i t mus t hav e a prim e diviso r z; a n d z mus t be differen t fro m eac h o f th e prim e n u m b e r s 2 , 3, 5 , 7 , . . . , x, smalle r tha n o r equa l t o x; h e n ce z must b e a prim e greate r t h a n x 5. Bu t y is eithe r prim e o r composit e 6. H e n c e x is n o t th e greates t prim e 7. T h e r e i s n o greates t p r i m e We hav e state d onl y th e mai n link s o f th e proof . I t can b e shown , however , tha t i n forgin g th e complet e chain a fairl y larg e n u m b e r o f tacitl y accepte d rule s o f inference, a s wel l a s t h e o r e m s o f logic , ar e essential . Some o f thes e belon g t o th e mos t elementar y par t o f formal logic , other s t o m o r e advance d branches ; fo r example, rule s a n d theorem s ar e incorporate d tha t be long t o th e "theor y o f quantification. " Thi s deal s wit h relations betwee n statement s containin g suc h "quanti fying" particle s a s 'all' , 'some' , a n d thei r synonyms . W e shall exhibi t o n e elementar y t h e o r e m o f logi c a n d o n e rule o f inference , eac h o f whic h i s a necessar y b u t silent p a r t n e r i n th e demonstration .

The Systematic Codification of Formal Logic 3 9 Look a t lin e 5 o f th e proof . W h e r e doe s i t com e from? T h e answe r is , fro m th e logica l principl e (o r necessary t r u t h ) : 'Eithe r p or non-jfr' , wher e 'p 9 i s calle d a sententia l variable . Bu t ho w d o w e ge t lin e 5 fro m this theorem ? T h e answe r is , by using th e rul e o f infer ence know n a s th e "Rul e o f Substitutio n fo r Sententia l Variables," accordin g t o whic h a statemen t ca n b e de rived fro m a n o t h e r containin g suc h variable s b y substi tuting an y statement (i n thi s case , 'y is p r i m e ') fo r eac h occurrence o f a distinc t variabl e (i n thi s case , th e vari able '/?') . T h e us e o f thes e rule s a n d logica l principle s is, a s w e hav e said , frequentl y a n al l b u t unconsciou s action. A n d th e analysi s tha t expose s t h e m , eve n i n such relativel y simpl e proof s a s Euclid's, d e p e n d s u p o n advances i n logica l theor y m a d e onl y withi n th e pas t o n e h u n d r e d years. 5 Like Moliere' s Monsieu r J o u r d a i n , who spok e pros e al l his lif e withou t knowin g it , mathe maticians hav e b e e n reasonin g fo r a t leas t tw o millen nia without bein g awar e o f all the principles underlyin g what the y wer e doing . T h e rea l n a t u r e o f th e tool s o f their craf t ha s b e c o m e eviden t onl y withi n recen t times. For almos t tw o t h o u s a n d year s Aristotle' s codifica tion o f valid form s o f deductio n wa s widely regarde d a s complete a n d a s incapabl e o f essentia l improvement . As lat e a s 1787 , th e G e r m a n philosophe r I m m a n u e l 5

For a more detaile d discussio n o f the rules o f inference an d logical principles needed fo r obtaining line s 6 and 7 of the above proof, the reader is referred t o the Appendix, no. 2.

40 GodeVs

Proof

Kant wa s abl e t o sa y tha t sinc e Aristotle , forma l logi c "has n o t b e e n abl e t o advanc e a singl e step , a n d i s t o all appearance s a close d a n d complete d bod y o f doc trine." T h e fac t i s tha t th e traditiona l logi c i s seriousl y incomplete, a n d eve n fail s t o giv e a n accoun t o f man y principles o f inferenc e employe d i n quit e elementar y mathematical reasoning. 6 A renaissanc e o f logica l stud ies i n m o d e r n time s bega n wit h th e publicatio n i n 1847 o f Georg e Boole' s The Mathematical Analysis of Logic. T h e primar y concer n o f Bool e a n d hi s immedi ate successor s wa s t o develo p a n algebr a o f logi c whic h would provid e a precis e notatio n fo r h a n d l i n g m o r e general a n d m o r e varie d type s o f deductio n tha n wer e covered b y traditiona l logica l principles . Suppos e i t i s found tha t i n a certai n schoo l thos e wh o graduat e wit h h o n o r s ar e m a d e u p exactl y o f boy s majorin g i n math ematics a n d girl s n o t majorin g i n thi s subject . Ho w i s the clas s o f mathematic s major s m a d e u p , i n term s o f the o t h e r classe s o f student s mentioned ? T h e answe r i s n o t readil y forthcomin g i f o n e use s onl y th e apparatu s of traditiona l logic . Bu t wit h th e h e l p o f Boolea n alge bra i t ca n easil y b e show n tha t th e clas s o f mathematic s majors consist s exactl y o f boy s graduatin g wit h h o n o r s a n d girl s n o t graduatin g wit h h o n o r s . A n o t h e r lin e o f inquiry , closel y relate d t o th e wor k of nineteenth-centur y mathematician s o n th e founda 6

For example , o f th e principle s involve d i n th e inference : 5 is greater than 3 ; therefore, th e square of 5 is greater than the square of 3.

The Systematic Codification of Formal Logic 4 1 All g e n t l e m e n ar e polite . N o banker s ar e polite . N o g e n t l e m e n ar e bankers . gtp bC p .:gCb # = 0 bp = 0 g* = 0 Symbolic logic was invented i n th e middl e o f th e 19t h cen tury b y th e Englis h mathematicia n Georg e Boole . I n thi s illustration a syllogis m i s translated int o hi s notation i n tw o different ways . In th e uppe r grou p o f formulas , th e symbo l ' C mean s "i s contained in. " Thus 'g C f say s that th e clas s of gentlemen i s included in the class of polite persons. In the lower group o f formulas tw o letters togethe r mea n th e clas s of things having both characteristics. For example, 'bp means the clas s of individuals who are banker s an d polite ; and th e equation l bp = 0' say s that this class has no members. A line above a letter mean s "not. " ( cp, fo r example , means "impolite.") TABLE 1

tions o f analysis , b e c a m e associate d wit h th e Boolea n p r o g r a m . Thi s ne w developmen t sough t t o exhibi t p u r e mathematic s a s a chapte r o f forma l logic ; a n d i t received it s classica l e m b o d i m e n t i n th e Principia Mathematica o f W h i t e h e a d a n d Russel l i n 1910 . Mathemati -

42 GodeVs

Proof

cians o f th e n i n e t e e n t h centur y succeede d i n "arith metizing" algebr a a n d wha t use d t o b e calle d th e "infinitesimal calculus " b y showin g tha t th e variou s no tions employe d i n mathematica l analysi s ar e definabl e exclusively i n number-theoretica l term s (i.e. , i n term s of th e integer s a n d th e arithmetica l operation s u p o n t h e m ) . Fo r example , instea d o f acceptin g th e imagi nary n u m b e r J — 1 a s a somewha t mysteriou s "entity, " it cam e t o b e define d a s a n o r d e r e d pai r o f integer s (0, 1) u p o n whic h certai n operation s o f "addition " a n d "multiplication" ar e performed . Similarly , th e irra tional n u m b e r J2 wa s define d a s a certai n clas s o f ra tional n u m b e r s — n a m e l y , th e clas s o f rational s whos e square i s les s t h a n 2 . W h a t Russel l (and , befor e him , the G e r m a n mathematicia n Gottlo b Frege ) sough t t o show wa s tha t all number-theoretical notions ca n b e de fined i n purel y logica l ideas , a n d tha t al l th e axiom s o f n u m b e r theor y ca n b e d e d u c e d fro m a smal l n u m b e r of basi c proposition s certifiabl e a s purely logica l truths . T o illustrate : th e notio n o f class belongs t o genera l logic. Tw o classe s ar e define d a s "similar " i f ther e i s a one-to-one c o r r e s p o n d e n c e betwee n thei r m e m b e r s , the notio n o f suc h a correspondenc e bein g explicabl e in term s o f o t h e r logica l ideas . A clas s tha t ha s a singl e m e m b e r i s sai d t o b e a "uni t class " (e.g. , th e clas s o f satellites o f th e plane t Earth) ; a n d th e cardina l n u m b e r 1 ca n b e define d a s th e clas s o f al l classe s simila r t o a unit class . Analogou s definition s ca n b e give n o f th e o t h e r cardina l n u m b e r s ; a n d th e variou s arithmetica l operations, suc h a s additio n a n d multiplication , ca n b e

The Systematic Codification of Formal Logic 4 3 defined i n th e notion s o f forma l logic . An arithmetica l statement, e.g. , ' 1 + 1 = 2' , ca n t h e n b e exhibite d a s a c o n d e n s e d transcriptio n o f a statemen t containin g only expression s belongin g t o genera l logic ; a n d suc h purely logica l statement s ca n b e show n t o be deducibl e from certai n logica l axioms . Principia Mathematica thu s a p p e a r e d t o advanc e t h e final solutio n o f th e p r o b l e m o f consistenc y o f mathe matical systems , a n d o f n u m b er theor y i n particular , b y reducing th e p r o b l e m t o tha t o f th e consistenc y o f formal logi c itself . For , i f the axiom s o f n u m b e r theor y are themselve s derivabl e a s t h e o r e m s i n forma l logic , the questio n w h e t h e r th e axiom s ar e consisten t i s equivalent t o th e questio n w h e t h e r th e fundamenta l axioms o f logi c ar e consistent . T h e Frege-Russel l thesi s tha t mathematic s i s onl y a chapter o f logi c has , fo r variou s reason s o f detail , n o t won universa l acceptanc e fro m mathematicians . More over, a s w e hav e noted , th e antinomie s o f th e Canto rian theor y o f transfinit e n u m b e r s ca n b e duplicate d within logi c itself , unles s specia l precaution s ar e take n to preven t thi s o u t c o m e . Bu t are th e measure s a d o p t e d in Principia Mathematica t o outflan k th e antinomie s ad equate t o exclud e all form s o f self-contradictor y con structions? Thi s c a n n o t b e asserte d a s a matte r o f course. Therefor e th e Frege-Russel l reductio n o f arith metic t o logi c doe s n o t provid e a final answe r t o th e consistency problem ; indeed , th e p r o b l e m simpl y emerges i n a m o r e genera l form . But , irrespectiv e o f the validit y o f t h e Frege-Russel l thesis , tw o feature s o f

44 Godel's

Proof

Principia hav e prove d o f inestimabl e valu e fo r th e fur t h e r stud y o f th e consistenc y question . Principia pro vides a remarkabl y comprehensiv e syste m o f notation , with th e hel p o f whic h al l statement s o f p u r e mathe matics (an d o f n u m b e r theor y i n particular ) ca n b e codified i n a standar d m a n n e r ; a n d i t make s explici t most o f th e rule s o f forma l inferenc e use d i n mathe matical demonstration s (eventually , thes e rule s wer e m a d e m o r e precis e a n d complete) . Principia, i n sum , created th e essentia l instrumen t fo r investigatin g th e entire syste m o f n u m b e r theor y a s a n u n i n t e r p r e t e d calculus—that is , a s a syste m o f meaningles s marks , whose formula s (o r "strings" ) ar e c o m b i n e d a n d trans formed i n accordanc e wit h state d rules , o f operation . Because o f it s historica l importance , Principia Mathematica wil l hencefort h constitut e o u r prototypica l ex ample o f a formalizatio n o f n u m b e r theory , a n d th e phrase "un d verwandte r Systeme " ("an d relate d sys tems") i n th e titl e o f Godel' s articl e wil l b e tacitl y in cluded wheneve r w e refe r t o Principia Mathematical th e whole famil y o f suc h system s wil l b e meant .

V

An Exampl e o f a Successfu l Absolut e Proo f of Consistenc y We mus t no w attemp t th e secon d tas k m e n t i o n e d a t the outse t o f th e p r e c e d i n g section , a n d familiariz e ourselves wit h a n important , t h o u g h easil y understand able, exampl e o f a n absolut e proo f o f consistency . B y mastering th e proof , th e reade r wil l b e i n a bette r position t o appreciat e th e significanc e o f Godel' s p a p e r of 1931 . We shal l outlin e ho w a smal l portio n o f Principia, th e elementary logi c o f propositions , ca n b e formalized . This entail s th e conversio n o f th e fragmentar y syste m into a calculu s o f u n i n t e r p r e t e d signs . W e shal l t h e n develop a n absolut e proo f o f consistency . T h e formalizatio n proceed s i n fou r steps . First , a complete catalogu e i s p r e p a r ed o f th e sign s t o b e use d in th e calculus . Thes e ar e it s vocabulary . Second , th e "Formation Rules " ar e lai d down . The y declar e whic h of th e combination s o f th e sign s i n th e vocabular y ar e acceptable a s "formulas " (i n effect , a s sentences) . T h e rules ma y b e viewe d a s constitutin g th e g r a m m a r o f th e 45

46 GodeVs

Proof

system. Third , th e "Transformatio n Rules " ar e stated . They describ e th e precis e structur e o f formula s fro m which othe r formula s o f give n structur e ar e derivable . These rule s are , i n effect , th e rule s o f inference . Fi nally, certai n formula s ar e selecte d a s axiom s (o r a s "primitive formulas"). The y serve as foundation fo r th e entire system . We shal l use th e phras e "theore m o f th e system" t o denot e an y formul a tha t ca n b e derive d from th e axiom s b y successively applyin g th e Transfor mation Rules . B y a forma l "proof " (o r "demonstra tion") w e shal l mea n a finit e sequenc e o f formulas , each o f whic h eithe r i s a n axio m o r ca n b e derive d from precedin g formulas i n th e sequenc e by the Transformation Rules. 7 For th e logi c o f proposition s (ofte n calle d th e "sen tential calculus" ) th e vocabulary (o r lis t of "elementar y signs") i s extremely simple . I t consist s o f variable s an d constant signs . Th e variable s ma y hav e sentence s sub stituted fo r the m an d ar e therefor e calle d "sententia l variables." They ar e th e letter s 'p' , 'q' , 'r' , etc. The constan t sign s ar e eithe r "sententia l connectives " or signs of punctuation. The sententia l connectives are: ' ~ ' whic h i s short fo r 'not ' (and i s called th e "tilde") , ——— 7

It immediatel y follow s tha t axiom s ar e t o b e counte d amon g the theorems .

An Example of a Successful Absolute Proof of Consistency 4 7 V whic

h i s shor t fo r 'or' ,

' D ' whic h i s shor t fo r 'if . . . t h en . . . ', a n d '•' whic h i s shor t fo r ' a n d ' . T h e sign s o f punctuatio n ar e th e left - a n d right-han d r o u n d parentheses , ' ( ' a n d ' ) ' , respectively . T h e Formatio n Rule s ar e s o designe d tha t combina tions o f th e elementar y signs , whic h woul d normall y have th e for m o f sentences , ar e calle d "formulas. " Also , each sententia l variabl e count s a s a formula . More over, i f th e lette r 'S ' stand s fo r a formula , it s forma l negation, namely , ~ (S) , i s als o a formula . Similarly , i f Sj a n d S 2 ar e formulas , s o ar e (Sj ) V (S 2 ), (Sj ) D (S 2 ), a n d (Sj ) • (S 2 ). Eac h o f th e followin g i s a formula : *p\ ' - (/>)' , '(/> ) D (q)\ '(( ) 3 (?) ) v ' i s a formula : n o t th e first, because , whil e '(/?) ' a n d '( — ( V q) D {p\l q) 9'. Or, i f w e substi tute actua l Englis h sentence s fo r 'p 9, w e ca n obtai n each o f th e followin g fro m 'p D p 9: 'Frog s ar e nois y D Frogs ar e noisy' ; '(Bat s ar e blin d V Bats ea t mice ) D (Bats ar e blin d V Bats ea t mice)'. 9 T h e secon d Trans formation Rul e i s th e Rule of Detachment (o r Modus Ponens). Thi s rul e say s tha t fro m tw o formula s havin g the for m S x a n d S 2 D S 2 i t i s alway s permissibl e t o de rive th e formul a S 2. Fo r example , fro m th e tw o formu las l pV ~ p 9 a n d '(/ > V ~ p) D (p D p) 9, w e ca n deriv e '/> D p\ Finally, th e axiom s o f th e calculu s (essentiall y thos e of Principia) ar e th e followin g fou r formulas : On th e othe r hand , suppos e th e formul a l (p D q) D ( ~ q D ~ py ha s already been established , and we decide t o substitute V for th e variable *p* an d *p V r' fo r th e variable 'q\ W e cannot, by this substitution, obtain th e formula l (rD (p\I r)) D (~ qD ~ r) \ because w e hav e faile d t o mak e th e sam e substitutio n fo r each occurrence o f th e variabl e 'q\ Th e correc t substitutio n yield s \r1(pVr))^(~(p\]r)Z)~ry. D ~ r)\ 9

An Example of a Successful Absolute Proof of Consistency 4 9 i. (pyp)

D

p

or, i n ordinar y En glish, i f eithe r p o r p, then p 2. pD (pV

q)

that is , if p, t h e n eithe r por q

1. I f (eithe r H e n r y VII I was a b o o r o r H e n r y VIII wa s a boor ) t h e n H e n r y VII I wa s a b o o r 2. I f psychoanalysi s i s fashionable, t h e n (ei ther psychoanalysi s i s fashionable o r head ache powder s ar e sol d cheap)

3. (pvq) D {qMp) that is , i f eithe r p o r q, t h e n eithe r qor p

3. I f (eithe r I m m a n u e l Kant wa s punctua l o r Hollywood i s sinful) , t h e n (eithe r Holly wood i s sinfu l o r Im m a n u e l Kan t wa s punctual)

4. (pDq)D )D D (rVq)) that is , i f (i f p t h e n 9 ) t h e n (i f (eithe r r or jfr ) t h e n (eithe r r or ^) )

4. I f (i f duck s waddl e t h e n 5 i s a prime ) t h e n (i f (eithe r Chur chill drink s brand y o r ducks waddle ) t h e n (either Churchil l drinks brand y o r 5 i s a prime))

In th e left-han d colum n w e hav e state d th e axioms , with a translatio n fo r each . I n th e right-han d colum n we hav e give n a n exampl e fo r eac h axiom . T h e clum -

50 GodeVs

Proof

siness o f th e translations , especiall y i n th e cas e o f th e final axiom , wil l p e r h a p s hel p th e r e a d e r t o realiz e th e advantages o f using a specia l symbolis m i n forma l logic . It i s als o i m p o r t a n t t o observ e tha t th e nonsensica l illustrations use d a s substitutio n instance s fo r th e axi oms a n d th e fac t tha t t h e consequent s bea r n o mean ingful relatio n t o th e antecedent s i n th e conditiona l sentences i n n o wa y affec t th e validit y o f th e logica l connections asserte d i n th e examples . Each o f thes e axiom s ma y see m "obvious " a nd trivial . Nevertheless, i t i s possibl e t o deriv e fro m t h e m wit h the hel p o f th e state d Transformatio n Rule s a n indefi nitely larg e clas s o f t h e o r e m s whic h ar e fa r fro m obvi ous o r trivial . Fo r example , th e formul a 'UpDq) D q)

((rDs) D s)

t)) D ((uD ((rD

s)D

u) D ((pDu) D

t)) (sDt))Y (s

can b e derive d a s a t h e o r e m . W e are , however , n o t interested fo r th e m o m e n t i n derivin g t h e o r e m s fro m the axioms . O u r ai m i s t o sho w tha t thi s se t o f axiom s is n o t contradictory , tha t is , t o prov e "absolutely " tha t it i s impossible b y usin g th e Transformatio n Rule s t o derive fro m th e axiom s a formul a S togethe r wit h it s formal negatio n ~ S . Now, i t h a p p e n s tha t 'p D ( ~ p D q) 9 (i n words : 'i f p, t h e n i f not-p t h e n q 9) i s a t h e o r e m i n th e calculus . (We shal l accep t thi s a s a fact , withou t exhibitin g th e derivation.) Suppose , then , tha t som e formul a S as wel l as it s forma l negatio n ~ S wer e deducibl e fro m th e axioms. B y substitutin g S fo r th e variabl e '/? ' i n th e

An Example of a Successful Absolute Proof of Consistency 5 1 t h e o r e m (a s permitte d b y th e Rul e o f Substitution) , a n d applyin g th e Rul e o f D e t a c h m e n t twice , th e for mula Y woul d b e deducible. 1 0 But , i f th e formul a con sisting o f th e variabl e ' V q)D D r)]

It ca n b e show n tha t thi s formulate s a necessary truth. The reade r wil l recognize tha t thi s formula state s mor e compactly wha t i s conveye d b y th e followin g muc h longer statement : If (i f p then r) , then [i f (i f q then r) the n (if (eithe r p or q) then r)] As pointe d ou t i n th e text , ther e i s a rul e o f infer ence i n logic called th e Rule of Substitution fo r Senten tial Variables . Accordin g t o thi s Rule , a sentenc e S 2 follows logicall y from a sentence S x which contain s sen tential variables , i f th e forme r i s obtaine d fro m th e latter b y uniforml y substitutin g an y sentence s fo r th e

116 Appendix variables. I f we appl y thi s rul e t o th e t h e o r e m just m e n tioned, substitutin g 'y is p r i m e' fo r 'p\ 'y is composite ' for ' q \ a n d 'x i s n o t th e greates t p r i m e ' fo r V , w e obtain th e following : (y is p r i me D x is n ot th e greates t prime ) D [(yis composit e D x is n ot th e greates t prime ) D ( (y i s p r i me V y is composite) D x is n ot t h e greatest p r i m e ) ] T h e r e a d e r wil l readil y n o t e tha t th e conditiona l sentence withi n th e firs t pai r o f parenthese s (i t occur s o n th e firs t lin e o f thi s instanc e o f th e t h e o r e m ) simpl y duplicates lin e 3 o f Euclid' s proof . Similarly , t h e con ditional sentenc e withi n th e firs t pai r o f parenthese s inside th e squar e bracket s (i t occur s a s the secon d lin e of th e instanc e o f th e t h e o r e m ) duplicate s lin e 4 o f t h e proof. Also , th e alternativ e sentenc e insid e th e squar e brackets duplicate s lin e 5 o f th e proof . We no w mak e us e o f a n o t h e r rul e o f inferenc e known a s th e Rul e o f D e t a c h m e n t (o r "Modu s Po nens"). Thi s rul e permit s u s t o infe r a sentenc e S 2 a nd from tw o o t h e r sentences , o n e o f whic h i s S x a n d th e other, S 2 D S 2. W e appl y thi s Rul e thre e times : first , using lin e 3 o f Euclid' s proo f a n d th e abov e instanc e of th e logica l theorem ; next , th e resul t obtaine d b y thi s application a n d lin e 4 o f th e proof ; and , finally , thi s latest resul t o f th e applicatio n a n d lin e 5 o f th e proof . T h e o u t c o m e i s line 6 o f t h e proof . T h e derivatio n o f lin e 6 fro m line s 3 , 4, a n d 5 thu s involves th e taci t us e o f tw o rule s o f inferenc e a n d a

Notes 11 7 t h e o r e m o f logic . T h e t h e o r e m a n d rule s belon g t o th e elementary par t o f logica l theory , th e sententia l calcu lus. Thi s deal s wit h th e logica l relation s betwee n state ments c o m p o u n d e d o u t o f o t h e r statement s wit h th e help o f sententia l connective , o f whic h ' D ' a n d 'V ' ar e examples. A n o t h e r suc h connectiv e i s th e conjunctio n ' a n d ' , fo r whic h th e d o t '• ' i s used a s a s h o r t h a n d form ; thus th e conjunctiv e statemen t c p a n d q' i s writte n a s 'p - q\ T h e sig n '— ' represent s th e negativ e particl e 'not'; thu s 'not-/? ' i s written a s '~p\ Let u s examin e th e transitio n i n Euclid' s proo f fro m line 6 t o lin e 7 . Thi s ste p c a n n o t b e analyze d wit h th e help o f th e sententia l calculu s alone . A rul e o f infer ence i s require d whic h belong s t o a m o r e advance d part o f logica l theory—namely , tha t whic h take s n o t e of th e interna l complexit y o f statement s embodyin g expressions suc h a s 'all' , 'every' , 'some' , a n d thei r syn onyms. Thes e ar e traditionall y calle d quantifiers, a n d the b r a n c h o f logica l theor y tha t discusse s thei r rol e i s the theor y o f quantification . It i s necessar y t o explai n som e o f th e notatio n em ployed i n thi s m o r e advance d secto r o f logic , a s a pre liminary t o analyzin g th e transitio n i n question . I n ad dition t o th e sententia l variable s fo r whic h sentence s may b e substituted , w e mus t conside r th e categor y o f "individual variables, " such a s V , ' / , 'z' , etc. , for whic h the n a m e s o f individual s ca n b e substituted . Usin g these variables , th e universa l statemen t 'Al l prime s greater tha n 2 ar e o d d ' ca n b e r e n d e r e d : 'Fo r ever y x, if x i s a p r i m e greate r tha n 2 , t h e n x i s o d d ' . T h e

118 Appendix expression 'fo r ever y x 9 is calle d th e universal quantifier, a n d i n c u r r e n t logica l notatio n i s abbreviate d b y t h e sign '(#)' . T h e universa l statemen t ma y therefor e b e written: (x) (x i s a prim e greate r tha n 2 D x is odd ) F u r t h e r m o r e , th e "particular " (o r "existential" ) state m e n t 'Som e integer s ar e composite ' ca n b e r e n d e r e d by T h e r e i s a t leas t o n e x suc h tha t x i s a n intege r a n d x is composite' . T h e expressio n 'ther e i s a t leas t o n e x' is calle d th e existential quantifier, a n d i s currentl y abbre viated b y th e notatio n '(3x)' . T h e existentia l statemen t just m e n t i o n e d ca n b e transcribed : (3x) (xi s a n intege r • x is composite ) It i s now t o b e observe d tha t man y statement s implic itly us e m o r e t h a n o n e quantifier , s o tha t i n exhibitin g their tru e structur e severa l quantifier s mus t appear . Before illustratin g thi s point , le t u s a d o p t certai n ab breviations fo r wha t ar e usuall y calle d predicat e expres sions or , m o r e simply , predicates . W e shal l us e T r (x) 9 as shor t fo r ' x i s a p r i m e n u m b e r ' ; a n d 'G r (x, z) 9 a s short fo r 'x i s greate r tha n z' . Conside r th e statement : 'x i s th e greates t p r i m e ' . It s m e a n i n g ca n b e m a d e m o r e explici t b y th e followin g locution : 'x i s a prime , and, fo r ever y z which i s a prim e b u t differen t fro m x, x is greater t h a n z . With th e hel p o f o u r variou s abbre viations, th e statemen t l x i s th e greates t p r i m e ' ca n b e written:

Notes 11

9

Pr (x) • (z) [(P r (z ) • ~ (x = z) ) D Gr (* , z)] Literally, thi s says: 'x is a prime and , for ever y z, if z is a prime an d z is not equa l t o x then x is greater tha n z' . We recogniz e i n th e symbolic sequenc e a formal, pain fully explici t renditio n o f th e conten t o f lin e 1 in Eu clid's proof . Next, conside r ho w t o expres s i n ou r notatio n th e statement ' x is not th e greates t prime' , whic h appear s as line 6 of the proof. Thi s ca n be presented as : Pr (x) • (3z) [P r (z ) • Gr (z , *)] Literally, i t says: 'x is a prime an d there i s at least one z such tha t z is a prime an d z is greater tha n x' . Finally, th e conclusio n o f Euclid' s proof , lin e 7 , which assert s tha t ther e i s no greatest prime, is symbolically transcribed by: (x) [P r (x ) D (3z ) (P r (z ) • Gr (z , x))] which says : Tor ever y x, if x is a prime, ther e i s at leas t one z such tha t z is a prim e an d z is greate r tha n x\ The reade r will observe tha t Euclid's conclusion implic itly involves the use of more tha n on e quantifier . We ar e ready t o discuss the step from Euclid' s line 6 to lin e 7 . There i s a theorem i n logic which reads : (p- q)D (PD q) or when translated , T f both p and q, then {\£p then q)\ Using the Rule of Substitution, an d substituting 'P r (x)' for c p', an d '(3z ) [P r (z ) • Gr (z , *)]' fo r '?' , we obtain :

120 Appendix (Pr (x ) • (3z ) [P r (z ) • Gr (z , x)]) D (Pr (x ) D (3z ) [P r (z ) • Gr (z , x)] ) T h e a n t e c e d e n t (firs t line ) o f thi s instanc e o f th e the o r e m simpl y duplicate s lin e 6 o f Euclid' s proof ; i f w e apply th e Rul e o f Detachment , w e ge t (Pr (x ) D (3z ) [P r (z ) • Gr (z , x)]) According t o a Rul e o f Inferenc e i n th e logica l theor y of quantification , a sentenc e S 2 havin g th e for m '(x ) ( . . . x. . . )' ca n alway s b e inferre d fro m a sentenc e S 1 having th e for m * ( . . . x . . . ) '• I n o t h e r words , th e sen tence havin g th e quantifie r ' ( x ) ' a s a prefi x ca n b e derived fro m th e sentenc e tha t doe s n o t contai n t h e prefix b u t is like th e forme r i n o t h e r respects . Applyin g this rul e t o th e sentenc e las t displayed , w e hav e lin e 7 of Euclid' s proof . T h e mora l o f o u r stor y i s tha t th e proo f o f Euclid' s t h e o r e m tacitl y involve s th e us e n o t onl y o f t h e o r e m s a n d rule s o f inferenc e belongin g t o th e sententia l cal culus, b u t als o o f a rul e o f inferenc e i n th e theor y o f quantification. 3. (pag e 54 ) T h e carefu l r e a d e r ma y d e m u r a t thi s point, wonderin g somethin g lik e this . T h e propert y of bein g a tautolog y ha s b e e n define d i n notion s o f truth a n d falsity . Ye t thes e notion s obviousl y involv e a reference t o somethin g outside th e forma l calculus . Therefore, th e p r o c e d u r e m e n t i o n e d i n t h e tex t i n effect offer s a n interpretation o f t h e calculus , b y supply ing a m o d e l fo r th e system . Thi s bein g so , th e author s

Notes 12 1 have n o t d o n e wha t the y promised , namely , t o defin e a propert y o f formula s i n term s o f purel y structura l features o f th e formula s themselves . I t seem s tha t th e difficulty n o t e d i n Sectio n I I o f th e t e x t — t h a t proof s of consistenc y whic h ar e base d o n models , a n d whic h argue fro m th e trut h o f axiom s t o thei r consistency , merely shif t th e p r o b l e m — h a s not , afte r all , b e e n suc cessfully outflanked . Wh y t h e n cal l th e proo f "absolute " rather tha n relative ? T h e objectio n i s wel l take n whe n directe d agains t the expositio n i n th e text . Bu t w e a d o p t e d thi s for m s o as n o t t o overwhel m th e r e a d e r u n a c c u s t o m e d t o a highly abstrac t presentatio n restin g o n a n intuitivel y o p a q u e proof . Becaus e m o r e venturesom e reader s ma y wish t o b e expose d t o th e rea l thing , t o se e a n unpret tified definitio n tha t i s n o t o p e n t o th e criticism s i n question, w e shal l suppl y it . R e m e m b e r tha t a formul a o f th e calculu s i s eithe r o n e o f th e letter s use d a s sententia l variable s (w e wil l call suc h formula s "elementary" ) o r a c o m p o u n d o f these letters , o f th e sign s employe d a s sententia l con nective, a n d o f th e parentheses . W e agre e t o plac e eac h elementary formul a i n o n e o f tw o mutuall y exclusiv e a n d exhaustiv e classe s K a a n d K 2 . Formula s tha t ar e n o t elementary ar e place d i n thes e classe s p u r s u a n t t o th e following conventions : i) A formul a havin g th e for m S 2 V S2 i s place d in clas s K 2 i f both S x a n d S 2 ar e i n K 2; otherwise , i t is placed i n K a.

122 Appendix ii) A formul a havin g th e for m S 2 D S 2 is place d in K 2, i f S j i s i n K a a n d S 2 i s i n K 2; otherwise , i t i s placed i n K P iii) A formul a havin g th e for m S j • S 2 i s place d in K 1? i f both S 1 a n d S 2 ar e i n K ^ otherwise , i t i s placed i n K 2 . iv) A formul a havin g th e for m ~ S i s place d i n K2, i f S is in K 2; otherwise , i t i s place d i n K x. We t h e n defin e th e propert y o f bein g tautologous : a formula i s a tautolog y if , a n d onl y if , i t fall s i n th e clas s K2 n o matte r i n whic h o f th e tw o classe s it s elementar y constituents ar e placed . I t i s clea r tha t th e propert y o f being a tautolog y ha s no w b e e n describe d withou t us ing an y m o d e l o r interpretatio n fo r th e system . W e ca n discover whethe r o r n o t a formul a i s a tautolog y simpl y by testin g it s structur e b y th e abov e conventions . Such a n examinatio n show s tha t eac h o f th e fou r axioms i s a tautology . A convenien t p r o c e d u r e i s t o construct a tabl e tha t list s al l th e possibl e way s in whic h the elementar y constituent s o f a give n formul a ca n b e placed i n th e tw o classes . F r o m thi s lis t w e ca n deter mine, fo r eac h possibility , t o whic h clas s th e nonele mentary c o m p o n e n t formula s o f th e give n formul a be long, a n d t o whic h clas s th e entir e formul a belongs . Take th e first axiom . T h e tabl e fo r i t consist s o f thre e columns, eac h h e a d e d b y o n e o f th e elementar y o r n o n e l e m e n t a r y c o m p o n e n t formula s o f th e axiom , a s well a s b y th e axio m itself . U n d e r eac h h e a d i n g i s indi cated th e clas s t o whic h th e particula r ite m belongs ,

Notes 12

3

for eac h o f th e possibl e assignment s o f th e elementar y constituents t o th e tw o classes. The tabl e i s as follows:

(pvp)

p

(pvp) = p

K1

K1

K2

K1 K1

K2

The first colum n mention s th e possibl e way s o f classi fying th e sole elementary constituen t o f the axiom. Th e second colum n assign s th e indicate d non-elementar y component t o a class, o n th e basi s o f conventio n (i) . The las t colum n assign s th e axio m itsel f t o a class , o n the basi s o f conventio n (ii) . Th e final colum n show s that th e first axio m fall s i n clas s K 1} irrespective o f th e class in which it s sole elementar y constituen t i s placed . The axio m i s therefore a tautology . For th e secon d axiom , th e tabl e is:

p

K1 K1

K2 K2

q

(pVq)

K2

K1

K1

K1

K1

K1

K1

K1

K1

K2

K2

pD (pVq)

K1

The first tw o column s lis t th e fou r possibl e way s o f classifying th e tw o elementar y constituent s o f th e ax iom. Th e secon d colum n assign s th e non-elementar y component t o a class , o n th e basi s o f conventio n (i) . The las t colum n doe s thi s fo r th e axiom , o n th e basi s of conventio n (ii) . Th e final colum n agai n show s tha t the secon d axio m fall s i n clas s K 2 fo r eac h o f th e fou r possible ways in which th e elementar y constituent s ca n

124 Appendix be classified . T h e axio m i s therefor e a tautology . I n a similar wa y th e remainin g tw o axiom s ca n b e show n t o be tautologies . We shal l als o giv e th e proo f tha t th e propert y o f being a tautolog y i s hereditar y u n d e r th e Rul e o f De tachment. (Th e proo f tha t i t i s hereditar y u n d e r th e Rule o f Substitutio n wil l b e lef t t o th e reader. ) Assum e that an y tw o formula s S 1 a n d S a D S 2 ar e b o t h tautolo gies; w e mus t sho w tha t i n thi s cas e S 2 i s a tautology . Suppose S 2 were n o t a tautology . T h e n , fo r a t leas t o n e classification o f it s elementar y constituents , S 2 wil l fal l in K 2 . But , b y hypothesis , S 2 is a tautology , s o tha t i t will fall i n K 2 fo r al l classification s o f it s elementar y constituents—and, i n particular , fo r th e classificatio n which require s th e placin g o f S 2 in K 2 . Accordingly , fo r this latte r classification , S x D S 2 must fal l i n K 2 , becaus e of th e secon d convention . However , thi s contradict s the hypothesi s tha t S 1 D S 2 i s a tautology . I n conse quence, S 2 mus t b e a tautology , o n pai n o f thi s contra diction. T h e propert y o f bein g a tautolog y i s thus trans mitted b y th e Rul e o f D e t a c h m e n t fro m th e premise s to th e conclusio n derivabl e fro m t h e m b y thi s rule . O n e final c o m m e n t o n th e definitio n o f a tautolog y given i n th e text . T h e tw o classe s K 2 a n d K 2 use d i n th e present accoun t ma y b e construe d a s th e classe s o f tru e a n d o f fals e statements , respectively . Bu t th e account , as w e hav e jus t seen , i n n o wa y d e p e n d s o n suc h a n interpretation, eve n i f th e expositio n i s m o r e easil y grasped whe n th e classe s ar e u n d e r s t o o d i n thi s way .

Brief Bibliograph y CARNAP, RUDOL F FINDLAY, J . GODEL, KUR T

KLEENE, S . C . LADRIERE, JEAN MOSTOWSKI, A . QUINE, W. V . O . ROSSER, J. BARKLE Y

TURING, A. M . WEYL, HERMAN N WILDER, R . L .

Logical Syntax of Language, New York, 1937. "Goedelian sentences : a non-numerica l ap proach," Mind, Vol. 51 (1942) , pp. 259-265. "Uber formal unentscheidbare Satz e der Principia Mathematic a un d verwandte r System e I," Monatshefte fur Mathematik und Physik, Vol. 38 (1931) , pp. 173-198. Introduction t o Metamathematics, New York, 1952. Les Limitation s Interne s de s Formalismes , Louvain and Paris, 1957. Sentences Undecidabl e i n Formalize d Arith metic, Amsterdam, 1952. Methods of Logic, New York, 1950. "An informal expositio n o f proofs o f Godel' s theorems an d Church' s theorem, " Journal of Symbolic Logic, Vol. 4 (1939) , pp. 53-60. "Computing machiner y an d intelligence, " Mind, Vol. 59 (1950) , pp. 433-460. Philosophy o f Mathematic s an d Natura l Sci ence, Princeton, 1949 . Introduction t o th e Foundation s o f Mathe matics, New York, 1952.

!25

Index absolute proofs of consistency, 2536,45-56, 109, 120-124 antinomies, 22-24, 60-63, 92-93 Archimedes, 3 Aristotle, 39, 40 arithmetic, formalized: incompleteness of, 4-5, 93-95, 102-106; consistency of, 57-58, 67, 94, 104-108; see also Principia Mathematica arithmetization: of algebra and calculus, 42; of formalized arithmetic, 68-78; of metamathematics, 80-87 axiomatic method, 2-4; limitations of, 4-5, 56, 58-59, 109-112 axioms: meaning of, 2-4; of the sentential calculus, 48-56; of Peano, 114; of PM, 48-50, 9394, 103-104 Bolyai, Janos, 9 Boole, George, 40-41 calculating machines and human intelligence, 111-112 calculus of signs, 25-36, 45 Cantor, Georg, 22, io7n chess and meta-chess, 34-36 class: notion of, 15 ; mathematical theory of, 22-23 completeness, 55-56, 102 computers, see calculating machines

consistency: formally characterized, 3°' 33-34' b°-5l> 10 4; problem of, 7-24; and truth, 13-19; of Euclidean geometry, 13-14, 1720; of non-Euclidean geometry, 14, 16-19; relative proofs of, 1420; absolute proofs of, 25-26, 32-36, 45-55, 57, 109; of the sentential calculus, 45-55, 120124; of formalized arithmetic (PM), 5, 30, 33-34, 58, 67, 94, 99-1oon, 104-108, 10 9 Correspondence Lemma, 72-73, 83, 85-87, 89, gon creative reason, powers of, 113 defined symbols in PM, 76n dem (x, z), 84-85 dem' (x, z), 85, 103 Dem (*, z), 85-87 demonstration, definition of, 46 Descartes, Rene, 5 descriptive predicates, 11 duality of points and lines, 64-65 elementary signs, 46, 69 elliptic geometry, 14, 16-19 essential incompleteness, 56, 5859' 93-94* 103-104 Euclid, 8-9, 37, 39, 56; his proof that there is no largest prime number, 37-39, 115-120 Euclidean geometry, 7-10, 13-15, 17-20, 56

127

128 Index finite models, 16, 20-21 finitistic proofs, 33, 3411, 58, loon, 10711, 108, 1091 1 formal logic: its codification, 37-44 formalization: of deductive systems, 10-14, 25-27, 33, 39-44; of t h e sentential calculus, 45-50; limits of, 112 formation rules, 45, 47 formulas: in a formal system, 28, 36; in sentential calculus, 45-56; in PM, 68-83 Frege, Gottlob, 42-44 Frege-Russell thesis, 42-43 fundamental theorem of arithmetic, 7gn G, see Godel sentence Gauss, Karl Friedrich, 9 Gentzen, Gerhard, 107 Godel, Kurt, 1-2, 4-6, 9, 22n, 56, 57-60, 66-67, 68-69, 7 2_73> 80, 83, 85, 89, 92-95, 96, 97, 98, ggn, 103, 104, 109-113; his Platonic realism, 110 Godel numbering, 68-80 Godel sentence, 92-94, 97-103, 105 Goldbach's conjecture, 59

informal reasoning, 112 intelligence of humans and robots, 111-112 intuitive knowledge, 13, 22, 11 2 Jourdain, M., 39 Kant, Immanuel, 39-40 Lobachevsky, Nikolai, 9 logic of propositions, see sentential calculus logical constants, 46-47 mapping: idea of, 57-67; of geometry onto algebra, 5, 19-20, 64; of meta-mathematics into arithmetic, 66-67, 68-91 mathematical induction, axiom of, 114 mathematics, nature of, 2-6, 8-14, 104, 110-111 "meaningless" symbols and meaning, 10-12, 15, 25-28, 34, 44, 45>71-73>85-86n>89>91 meta-mathematics, 27-33;anc* metachess, 34-36 meta-zoology, 31 models: and proofs of consistency, 15-25, 120; finiteand nonfinite, 20-2 5 Modus Ponens, 48, 51, 116, 120 Moliere, 39

hereditary property, 52, 54, 55n, 124 Hilbert, David, 11-12, 19-20, 25, 27, 32-34, 36, 64, 107-108, 10 9 names of expressions and of numhuman brain, limitations of, 111bers, 28-29,30-3 in, 89-90^91112 92n, 96n, 97-g8n negation, formal imitations of, 33imaginary numbers, 42 34, 41, 46, 71-72, 100 implicit definition, i2 n incompleteness, 56; of formalized non-Euclidean geometry, consistency of, 10, 14, 16-18 arithmetic, 4-5, 67, 93-95, 102non-finite models, 20-24 106; essential, 58-59, 93-94, normal classes, 23-24 103-104 number, definition of, 42-43 inference, rules of, 38, 39, 44, 69, number theory: defined, 4n; re71,84,93-94

Index 12 9 duced to logic, 42-44; see also arithmetic, formalized numerals vs. numbers, 8o,-o,on numerical variables, 73, 74

Richard, Jules, 60 Richard Paradox, 60-63, 66, 92-93, 98~99n Riemann, Bernhard, 9 robot mathematicians, 111-112 omega-consistency, 99-ioon Rosser, J. Barkley, ggn rule: of substitution, 39, 47-48, 50Pappus of Alexandria, theorem of, 51, 115, 119; of detachment, 64-65 48, 51, 106, 116, 12 0 paradoxes, 22-24, 60-63, 66, 92rules of inference, see transforma93'98~99n tion rules parallel axiom: Euclidean, 8-9, 14, Russell, Bertrand, 1,12, 22n, 2319, 20; non-Euclidean, 14, 17 24,37,41,42,71,94, 104 patterns of meaningless symbols, 26- Russell's Paradox, 23-24, 62 27> 33> 34-35» 81, 83, io7n self-evidence: of Euclid's axioms, 8Peano, Giuseppe, 114 10; and intuition, 13 Platonic realism in mathematics, sentential calculus, 45-56, 120-124 110-mn sentential connectives, 46-47 'polysyllabic' as self-describing adsentential variables, 46, 74, 121 jective, 6in strings of symbols, 26-27 predicate variables, 74-75 sub (x, 17 , x), 87-89 prime number: defined, 37, 60; Sub (x, 17 , x), 90-91 proof that there is no greatest, 37-39, 115-120; sums of, 59; in supermarket tickets, 80-81 Richard Paradox, 60, 61; use of in Godel numbering, 74-80, 81- tautologies, 53-56, 120-124 82, 83, 88n; expressed in PM no- theorem, definition of, 46 transfinite induction, principle of, tation, 86n io7n primitive formulas, 46; see axioms transfinite ordinals, io7n primitive recursive truths of arithtransformation rules, 25-28, 33, 35, metic, 72-73, 83, 85, 86n, 89 Principia Mathematica, 1 , 37, 41-44, 38~39' 44' 46, 47-48' 52, 5 8» 45' 57"59' 68, 71-73, 75' 7&n, 69, 77n, loon, 109-111, 115120 77-78, 80, 82-87, 90-108, 109 trisection of angles, 7, io7n projective geometry, 64-66 truth tables, 121-124 proof: Hilbert's theory of, 32-36; definition of, 46, 77n undecidability, 93,100-103 use and mention, 26-34, 89-90^ quantification theory, 38, 12 0 9i-92n, 97-g8n quantifiers, 70, 118-120 quotation marks in names for expressions, 3 in variables, 29, 39, 73~75 reduction of arithmetic to logic, 42 relative proofs of consistency, 17-20

Whitehead, Alfred North, 1, 22n, 37,41,71,94, 104