MARTIN LOF Bibliopolis Book

8/6/2019 MARTIN LOF Bibliopolis Book

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STUDIES IN PROOF THEORY

Advisory Editors

BIBLIOPOLIS

. PER MARTIN-LOF

Notes 'by Giovanni Sambin of- a series 0/given in Padua, June i980

INTUITIONISTIC TYPE TH

P. MARTIN-LOFG .E . MINCW. POHLERSD. SCOTTW. SIEGC. SMORYNSKIR. STATMANS . TAKASUG . TAKEUTI

D. PRAWITZH. SCHWICHTENBERG

C. CELLUCCIJ..Y. GIRARD

P. AczELC. BOHM .

W. BUCHHOLZE. ENGELER

S . FEFERMANCH . GoAD

W. HOWARDG.HUET

D. LEIVANT

Managing Editors


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I . CONTENTS

Natu ra l numbers . . . . ... . . . .. . . . .. . ..... . . . . . . . ..

Pr-opos Lt ional equa l i ty ... ... .. . . ...... . . : . . . . . ...

Lis t s ; . . . . . . . . . . . Wel lorder ings ........ . ..... . . . . . . . . . . .........Un iverses .... .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

, ,.. "

Fin i t e se t sConsis tency

Di s j o i n t union of " two se t s . . .... . .... . . . . ..... . . . . .

Introductory remarksPropos i t ions an d judgements "Exp lana t ions of th e form s of judgement 'Pr op os i t i onsRule s o f e qua l i ty , . . . . . . . . . . . . . . .. . . . . . . . . . . . .Hypothetical judgements an d s u bs t i t u t i on ru l e sJudgements wi t h more than one as sumpt ion an d con tex tSe ts and ca te go r i e s _General remarks on th e rule sCarte s ian product of a family of s e t sDe f i n i t i ona l equa l i t y :App Lf ca t Lcns of th e car te s ian produc t ;Disj o in t uni on of a family of se t s .. . . . .... ..... . .Appli ca t i on s of th e ' d i s jo in t uni onThe axiom of ' choice ...... . . ..... . . ... .. . ..... .The n ot io n o f such t ha t .. . . . . . . . .. . . . . . . . . . . .... .

I\

1l

IitI 1984 by Bibliopolis, edizioni di scienzeNapoli, via Arangio Ruiz 83 , '; . :,' " " , ,All rights reserved, No part of'ihis>-,1;;qp!i:,:ti,ay 'be reproduced, in anyform or by any means without permission'-inwriting from. the publisherPrint ed in I taly by Grafitalia Via Censi dell 'Arco, 25 Cercola (Napoli))

ISBN 88-7088-105-9


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It1,I

I

PrefaceThese lectures were given in Padova a t th a Labo

Ricerche 'd i Dinamica d e i S is tem i e di Elettronica Bi,Cons i g l i o Nazionale delle Ricerche dur in g t he monthI am indebted to Dr'., Enrico Pagello of t h a t laboratoportuni ty of s o do in g . The audience was made up by pmathematicians and computer s c i e n t i s t s . Accordingly,something which might be of to each of thesegories. Essential ly th e same l e c t u r e s , a l b e i t in aproved and more advanced form, were given l a t e r in tas par t of the meeting on Konstruktive Mengenlehre uwhich was organized in Munich by Prof. Dr . Helmut Scte whom I am indebted fo r th e i n v i t a t i o n , d ur in g t he

- 3 October 1980.The main improvement of th e Mun ich .l e c t u r e s , asthose given in Padova, was t o e adop ti on o f a sy st em

(Ger. S tu fe ) n o ta ti on which allows me t o write simplFl (A,B), L(A,B), W(A,B) , ),(b),E(c,d) , D(c,d,e) , R(c , d , e ) , T(c;d)

instead of(Tl x EO A)B(x), ( L x e A)B(x) , (Wx e iI)B(x) , (AXE ( c , ( x, y ) d ( x, y , D ( c , ( x ) d( x ) , (y ) e ( y ; R(c,d,(T ( c , ( x , y , z ) d ( x , y , z ,

respectively. Moreover, th e use , of higher l e vel va ris t a n t s makes i t possible to formulate th e el iminatiorules fo r the cartesian product in such a way that t


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IT-el iminationcE I : (A ,B)

c = (p(c) , q ( c) E E (A,B)

c (Ax)Ap(c,x) E n(A,BJC E n (A,B)

F( c ,d ) -= d( (x)Ap(c , x .

s tockholm, January 1984,

Moreover, th e second of these, tha t i s , th e r

ca n be derived by m ea ns of th e in th e same w

is .der ived byway of example on p . 62 of th e main teth e new elimination and equal i ty ru les can be deriveones by making th e defini tion

So , ac tua l ly , they ar e equivalent.I t only remains fo r me to thank Giovann i Sambin

undertaken, a t hi s own s ugge st io n, t he c on si de ra bl eand t yp in g t he se n ot es , t he re bf making th e lectureswider aUdience.

l!1Idey) 6 C(A(y

d(yl ' 6 cO.(y ,(y Lx ) E. B(x) (x oS A

(y(x) E. B(x) (x E. A

F ( c " d ) 6 C( c )

F(A,(b) ,d )

. (x E A)b Cx ) 6 B(x)

C E. n (A ,B )

r-espect.tvely. Here y is a bound function var iab le , F i s a new non-(eliminatory) operator by m ean s of which th e binary ap -

pl ica t ion operation can be defined, putting

same ,p a t t e r n as t h e e l imina ti on and e qu al it y r ul es f or a l l th e othertype forming opera t ions . In the i r new f o rmu la t ion , t hese rules read

Ap(c,a) == F(c, (y )y (a , Pel' Martand y(x) E. B(x) (x E. A) i s an assumption, i t se l f hypothetical , whichhas been pu t within parentheses to indicate tha t it is being dis-charged. A program of th e new form F(c,d) ha s provided c ha svalue A (b ) and deb) ha s value e . This rule fo r eva lua t ing F(c, d )reduces to th e l a zy eva lua ti on rule fo r Ap(c,a) when th e above defi-nition i s being made. Choosing C(z) to be B(a), t hus independen t ofz, and d(y) to be y(a) , th e new eiimination rule reduces tp th e ol don e and th e new equal i ty r u le to the f i r s t of th e two ol d equal i ty

1jI1


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- 1 -

Introductory remarks

,Mat hemat i cal logic and relation between logihave been i nt e rp re t ed i n a t three diffe rent way

(1 ) mathematical logic, as symbolic log ic , or logmatical .ymbolism;( 2) ma th ema ti ca l logic 'a s foundations to r philosmatics;(3 ) mathematical logic as l ogi c s tud i ed by matheas a branch of mathematics. '

We sha l l here mainly be interested in mathematical losense. What sha l l do is also mathematical lo gic inbu t certainly no t in the third .

'The principal problem tha t remained a f te rP ' r inciwas completed was, according to i t s authors, that ofaxiom of reducibi l i ty (or , as we would now say, th e iprehension axiom) '. The, r ami fi ed t heo ry of types was pi t was no t suff ic ien t fo r deriving even elementary pSo axiom of reducibi l i ty was on th e pragmawas needed, although no . a ti s fa c t o r y ju s ti f ic a t io n (ecould be provided. The whole point of the ramificatioso tha t i t might just as w ell be abolished. What thent he s impl e theory o f t yp es . I t s o f fi c ia l j u st if ic a ti o

rests on th e in te rpre ta t ion of proposit ions aand proposi tional f un ct io ns ( of one or s ever a l va ri abfunctions. The laws of th e c lassica l proposi tional l oclear ly val id , and so ar e th e quantifier laws , as lont ion is rest r ic ted to f in i te domains. However, it doeible to make sense of quantif ication over in f in i te do


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ment: '

separ-a te s t eps :

Here th e , proposi t ion (Get ion or judgement ' (Ge r . Ur te i l ) i s e s se n ti al . Whaof' th e log ica l opera t ions ( . l : :> & v V 3 '), , , , " andproposi t ions. When we hold a proposi t ion to be t ru

speak, they f'orm an open concept, Inp r es e nt at io n s o f f i r s t order logic , we d 'ca n

The dis t inc t ion between proposi t ions and jUdgfrom Frege to Pr inc ip ia . These no t ions have l a t e rfo rma l i s t i c n ot io ns o f formula an d theorem (i n a fspec t ive ly . Contrary to formUlas, proposi t ions ar et ive ly . So to

(2 ) s p e ci f 'i c a ti o n o f aXioms an d ru les o f i nf er

Proposi t ions an d Judgements

(3 ) semantical i n t e rp r e t a t i on .

proposi t ion

In pa r t i cu l a r , ' t h e premisses an d conclusion of a ljudgements .

(1 ) induct ive def ini t ion of te rms and formulas

- 3 -

Formulas and dd t 'e uc ar e given meaning only throwhich i s usua l ly done follOWing Tarsk i an d assumingWhat we do here i s t t, mean 0 be closer to ord in

We wil l a vo id k e ep in g form a nd meaning (cs tead we w ill a t th e same t ime display ce r t a in formin fe rence tha t a re used in mathemat ica l d, " proofs ant i c a l l y . Thus we make exp l i c i t h t

- -

l ' C. A. Hoare , An ax iomat ic b asis o f computer programming, Com-munications of th e ACM, Vol. 12,196'9, pp . 576-580 an d 583.

2 E. W. Dijks t ra , A d i sc ip l i ne of Programming, P r en t ic e Ha l l,Englewood Cl i f f s , N. J ., 1976.

3 "P. Martin-Lof, Cons t ruc t ive mathematics a nd c ompu te r p ro gr am -ming, Logic, Methodology and Philosophy o f Science VI ; Edited ,byL. J . Cohen, J . Los, H. Pfe i f f e r an d K. -P . Podewski, North-Holland,Amsterdam, 1982, pp . 153-175. '

domain of natura l numbers, on th i s i n t e rp r e t a t i on of t he n ot io ns ofpropos i t ion an d propos i t iona l For th i s reason , afuong o the r s ,what we develop here i s an i n t u i t i on i s t i c theory o f t ypes ; which i salso p r ed i ca t iv e ( or ' ram i fi e d) . It i s f r ee from th e d e fi ci en cy o fRusse l l ' s ramif ied t he or y o f types , as r eg ar ds t he poss ib i l i t y of de-veloping elementary par t s of mathematics , l ike th e theory of ' r e a l num-ber s , because of the presence o f th e o pe ra tio n which a l lows us to formth e car tes ian product of ,any given family of s e t s , in pa r t i cu l a r , th es e t of a l l func t ions from one se t to another .

In two a reas , a t l e a s t i o u r l an gu ag e to have advan t age s -over t r adi t ional foundational languages. Fi r s t , Zermelo-Fraenkel se ttheo ry cannot adequately dea l with t h e f o un d at io n a l problems of ca t egory theo ry , where th e category of a l l s e t s , t h e c a te g or y of a l lg ro up s, t he , category of func to rs from o ne such category t o a no th ere t c . ar e cons ide red . These problems coped with by means o f th ed i s t i nc t i on between se t s an d c at eg o ri es ( in th e l og i ca l o r p h il os o ph i-cal sense , no t in th e sense of c a te g or y t h eo r y) which i s made in i n tu -i t i on i s t i c type theo ry . Second, present l og i ca l symbolisms ar e inad-equate as programming languages , which explains why computer sc ien -t i s t s have developed the i r own languages (FORTRAN, ALGOL, LISP,PASCAL, .. . ) an d systems of proof ru le s (Hoare 1 , Dijks t r a2 , . ) . Wehave shown elsewhere 3 how th e addi t ional r ic hn es s o f ' t y p e theo ry , ascompared with f i r s t order pred ica te logic , makes it usable as a pro -gramming l anguage .


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A

A v B

a E A

a i s a method of solving thpro b l em (doing t he t as k) A

a is a method of( rea l izing) th e intention( expec ta t ion ) A

A i s a se t a i s an 'eLemen t of th e se t .A i s a propos i t ion a i s a pr oof (cons t ruc t ion)

th e pr opos i t i on A

A se t

( 2 ) A and B equa l (A B),

(4 ) a and b ar e equal elements of th e se t A( 3) a i s .an element of th e se t A (a A) ,

. We us e f ou r f orms of(1 ) A i s a se t (a bb r . A

i n th e ta.ble:

A is a problem( task)

A i s an i ntent ion(expectati on)

( I f we read l i t . r a l l y a s , then weA = BESe t , a e El (! ) , a = b E El(A ) , respectivsyntactic variabl e s co u ld be us ed th e fse 0 smements and c a pi t a l le t ters f or ae'ts i s onl y t oin o r d i n a r y se t t h eo ry , a E b and a = b ar e pr opa r e j u dgements here. A jUdgement of th e f orm A =less we already kn ow A and B t o sets . Likewisf orm a E A pr esupposes tha t A i s a se t , and a jua = b E A pr esuppose .s, f i r s t , tha t A i s tse , ab ar e elemen t s of A .

Each f orm o f judgemen t admits f 1severa di

A tr-ueprop .

l-AyB

A v B t r ue

A, B prop . .1- A

A prop.

or

where we use, l ike Frege, th e symbol l- to th e lef t of A to signifythat A i s true . In our system o f r ule s , . t hi s wi l l always be expl ic i t .

A r ul e of inference is j ust i f ied by exp la ining the conclusion onth e assumption that th e pr emi sses ar e known. H.ence, before a ru le o finfere nce can be just i f ied , i t must be explained wha t i t i s that wemust know "in o r d e r to have th e r ight to make a judgement of a ny oneof t he v ar io us f orms that th e premisses and cOnclusion can have.

takes f o r g ra nt ed that A and B are f ormulas, and o n l y then doe s i t sa ythat we can i n f e r A v B to be true when A i s true . .I f we are t o give af ormai rule, we have to make t hi s e x pl lc it , writing

granted. When one t re a t s lo g i c as arty othe r br anc h of mathematics, asi n th e metamathematical t r ad i t ion originated by Hilbert , such ju dge -ments and in ferences are onl y part ia l ly and f ormal l y re pr ese nted inth e s o- cal led ob jec t l an gu ag e, w hi le t he y ar e implicit ly used; a s inany other branch of mathematics, in the s o-cal le d metalanguage.

Our main aim is to build up a system of f orma l rules representingin th e bes t possible way informal (mathematical) re asoning. In th eusual natu ra l deduction sty le , th e rules given are no t quite . f ormal .For i ns t a nc e , th e rule .


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4 A. Heyt ing , Die in tu i t ionist ische Grundlegung de r Mat hemat ik ,Erkenntnis, Vol. 2, 1931 , pp . 106-115 . .

5 A. N. Kolmogorov , Zur Deutung de r in tu i t ion is t i schen Logik,Zei tschr i f t , Vol. 35 . 1932, pp .

6 H. B. Curry and R. Feys, Combinatory Logic, Vol. 1, North--Holland ; Amsterdam , 1958 , pp , 312-315.

7 w., A. Howard , The formulae-as-types not io n o f cons t ru c ti on, ToH. B. Curry : Essays on Combinatory Logic, Lambda Calculus arid Formal-ism , Academic Press, London , 1980 , pp. 419 -490 .

The se cond, logical in te rpre ta t ion is discussed toghether with rulesbelow . The third was suggested by Heyting4 and t he fourth by Kolmo-gOrov5 . The las t i s very close to programming. "a i s a method . . . " ca n 'be read as "a is a program . . . ". Since programming languages have aformal notation fo r th e program a , b ut not fo r A, we complete th e se n-tence with " . . . which meets th e s p e c i f i c at i on A" . In .Kolmogorov 's in - t e r p r e t a t i o n , th e word problem re fers to something to be done and th ewor d program to how to do i t . The analogy be tween t he f i r s t and th esecond interpretation is . implici t in th e Brouwer-Heyting in te rpre t -a t ion logica l I t was made more expl ic i t bY ,Curry ,andFeys6 , bu t only fo r th e impl ica t iona l fragment , and i t wa s extended t o, . 7in tu i t ion is t ic f i r s t order ari thmetic by Howard _ I t is th e only knownway of logic so tha t axiom of 6hoic ebecomes , valid .

To dist inguish between proofs. of judgements ( u su a ll y i n t ree - l ikeform) and proofs o f propos i t i on s ( he re identified with elements , thusto th e l e f t of E . ) we reserve th e word cons t ru ction for the l a t t e r anduse it when confusion might occur.

Explanations of the forms of judgem

by which i t s elements a re const ruc ted . However"definit ion i s c lea r : 1010 , fo r instance, thought h eg i ven ru les , is clearly a n e leme nt of N, sincan bring it to th e form a ' fo r some a e ' N. W

th e elements which have a form by which wthat they a re the resul t of one of the rules , aca l , from a l l other e lements, whi c h we wi l l ca l

a 'E No E Na N

The f i r s t ,i s th e ontologi ca l (ancien t Greek) , t(Descartes, Kant, . ) and th e third

ern) way of posing essent ia l ly th e same questioncould assume that a se t i s defined by prescribinar e formed. This we do when we say tha t th e se tN i s defined by giving th e rules :

What does a judgement of the form "A i s a

What i s it that we must know in order to hsomething to be a set?

What is a set?t ions :

For each one of th e . four forms of judgemena judgement of that form means . We can explainof th e f i r s t form, means by answering one of th e


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b Eo B

( 2 ) two s e t s A and B a r e equal ifa ! A a E A a--- ( t h a t i s , and -a ! B a ! B a

(4 ) two a r b i t r a r y elements a, b of a se t Aexecuted, a and b yield equa l c a non ic a l el

This is th e meaning of a judgement of the formwe assume t he n ot io n of method as primitive. Th

of th e present language w i l l be sucof an element a of a s e t A terminates with a .vaoutermost form o f b t e l l s t hat, it . i s a canonico rd er or la zy evaluation) . Fo r instance, th e cog iv es th e v al ue (2 + 1 ) ' , which i s a canonical2 + 1 E N.

Finally:

and

fo r a r b i t r a r y canonical elements a , b.

abE A

(3 ) an element a of a s e t A i s a method (oexecuted, yie lds a canonical element of A

This is th e meaning of a judgement of th e formWhen we explain what an element of a s e t A

we know tha t A i s a s e t , t h a t i s , in p a r t i c u l a r. elements ar e formed. Then:

b ' ! N'

b = d ! B

b ! B

and

a


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This i s th e meaning of a judgement of the form a = b ! A. This d e f i -n i t i o n makes good sense since it i s ' p a r t of the d e f i n i t i o n ' of a s e twhat it means fo r two canonical ,e lements of th e s e t to be equal .

Example. I f e , f ! A x B, then e and f ar e methods which yieldcanonical elements ( a , b ) , (c,d) E A x B, r e s p e c t i v e l y , as r e s u l t s ,and e f E A x B i f (a,b) ( c , d ) E A x B, which in turn if

Proposit ions

8 D. Prawitz, I n t u i t i o n l s t 1 c l o g i c : a philosLog ic and Philosophy, Edited by G. H. vonThe Hague , pp . 1-10.

a proposit ion is defined by laying down whaof th e proposi t ion ,

a proposit ion i s t r u e i f it ha s a proof, t h8it ca n be given

and t h a t

C l a s s i c a l l y , a proposit ion i s no th i ng but aan element of th e s e t o f t r u th values, whosetrue and th e f a l s e . Because of th e d i f f i c u l t i

th e r u l e s fo r forming proposi t ions by 'means of qi n f i n i t e domains , when a proposi t ion i s understoot h i s explanat ion i s r e j e c t e d by th e i n t u i t i o n i s t ssaying t h a t

Thus , i n t u i t i o n i s t i c a l l y , t r u t h i s i d e n t i f i e d wio f c ou rs e' n o t (because of incompletenessb i l i t y within any p a r t i c u l a r f ormal sy st em.

The explanat ions of the meanings of the l o gf i t together with th e i n t u i t i o n i s t i c conception oi s , ar e given by th e standard t a b l e :

d B;= c E A and b


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' t h e f i r s t l ine ' of which should be interpreted a s s ay in g tha t therei s nothing tha t counts as a proof .

The above table can be made more expl ic i t by saying:

a p ro of o f t he p ro po si ti on.L

A & BAV BA :> B

( Vx)B(x) ,(3 x)B(x)

a p ro of o f t he p ro po si ti on.i ,

A & B

A v B

A :> B

('o'x)B(x)

(3 x)B(x)

consists of

a proof of A and a proof of Ba proof of A or a proof of Ba method which takes any proofof A into a proof of Ba method which takes an arb i t ra ryindividual a into a proof of B(a)an individual, a and a proof ofB(a)

has th e form

(a ,b) , where a i s a proof of Aand b is a p ro of o f Bi (a) ; where a is a proof of A,or j{b) , where b is a proof of B(Ax)b(x), where b(a) i s a proofof , B provided a a proof of A(Ax)b(x), where b(a) is a proofof B(a) prOVided a is an individual(a ,b) , where a is an individualand b i s a p ro of o f B(a)

As i t stands, this table i s n o t s tr ic tl y cor-r-ect.,proofs of canonical form only. ' An arbitary proof,arb i t ra ry element of a se t , is a method of producinnonical form.

I f we take ser iously , th e idea tha t alaying down how i t s canon ica l p roo fs are f'or-me d (atable above) and accept tha t a se t is defined by pcanonical elements ar e formed, then i t is c le a r t hlead to unnecessary dup li c at i on to keep t he n ot io nand se t (and th e associated notions of proof, of aement of a set) apar t . Instead, we simply identifyt reat them as one an d th e same notion. This i s the(proposi tions-as-sets) in te rpre ta t ion on Which in ttheory is based.

- 15 -- -


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We now begin to build up a system of r u l e s . Fi r s t , we g i ve th efo l lowing r ule s o f e qu al it y , which ar e ea s i l y explained using th efa c t t h a t t he y ho ld , by de f in i t i on , fo r can on ic a l e l eme nt s :

Fo r in s t ance , a de t a i l e d explanat i on of t r an s i t i v i t y i s : a = b E Ameans t ha t a an d b y ie ld canonical elem en t s d an d e , respect ive ly , an dt ha t d = e ! A. S imi la r ly , if c y ie lds f , e = f EA . S in c e we assumet r an s i t i v i t y fo r canonica l elements , we obtain d = f E A, which mean st ha t a = c ! A.

a = bb e A

b .. A

a E Aa


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means t h a t b(a) and d (a ) a re equa l e lemen t s of th e s e t B(a) fo r anye lement a of the s e t A. We then have

A2(X,) i s a family of s e t s ove r A"

(1 ) A(X" . . . xn) set (x , E A, . x2 e A2 ( x , ) , .Xn

A, i s a s e t .

A3(X"x2) i s a family of s e t s with two indicex2 ! A2 ( X, ) ,

Jud gements with more than one assumption and

An(x, ..,oo. , xn _ , ' i s a family o f s e t s with n- 'X2 E A2 (x,) , . . . , xn_ , E An_,(x, ' . . . xn_2) .

means tha t A(a" . . . ,an) i s a s et whenever a , ! A"anE . .. . an_') and tha t A(a" . . . ,a n) = A(b "whenever a , = b, e A" . . . , an = bn ! An(a " .. . ,a nA(x" . . ,x n) i s a family of s e t s with n indices. Ta judgement of th e form ( , ) c o n s t i t u t e what we c aplays a role analogous to th e s e t s of for mul ae rappearing in Gentzen sequents . Note also tha t anycontext always a context. Because of th e meanin

Then a judgement of th e form

We may now f u rth e r gene r al iz e judgements t o ijudgements with an a r b i t r a r y number n of assumptiomeaning by induction. tha t i s , assuming we understjudgements with n - ' assumptions. So assume we know

(x E A)b(x) E. B(x)

b(a) = b(c) ! B{a)c ! A

(x ! A)b(x) = d(x) e B(x)

(x ! A)b Cx) ! iHx)

a ! A

b(a) E. B(a)Final ly, a judgement of th e form

b(a) = d(a) ! B(a)

Substi tution

a ! A

(4) b (x ) = d(x) ! B(x) (x ! A)

Substi tution

A hypothetical of th e form(3 ) b(x) E B(x) (x A)

wQi c h i s th e l a s t s u b s t i t u t i o n rule .

means t h a t we know b(a) to be an element of th e s e t B(a) assuming weknow a to be an element of th e s e t A, and t h a t b(a) = b I c ) E. B(a)wheneve r a and c ar e equal elementa of A. In other words , b(x) i san ex t ens iona l func t ion with domain A and .r ange B(x) depending onth e argument x . Then th e following r ul es a re j u s t i f i e d :


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t he cat ego ry o f f am il ie s o f sets B(x) (x E A)'

t he c at eg ory of elements of a given se t (o r prt ion) .

t he c at eg or y o f sets (o r proposit ions).

e tc .sets , .

t he ca tegory of functions c(x.y) E C(x.y) (x EA i s a set . (x E A) and C(x.y) . (x "- A. y

t he c at eg or y o f f am il ie s o f sets C(x .y ) (x EAA set . B(x) se t (x E A).

Sets and categories

t he c at eg or y o f f un ct io ns b (x ) E a(x) (x E A).B(x) se t (x E A) ,

In addition to th es e , t he re a re higher categories .o f b in ar y f un ct io ns which take two sets into anothex . which takes two sets A and B into their cartesi

A category . i s defined by explaining what an o begory is and when two such objects ar e equal . A catea se t , since we can grasp what it means to be an o bcategory even without exhaustive r ule s fo r forminginstance. we now grasp what a set i s and when two swe have d ef in ed t he c at eg or y of sets (and. by th e sacategory of proposi tions) , bu t it i s no t a se t . Sofined severa l ca tegories :

(4 ) a(x, . xn) = b(X, . ,x n) E A(x, . xn)(x EA. X E A (x , . ....x ," n n n-( equa l func t ions with n arguments).

{3 ) a(x" . . . , xn ) E A(x, , xn ) (x , e A, . . . . .xn E An(x , . xn_,(function with n a rguments),

(equal families of sets with n indices) ,

and we assume th e corresponding subst i tut ion rules to be given.

judgement of th e ( ' ) , we se e that th e f i r s t two rules of subst i -tu t io n may be extended to t he c as e of n assumptions,and we understandthese extensions t o be given .

I t is by now clear how t o e xp la in th e meaning of the remainingforms o f hypothet i ca l judgement :

- - -


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i s an example of an o bje ct o f tha t c a te g o r y .We wi l l sa y ob j ect of a category bu t e lement of a se t , wh i c h r e-

f l ec t s th e difference between ca tego r ie s and se t s . To ca t -egory it i s not necessary to presc r ibe how i t s ob jec t s ar ebut Jus t to grasp what ( a rb i t r a ry ) o bj ec t o f t he c at eg o ry i s . Eachs e t determines a ca tegory , namely th e category of e l ements o f th e s e t ,bu t not conversely: fo r in s tance , th e c a t eg o r y of se t s an d th e ca t -egory o f p ro po si ti on s a re not s e t s , s ince we cannot desc r ibe how a l lt he i r elements a re formed. We ca n now s a y t h a t a judgement i s a s t a t e -ment to th e e f f e c t t ha t s o me t h i n g i s an obje c t of a c at e go r y (a 6 A,A s e t , . . . ) or tha t tw o objects o f a c a te g or y a re e qu al (a = b E A,A B, ) .

What about t h e word type in th e logical s e n s e . given to it byRussel l with h is ramif i ed (resp. s imple) th e ory o f type s ? Is type syn-onymous with cate gory or with s e t ? In some cases with th e one , its e e ms , and in o the r cases with th e o the r . And it i s th i s confusion oftwo d i f f e r en t concepts which ha s l ed to th e impredicat i vi ty o f th es imple theory of t ypes . When a type i s defined a s th e range o f s ign i -f icance of a propos i t i ona l funct ion, so t ha t types ar e wha t th equan t i f i e r s range over , then it seems t ha t a type i s th e same th inga s a s e t . On th e o the r hand , when on e s pe a ks a b ou t th e s im p le t yp eso f p ro p os it io n s, p ro p er ti es o f ind iVidua ls , r e l a t i ons between ind iv id -uals e t c . , it seems as if t ypes and c a te g or ie s a r e th e same . The im -p o rt a nt d i ff e re n ce between th e ramif ied types o f p r op o si ti o ns , prop-e r t i e s , r e l a t i ons e tc . of some f i n i t e order and th e s imple types ofa l l proposi t ions , proper t ies , r e la t io n s e t c. i s prec i se ly t ha t th eramif ied t ypes a re (o r ca n be unders tood as) s e t s , so t ha t i t makessense t o q u an ti fy over them, whereas th e s imp le t yp e s ar e mere ca t -egor ie s .

Fo r example, BA i s a s e t , th e s e t of func t ions from th e s e t A to

t h e se t B (BA wi l l be in troduced as an abbreviwhen B(x ) i s constant ly equal to B) . In par t i cbu t it i s no t th e s ame th ing as which ireason tha t BA ca n be cons t rued t ias a se s t ho f function as pr imi t ive , in s t ead o f d e fi ni ngordered p air s or a b ina ry re la t ion sa t i s fy inguniqueness cond i t i ons , which would make it a ci ns te ad o f a . se t .

When on e s pe ak s a bo ut da ta types in compuju s t as wel l sa y data s e t s . So here type i s alws e t an d n ot w it h category .


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(x E , A)

x e.B(x)

(x E C)

b(x} e D(x)

A = CC = A

x E Ab Lx) e. B(x)

(x E C)

rec t ion i s s imi l a r .

(AX)b(x) E ( D X E C)D(x)

name of th e former . Since , in g en era l, th ere aref o r g e ne r at i ng a l l func t ions fro m o ne s e t to anowe cannot generate induc t ive ly a l l th e elements o(n x IS, A)B(x} (o r , in par t icular , of th e 1,form 'BA

We ca n now jus t i fy th e second ru le o f se t fo(AX}b(x) be a canon ica l element of (TIx E A)B(x}(x E A) . Therefore, assuming x E C we ge t x E A bi t y of se t s from th e premiss A C, an d hence b(xt he p remi ss B(x} = D(x} (x E A) , again by e q ua liassumed to hold a lso for fam il ie s o f se t s ) , we oand hence (Ax}b(x) e: ([ 1 x e C)D(x} by n- i n t r odu

tha t these ru le s i n tr o du c e c a no n ic a l elemenelements, even if b(a} i s not a canon ica l eleA. Also , we assume t ha t t he u su al var iab le r e

i . e . t ha t x does n ot a pp ea r free in an y a ss ump ti oE A. Note t ha t it i s neccessary to un

hex) e B(x} (x E A) 'i s a funct ion to be ab le to felement (AX}b(x) E ( f ix e , A)B(x); we c ou ld s ay t h

D(x)(x c;; A)

B(x)(Tl x E A)B(x) = (Tix E C)D(x}

(x e A)BOc} se ts e t

( f ix E A)B(x} s e t

(AX}b(x) = (Ax;}d(x) e ( f ix ' e A)B(x}b(x) = d(x} e B(x)

(x e A)

(AX)b(x) e (T ] x E A)B(x)b(x} E. B(x}

Car tes ian product of a family of se t s

n - in t roduc t ion

D-formation

Given a se t A an d a ,f a mi l y of se t sB(x} over th e se t A, we ca n

The second rule says tha t from equal arguments we ge t equal values .T he s am e holds fo r a l l other se t forming operat ions, and we wi l l never

i Th conclus ' on of th e f i r s t ru le i s tha t somethingspe l l it out ag a n. e i s a s e t . To unders tand which se t it i s , we must know how i t s canoni-ca l e lement s and i t s equal canon ica l e lement s a re formed. This i s ex-p la ined by th e in troduct ion ru les :

form th e product:

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(x E A)b(x) E B(:c)

C E (TIx E A)B(x)

a E A

(Ax)Ap(c,x) E (nx E A)B(x)

ApAX)b(x) ,a ) = bra) E B(a)

n-equa l i ty

We have to e xp lai n th e meaning of th e new constant AApplication) ; Ap(c,a) i s a method of obtaining a canB(a) , and we now explain how to execute i t . We knowc (T l x E A)B(x) , that i s , that c is a method whichca l element (AX)b(x) of (TIx e A)B(x) as resul t . Now

i t fo r x i nb(x) . Then .b ( a ) E B(a). Calcuobtain as resul t a canonical element o f B(a), as reqin th is explanation ; no concrete computation i s carrth e character of a thought experiment (Ger. Gedankenuse Ap(c,a) i ns te ad o f th e more common bra) to distinof applying th e binary application function Ap to thand a from th e resul t o f a pp ly in g b to a. Ap(c ,a) coapplication operation (ca) in combinatory logic. But

lo gic th er e a re no type rest r ic t ions , sinways form (ca) , fo r any c and a.

. .. . .th e pr-ogr-am th e. f i r s t rule sa y.s that . appla program to argument yields th e same as egram with that ,rgument aj Similar ly, . th e sec

The f i r s t equal i ty rule shows how . the new function Acanonical elements of (F l x E A)B(x): Think of (AX)b(

A) ,

a = b E A

a E A

X ! A

B[x) = D(x)

(x 6 C)

(Ax)d(x) '" (D x '" C)D(X)= (Ax)d(X) E (TIx E A)B(x)=

C = AA = C

X E A

c e; (n x ! A)B(x)Ap(c,a) ! B(a)

c = d E (nx ! A)B(x)

(Ax)b(x)

Ap(c,a) = AP(d,b) E B(a)

(Ax)b(x) = O.x)d(x) ! (TIx E C)D(x)b(X) = d(X) ! D(X)

A = C

(x ! C)

(Ax)b(x) and (Ax)d(x) be equalsame assumptions. So le t Then b (X) = d(x) E B(x) (x Ee lemen ts o f. (n x 6 A)B.,(x) .

n _elimination

1 canonical elements ofand (Ax)d(X) are equashows tha t (Ax)b(x)(f\x e; C)D(x).

and therefore th e derivation

corresponds directly to th eWe also have to prove that

under th e. canonical

. d . s a formalderivation cannot be considere aWe remark that th e above 0in type theory i t se l f S1nce . of the second T1_formation ruleproof 0 l o ty between two se ts whichformal rule of p ro v1 ng a n e qu a 1th e re is no exp lanat ion o f what such an equal i ty means.

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B t ruet rue

A & B t rue

Def in i t iona l . equa l i ty i s i n te n s io n a l e q u a li ty ,meaning (synonymy). We us e th e symbo l : : or =def . (wintroduced by Defini t ional equal i tybetween l ingu i s. t ic expr-e sa t on s ; it should no t be coit y between ob jec t s ( se t s , e lement s .o f a s e t e t c . ) wby =. De f i n it i o n al e q u a l it y i s th e equivalence re la tabbreviatory de f in i t i ons , changes of bound var iab leof sUbs t i tu t ing equals fo r equa l s . Therefore it i sin th e sense tha t a == b V .., (a == b) holds , s imp l ynot a p ro p os it io n i n th e sense o f th e present theorequa l i ty ca n be used to rewr i t e expressions, in whia bi li ty i s es sen t i a l in checking th e formal correctnIn fac t , to check th e c o rr e ct n es s o f an inference l i

De f in i ti o n al e q u al it y

fo r in s t ance , we must in par t icular make sure t ha tth e express ions A an d B above th e l ine an d therences below ar e th e same, t ha t i s , tha t they a re dequal. Note t ha t th e r ewr i t ing o f an expression i sformal inference.

ru les fo r BA, which i s th eBA . to

(AX)AP(c .x ) B(x) (x E A)

provided aAp(c,a) yie lds th e

equal elements, we needru le er n _introduct ion fo r ) . B( )By th e (a ) Ap(c,a E a) This means b a =

b(X) -_ Ap(c .x ) E B(x) . (x EA . (' )b(X) and hence. ce c y i e l d s AX .E A . But th is i s t r ue ,same value as b(a ) .

d c ts c on ta in t heThe ru les fo r pr o u t B In fac t , we taketh e se t A to th e se .s e t o f f un c ti on s from Here th e concept ofB does no t depend o n x .be (ll .x E A)B, wheren i t i ona l equal i ty is use fu l .

f which we k now only( ) fo r a program 0t o o bta in a nota t ion, Ap c ,x , . f l lows . .Recall t ha tl e ca n be as 0c The second ru t s as r e -th e name y ' e ld equal canonical elemen, . 1 if they )two elements ar e equa It (Ax)b(x) , where oCx) E B(Xi Id s th e resu

Su l t s . eye t to prove i si s canonical , what we wan(x A). Since (Ax)Ap


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Appl ic at i on s o f th e car tes ia n product

F i r s t , us .ing def in i t ional equal i ty , we ca n now def i ne BA bypu t t ing

V - in t roduct ion(x E A)

B(x) t rue

prov ided B does no t depend on x , We n ex t c on si de r th e n - ru l e s inth e i n t e r pr e t a t i o n o f p ro po si ti on s a s se t s . I f , i n th e . f i r s t rUle ,n - fo rmation, we th in k of B(x) as a p ro po si ti on i ns te ad o f a se t ,then , a f t e r th e def ini t ion

(V x 6 A)B(x) _ (Tl x G A)B(.x) ,it becomes th e ru le

a(x 1 , ,x ) .0 E i\{x 1 , ,x ) (XE Am 1 1 ' " ., X Ex E A ( mm+l m+ l x 1 , .. ,xm) , .. . , x .0V -formation

('r;fx E A)B(x) prop .true (x E A1 1 ' . ' X E A (xA m m l'

m+l(x 1 , ,x ) t rue A (m ' . . . , .0 x1

prop . (x E "i1 1 , .. , X G A ( xA m m l ""m+l(x 1 , . . ,x ) t rue A (m ' . . , .0 x 1 "

A(X1 , ,x j. m

A(x 1 , . .. ,x )m

Sim i l a r ly , we wri te

that A(a s , x1 ' . . . ,x ) i s a px . m roposi t ion provided em e Am (x 1" , x ) and ' A xlm- l ' m+l(X 1" " 'x ) A (m ' . . . , .0 x1:

. namely, ' suppose tha t A. . . 1 up to A an d AThen ,. if we ' a .' m+ .0 depend onlyre merely i n te r es te d i n th einesserit '; a 'l t ru th of A(to wri t .e expl i c i t symbols fo thA . s . r e elemen. n' 0 we abbreviat e i t with(x 6 A)

B(x) prop . A s e t

whic h says tha t un i ve rs a l q u an ti f ic a ti o n forms propos i t ions. A s e tmere ly says tha t th e domain over which th e universa l quant i f i e r rangesi s a s e t and th i s i s why we do no t change it in to A prop. Note t ha tth e r u l e of V - f o r ma t i o n i s jus t an ins tance of n- fo rma t ion. Wes im i l a r ly have


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ar e a l l t r ue , as an a b b re v ia t io n o fx E Am(X" . . . ,xm_, ) ,(x x ) prop. (x , e A" .. . , m1 ' , m . x E A (x " . . , xm) x EO A , (x" .. . , xm) , .. . , n nm+l m+

::> a re :

::>

whic h i s a gene ra l i za t ion of t he u su al ru le of formwe may als o us e th e assumption A t rue to prove B .pri z a t i o n i s p e r h a p s "more ev iden t in the Kolmogorov iwhere we might be in th e p o si ti on t o jUdge B to be.der th e assumpt ion tha t th e problem A ca n be solvedsuf f ic ient fo r th e ,problem A: :> B, t ha t i s , th e probprovided tha t A ca n be so lved , t o make sense . The i

(V x eA)B(x) t rueE AV - e l imina t ion

th e ru le of n - e l imina t ion ,back t o th e V - ru les , fromTurningwe have in pa r t i cu l a r

B(a) t rue (A t rue )f of (V x A)B(x),we -se e t ha t , if c i s a prooRes to r ing proofs , (V it E. A)B(x) i sPr o o f of B(a) ; so a proof oft he n A p (c ,a ) i s a A . t a proof o f B (a ),"t y element of 0a method which t akes an ra r " of th e un ive r sa li n t u i t i on i s t i cin ag reemen t w i t h th e

quan t i f i e r .I f we now def ine

B t rueA ::> B t rue

wh i ch c omes f rom th e ru le o f O- in t roduc t ion by sup a n d

::> -e l iminat ion(n x E. A)B,

A :::> B prop .

::> - fo rmat ion

we obtain from th e I1-rules th e . ru le swhere B does not depend on x , trule of n - f o r ma t i on , assuming. B does nofo r implicat ion . From th e

depend on x , we ob ta in which i s obtained from th e ru le o f n - e l imina t ion byExample ( the combinator I ) . Assume A s e t an d x

n- in t roduc t i on , we (AX)x E. A --. A, an d t h er efos i t io n A, A ::> A t rue . This e xp re ss es t he fac t t h a ti s th e method: take th e same proof (const ruct ion) . Wcombinator I pu t t ing I =(AX)X. Not e th e same Ise t o f th e form A -+ A, s ince we do not have d i f f e r end i f f e r en t types .

A t rueB t rue

A ::> B t rue

Bprop .(A t rue )

A prop .

- 37 -- -


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of B.

S =: (Ag)(Af)(Ax)Ap(Ap(g,x) ,Ap(f,x

(Y x E A)(Vy E B(xC(x,y)::>(Yf e (n x e A)B(x(Y x E A)C

(A ::> (B ::> C ::> A::> B) ::> (A ::> 15 tru

(V x E A)(B(x) ::> C(x ::> Y x E A)B(x) ::> (Y x E

S E (Tl x E A)(S(x) -C ( x - ( f Tlx E A)B(x) _ (

which i s t he u su al combinator S, denoted by Agfx.gt or y l og ic . In th is way , we have assigned a type (snator S. Now th ink o f C(x,y) as a proposi tional funhave proved

(V x e A)(Vy e: B(xC(x,y)::> (V f e fT B ) (VxxA 'xIf .we assume that C(x,y) does no t depend on y , then(n y e B(xC(x,y) =B(x ) -C(x ) and therefore

.whi ch is t radi t ional ly written

So , .i f we think of B (x) a nd C(X) as proposi tions, we

Now assume that B(x) does no t depend on x and that Cdepend on x and : y. Then we obtain

that i s , in th e logical in te rpre ta t ion ,

This i s just th e secone axiom of th e Hilbert sty le prcUlus. In th is las t th e proof above, when writt

So , again by

Assume A se t , B(x) se t (x E A),le t x E A, f E (Tl x E A)B (x ) andThen Ap(f,x) E B(x) and

We may now pu t

) E (Tl x E A)(ny E B(icC(x,y)Ag) (.Ar) (Ax)Ap(Ap (g,x) ,Ap ( f ,x )- (T I fE ( nX EA)B(x(nx E A)C(x,Ap(f,x).

Since th e set to th e r ight does no t depend on g, abstracting on g, weobtain

(Af)(Ax)Ap(Ap(g,x),Ap(f,x!( n f E ( n x eA )B ( x ( n x eA)C(x,Ap(f,x.

and, by A -abstraction on f,(Ax)Ap(Ap(g,x) ,Ap(f .x ) e (nx E A)C(x ,Ap(f , x ,

Now, by A -abstraction on x, we obtainAp(Ap(g,x),Ap(f,x E C(x,Ap(f ,x.

n-el iminat ion,

Example (the combinator S).C(x,y) set (x E A, y E B(x andg e (Tl x E A)(ny e B(xC(x,y) .Ap (g ,x)! (n y ! B(xC(x,y) by n-el iminat ion.

Example (the combinator K). Assume A se t , B(x) se t (x E A) andA B(x) Then byA -abstraction on y, we obtaine t x . . ,y ! . ' .

(Ay)X E B(x) - A, and, by A -abst ract ion on x,d f i e th e combinator KAX)(AY)X E (n x E A)(B(x) - - A). We can e n

fAd B as proposi tions, whereutting K == (Ax) (Ay)X. I f we think 0 anB does no t depend on x, K a pp ea rs a s a proof of A ? (B ? A); soA=> (B ? A) i s t rue. K.expresses th e me:hod: given any proof x of A,take th e function from B to A which is constantly x fo r any proof y


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- 40 - 41 -


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a E C and b E D(a) by equal i ty of sets and subst i tu t ion . Hence(a j b ) E: (Ex E C)D(x) by L-int roduct ion. The other direction iss imi lar .

E -el imination

L-equal i ty

a E; A b e fiCa)(x E A, y eB(xd(x,y) E CUx,y

E(c,(x,y)d(x,y E C(C)

d(a,b) E Ca,b

A second rule of L -equal i ty , analogous to th e sen-equa l i ty , is now derivable, as we shall se e l a te r .

is -correct.

(Here, as in L -el imination,. C( ). ( ...z se t z e (L.X E A)Bpl ic i t premiss.) Assuming that we know th e p r emi sse s,j us t i f i e d by imagining Ea,b) , (x ,y)d(x,y to be

we f i r s t execute (a ,b) , which yields (a,b) i t se I f as .subst i tu te a, b fo r x, y in d(X,y) , ob ta in ingd(a ,b)execute d(a,b) un t i l .we obtain a canonical element esame canonical element is produced by d(a,b) , and t hu

d(x,y) E C(x,y(x ! A, y ! B(x

C E (L x E A)B(x)

where we presuppose th e premiss C(z) se t (z E (Lx E A)B(x, althoughit i s no t written ou t explici t ly . (To be precise, we should alsow ri te o ut th e premisses A set and B(x) set (x E A).) We explain th e. rule of L -el imination by showing how th e new constant E operateson i t s arguments. So. assume we know t h e p r emi sse s. Then weE(c,(x,y)d(x ,y as follows. Firs t execute c, which yields a canonicalelement of the form (a,b) with a E A and b E B(a). Now. subst i tute aand b fo r x and y, r es p ec t iv e ly , i n th e r ight premiss, obtaining. d(a,b) e: Ca ,b . Executing d(a,l we .ob t a i n a canonical element e ofCa , b . We now want to show that e i s also a canonical element ofC(C) . I t is a general fact that , i f a E A ahd a has value b, thena = b E A (note, however, that does no t mean tha t a = b E A i snecessarily formally derivable by some part icular set of formal ru les) .In ou r case , c = (a,b) E (Lx E A)B(x) and hence, by subst i tut ion,C(C) = Ca , b . Remembering what it means fo r two se ts to be equal,we conclude from th e f ac t t ha t e is a canonical element of Ca,btha t e i s also a canonical element of C(c) .

Another not at io n f or E(Ci (X ,y) d( x,y cou ld be (Ex,y)(c ,d(x,y ,bu t we prefer th e f i r s t since it shows more clearly that x and y be -

bound only i n d (x ,y ).

3-ei iminationAppl ic at io ns o f t he disjo in t union


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C true

B prop.(A t rue)

C true

A prop.

(: I x E A)B(x) t r-ue :

This rule i s an instance Of and a gener ausual rule of forming proposit ions of th e form A & B,know tha t B is a proposit ion onl y under th e assumption

A & .B vpr -op ,

&-forma tion

(x E B(x) true)

A & B = A x B == (L.X E A)B,where B does no t depend on x. We derive here only th ejunction.

Here, as usual , no assumptions , excep t those expl ic i t lmay depend on th e variable x. The rule of I :-el iminatithan th e :3 -elimina tion ru le , which i s obtained from ip ro of s, s in ce we take into cons iderat ion a l so proofs (which i s no t po ss ib le w it hi n t he language of f i r s t ord

1 > logic. This addi t iona l s t reng th wil l be vis ib le when tand r ight projections below.

The ru le s o f disjo in t union del iver a lso t he u su a. j un c t i on and t he u su al p ro p er ti es o f th e car tes ian prose t s i f we define

(x E A)

B(a) true

B(x) prop .

( 3x E A)B(x) truea ! A

.A se t

3- in troduc t ion(3 x ! A)B(x) prop.

3 -formation

(3X ."E A)B(x) :: (E x E A)B(x),

I n ac cord anc e w it h th e in tu i t ion is t ic of th e existen-t i a l quant i f ie r , th e rule of 'L- introduction may be interpreted assaying that a (canonical) proof of ( 3 x E A)B(x) is a pai r (a ,b) ,where b is a proof of th e f ac t t ha t a sa t i s f ie s B. Suppressingproofs, we obtain th e rule of 3-in t roduc t ion , in however,th e f i r s t premiss a E A i s usually no t made expl ic i t .

then, from th e E- ru les , in te rpre t ing B(x) as a proposi tional ' func-tion over A; we obtain as part icu la r cases:

As we have already done w ith t he car tes ian product,_we sha l lnow se e what are the logical interpretations of the dLs j oLn t una'on .I f we pu t

fore, taking C(z) to be A and d (x ,y ) t o be x in th e r&-introduction


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b E B(a)pa,b = a E A

' a E Ap(c) e A

Left projection

C E (3 x E A)B(x)p f c ) e A

I f we now turn to th e logical in terpre ta t ion, we

(3 x)B(x) true(Ex)B(x) individual

a coun ter pa rt o f which we have ju s t proved, and a rul(3 x)B(x) true

Bex)B(x true

holds, which means that from a proof of . (:3 x ! A)B(x)element of A fo r which t he p roper ty B holds. So we hadescription operator (1x)B(x) (the x such that B(x) hchoice operator (Ex)B(x) (an x such that B(x) holds),in tu it ion is t!c 'poi nt of view, (3x E A)B(x) ' i s true wproof of it . The dif 'f iculty witQ an epsi lon term (Ex)is const rued as a function of t he p roper ty B(x) i t se lproof of (3 x)B(x). This is why Hilbert had to postulof th e 'form

Ln a t Lon and r:-equal i. ty, we o bt ai n a s derived rules:

C true

B true

C true

A & B trueA true

'A & B true

A & B true (A true) A & B true (B true)A true B true

Example ( left projection). We definep(c) - E(c, (x ;y)x)

(A t rue , B true)&-elimination

and cal l i t th e le f t pro jec ti on of c since i t i s a method of obtain-ing th e value of the f i r s t ( left ) coordi na te o f . th e pair produced byan arbi t rary element c of (Ex E A)B(x). In fac t , i f we take th e termd(x,y) in t he e xp la na ti on o f r : -e l imina ti on to be x, then we ,see that

obta1"n th e pair (a,b) with a E A and b E B(a)to execute p(c) we f i r s twhich is th e value of c, and tlien subs t i tu te . a, b fo r x, y in x, ob -taining a, which is executed to yield a canonical element of A. There-

From this rule of &-elimination, we o bt ai n t he standard &-eliminationi C to be A and B themselves:rules by choos ng

Restoring proofs, we see that a ( canon ic al ) p roof o f A & B i s pai rb a.re given Proofs of Aand B respectively.(a,b) , 'where a and

which ha s a counterpart in th e f i r s t _of th e ru le s o f r ight proje c t ion


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that we sha l l se e in th e n ex t example .Example ( r ight projection) . We define

q Cc ) =-E(c,(x ,y)y) .Take d (x ,y ) to be y in -the r ule of L -e l im i na t ion . Fr om x E A,Y E B(x) we obtain px ,y = x E A by l e f t projec t ion , and there-fore B(x) = B(px , y ) . So , by th e ru le equali ty of se ts ,y E B(p( ( x ,y ) . Now cho os e C(z) -se t -( z E (L.x E A)B(x to be th efa mily B(p(z se t (z E (L.X E A)B(x . Then th e rule of L. - e l imi n-a t ion gives q(c) E B( p( c . More f or ma l l y :

( 3 x E A)B(x) , then q Cc) is a construction of B(p(cpro jec t io n , p(c) E A. Thus , supp r e ss i ng t he cons t r uc tc lu s io n , B( p( c) is t rue . Note , however , t ha t , in c ason x , i t is impossi ble to su ppress th e con s tr u c t i on ins i n c e th e conc lus i on depends on it .

Fi nal ly, when B(x) does no t depend on x, so t hati t s imply as B, and both A and -B are thought of as prf i r s t rule _of r i gh t projection reduces to

&-el i minati onA & B true

(x E A) (YEB(x B true

So we have :

p x , y = x E. A

qtc ) == sre , (x ,y)y) E. B(p(c

( ' Ix E A)( B(x):::> (3 x E A)B(x t rue ,

A ::> (B => ( A & B true .

by s up pressi ng t he co ns t r uc t i ons in both th e premisssion.

f rom which , in particular, when B does no t depend on x

Example (axioms of con j u n c t i o n ) . We f i r s t deriveA (B (A & B t rue , which i s th e axiom corre spondof &- in t roduction . i ssume A se t ; B(x) se t (x E A) andY E B( x ) . The n (x,y) E (Lx E A)B(x) by L.-introdu ctioI l - i n t r od uc t i o n , O,y )( x ,y ) E B(x) (L x E A)B(x) (no(Lx E A)B(x) does no t de pe nd on y ). and (Ax) O..y)(x,y)E(T ] x (Lx E. A)B(x .The l o g i c a l reading

We now use th e l ef t and -right projec t i ons to de r iv e Aand A & B::> B t r ue . To ob ta i n t he f i r s t , as s ume z E. (

bE . B(a)b E Bf a )

B(x) = B( p( l x , y )

a E A

- x = px,y E A

qa ,b

Y E _B(p( (x, y )(y E B(x

C E (L x E. A)B(x )q Co ) E. B'(p Lc)

Ri ght pro jec t ion

C E_(L X E A)B(x)

Th e s ec on d o f th es e r ule s is derived by L -e qual i ty in much the sameway as th e f ir s t was derived by- L -e l i min a t io n .When B(x) i s th ought -o f a s -a pr oposi t i ona l f un ct i on , th e f ir s trule of r ight proje ction say s t ha t , i f c i s a cons t ruc t io n of


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Then V(z) A by l e f t projectiori, and, by on z,(Az)p(z) E: (L.X E A)B(x) - A,

In p a r t i c u l a r , when B (x ) doe s no t depend on x, we obtainA & B :::> A t rue .

,' To obtain th e second , from 'z E. (I : x e. A)B(x), we have q (z ) E: B(p ( z by r ight projection , and , hence, by A-abstraction,

(Az)q(Z) E: (Tl e E. (L x E. A)B(x B ( p ( z (note that B( p( z d ep en ds on z) . In p a r t i c u l a r , when B(x) does notdepend on x , we obtain

A & B :::> B t rue.Example (another appl ic a ti on o f th e d i s j o i n t union). The rule of

L - e l i ; i n a t i o n says t h a t any function ,d(x,y) with arguments in A andB(x) gives also a function (with th e same values, by L -equali ty) witha pair in (Lx E: A)B(x) as single argument. What we now prove is anaxiom correspond in g to t h i s rule . So , assume A s e t , B(x) s e t (x E A),C(z) s e t (z E (Lx E. A)B(x and l e t f E: (TIx E A)(Dy E: B ( x C q x , y .We want to find an element of

(TIx E A)(TIy E. B(xCx,y - ( O z E ( L x e A)B(lCC(Z).We define Ap(f ,x,y) = Ap(Ap(f,x),y) fo r convenience. Then Ap(f,x,y)i s a ternary functiori , and Ap(f,x,y) E Cx ,y (x E A, y ! B(x . So ,assuming z E (Lx EO 'A)B(x ) , by L : - e l imination, we obtainE(z,(x,y)Ap(i ,x,y ! C(z) (discharging x ! A and y e. B ( x , and, byA-abstraction on z , we o bt ai n t he f un ct io n

(Az)E(z,(x,y)Ap(f ,x,y EO (TIz E (L x eA)Bwith argument f. So we ,s t i l l have th e assumption

f E (D x e A)( Ilv e B(xC(x,y) ,which we discharge by A-abstraction , obtaining

(Af) (Az)E(z ,(x,y)Ap(f ,x,y !(O x e A)(Oy e B(xCx , y -+ (TIz ! (

In th e logical reading, we have(Y x e A)(Vy ! B(xCx,y: :> (Y z !( L x ! A)

which reduces to th e common(Y x e A)(B(x)::J C) ::J 3 x e: A)B(x) ::> C

when C does no t depend on z, and to

(A :::> (B::>' C:::> A & B) '::> C) truewhen, i n a d di ti on , B Is independent of x.

The axiom of choi ce have


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We now show tha t , w ith t he ru les in t roduced so far , we ca n givea proof of th e axiom of choice , which in o ur s ym bo li sm read s :

( \ Ix E A)(3 y E B(xC(x ,y )=:>(3f e ( n x e A)B(x)( ' Ix E A)C(x ,Ap(f ,x t rue .

The usua l argument in ma t h ema t i c s , based on th e i n -tu i t i o n i s t i c i n t e rp r e t a t i on of th e logica l constants , i s r ou gh ly a sfollows : to prove ('II x ) (3 y)C(X,y) ::> ( 3 f ) (v x )C(x , f ( x , assume tha twe have a .p r oo f o f th e antecedent . This means tha t we have a methodwhich, a pp li ed t o an arbi t ra ry x , y ie ld s a proof of (3 y)C(x ,y ), t ha ti s , a pair consis t ing of an element y an d a proof o f C (x ,y ). Le t fbe th e method wh i c h , to an a rb i t r a r i ly g iven x, ass igns th e f i r s tco mponent of t h i s pa i r . Then C (x ,f( x h ol ds fo r an arbi t ra ry x , an dhence so does th e consequent . The s ame idea ca n be put in to symbols ,ge t t i ng a fo rma l proof in i n tu i t i on i s t i c type theory . Le t A se t ,B(x) se t (x e A) , C(x ;y ) s e t (x E A, Y E B(x , a nd a ss um eze (Tl x E, A) (Ly e B(xPC(x ,y ) . I f x i s an arbi t ra ry elemen t ofA, L e . x e A, t hen , by n-e l imina t i on , we obtain

Ap(z,x) ! (E y e B(x))C(x,y) .We now apply l e f t t o o bt ai n

p(Ap(z ,x E B(x)

an d r i gh t p r oj e c ti on to obtainq(Ap(z,x e C(x,p(Ap(z ,x) .

By A-abst rac t ion on x (o r n . - in t roduc t ion ) , d ischarging x E A, we

(Ax) p (Ap (z .x ) E (Il x E A)B(x ) ,and, by TI-equali ty,

ApAx)p(Ap(z,x . x ) - p(Ap(z,x E Bf xBy sUbst i tu t ion , we g et

C(x,ApAx)p(Ap(z,x .x ) = C( x ,p(Ap(z , xan d hence , by e qu al it y o f s e t s ,

q(Ap(z ,x}) ewhere (Ax) P(Ap(z , x ) i s independent of x .. By abst rac

(Ax)q(Ap(z ,x E (TIx EA)C(x ,ApAx)p(Ap(z ,We now .use th e r ule o f p ai ri ng ( tha t

Ax)p (Ap(z ,x , (Ax)q (Ap(z ,x) !( L . f E. (TIx E A)B(x)(TIx e A)C

(note tha t , in th e l a s t s tep , th e new var iable f i ssUbst i tu ted fo r (Ax)p(Ap(z,x in th e r i gh t member) .abst rac t ion on z , we

(Az)Ax)p(Ap (z ,x,(AX)q(Ap(Z ,X )E (TI x E A)=:> ( E f E (TIx E A)B(x (nx

In Zermelo-Fraenkel se t theory , t he re i s no proo f c ho ic e, so it must be taken as a n a xi om , fo r whicseems to be d i f f i cu l t to c l a i m se l f - ev idence . Here a


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is clear when developing in tu i t ion is t ic

Cauchyf a ) _ (V e E Q)(e > 0 :=> (3 in E N)(Vn E N)(

The n ot io n o f such tha t

R =: (LxEN-Q)Cauchy (x )i s t he de fi n it ion of reals as the se t of sequencesnumbers satisfying th e Cauchy condition ,

11 S . Feferman, Constructive theories of functionsLogic Colloquium .78 , Edited by M. Boffa, D. van Dalen aNorth-Holland, Amsterdam, 1919 , p p. 15 9-2 24.

where a i s th e sequence aO' a 1 , In th is way, a realsequence of ra t ional numbers with a proof thath e Cauch1 condition. So, a ssuming c E R, e E Q and d 'E

In a dd it io n t o disjo in t union ; exis ten tial quan t.Lfcartesian product A X B and . conjunction A & B, t he o p ea f i .fth Lnt.er-pr-e t at.Lon : th e se t of a l l a EO A such thatLe t A be a se t and B{x) a p roposi t ion fo r x EO A. We wanih e se t of a l l a& i tha t B(a) holds (which i s usu[x E A: B(x.ll> . 'ha ve an ' element a EO A such tha t B(a)to have an element a EO A together with a p ro of o f B (a ),element b E B(a). So th e elements of th e se t of a l l elesa t isfy ing B(x) ar e. p ai rs ( a, b) .wi t h b E B(a) , Le ele(Ex E A)B(x) . Then th e L-ru le s play th e role of th e caxiom ( or t he s ep ar at io n principle in ZF). The informab B(a) i s cal led th e wi tness ing informat ion by Fefermca l app li c at i on is t h e fo ll owing .

,Bxampl e (the reals as Cauchy sequences).

of a sequence 'choicefo r i n s t a n ce, i n finding th e l imi t

function.

th e axiom of choice has been provided in th e formj us t i f icat ion of f In mani sorted languages, th e axiom of,choice i sof the ,above proo " .. t the re is no mechanism to prove 1 t. For instance, 1nexpress ib le bU f f in i te type , it must be taken as .an axiom . TheHeyting ar i thmet ic 0'" i om ofneed fo r th e a

mathematiCs ' a t d ep th ,t i 1 inverse of a surjectiveof rea ls or pa r a

- -- 54 -


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b = d EB

b E B

j ( b ) = j (d) E A +

j ( b ) E A + B

B s e tA + B se t

A se t

a Ai ( a ) E A + B

i (a ) = i (c ) E A + B

+-formation

. +-introduct ion

where i an d . j ar e two new pr imi ti ve cons tan t s ; t he i r ut h e i n fo rmat io n tha t an element of A + B comes from Aof th e tw o is th e case . I t goes wi th o ut s a yi ng t ha t wru les o f + -i nt ro du ct io n f or equal e lemen ts :

Dis jo in t union of tw o s e t a

The canon ica l elements of A + B a re formed using:

We now g iv e t he ru le s fo r th e sum (d i s jo in t unionof tw o s e t s .

Since an a rb i t r a ry element c of A + B y ie lds a canon ith e form i (a ) or j ( b ) , knowing c e A + B means .t h a t w

t e r mi n e f rom whi ch of th e two se t s A an d B the elemen

Only b y m ea ns of t he p ro o f q (c ) do we know how far to go th eapproximation des i r ed .

Ap(p(c) ,p(Ap(Ap(q(c) ,e ) , d ) E Q.

Ap(Ap(q(c),e),d) E (3 m E N)(Vn E N)(lam+n-aml e ).

! N,

and we now ob ta in am by applying p(c) to it ,

Applying l e f t pro jec t ion , we obtain th e m we need , i . e .

an d

. d d' proof of the pr oposi t ion e > 0) , then , by meanso the r wor s ' . 1S aof th e pro jec t ions , we obtain p Cc ) an d q(c) E Cauchy(p(c.Then

- - - 1 -


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+-elimination

C E A + B(x E A)

d(x) C(i(x(y e B)

e(y) E C(j(y

, ,become ' e vi den t .of two proposit ions is now interpr

sum ,o f two se t s . We therefore put :

C true(B true)

B t rueA V B t rue

B prop.

C true(A t rue)

C true

A V B prop.A prop.

A trueA V B true

A v B true

V -formation

V - introduction

v -elimina tion

A V B == A + B.From th e formation and i nt ro d uc ti on r u le s f or +, we thecorresponding rules ' f or V :

Note th a t , i f a i s a proof of A, then i (a ) i s a (canoniA v B, and similar ly fo r B.

follows from th e rule o f + -e liminat i on by choosing a famC == C(Z) (z E A + B) which does no t depend on z and supproofs (constructions) both in t he p rem is se s, i nc lu d in gt ions, and t h e conc lus ion .

(y E B)

(y E B)

e I y ) E C( j (y

e(y) E 'C( j ( y

e I b ) E C(j(b

d f a ) E C(i(a(x E A)

d Ix ) E C(i(x

D(c,(x)d(x), (y)e(y E C(c)

D(j(b) , (x)d(x) ,(y)e(yb E B

i (a ) , then c = i t a ) E A + B and hencevalue j ( b ) , then c = j (b) E A + Bandth is explanation of th e meaning of D,

+-equal i ty(x E A)

a E A d(x) E C(i(xD(i(a) , (x)d(x) , (y)e(y

where th e premisses A 's et , B se t and C(z) se t (z E A + B) a re p re -supposed, a lt hough not explicitly wr it te n o u t. We must now explainhow to exeaute a program of th e new form D(c ,(x)d(x) ,(y)e (y. 'As -sume we know c E A + B. Then c wil l yield a canonical element i t a )with a E A or j (b) with b E B. In th e f i r s t case, subst i tu te a fo r xin d (x ), obtaining d(a) , and execute i t . By th e second premiss,d(a) e C( i ( a , so yields a canonical element of C( i ( a . Simi-l a r ly , in th e second c as e, e (y ) i ns te ad o f d(x) must be used to ob -t ai n e (b ), which produces a canonical element o fC( j ( b . In e i thercase , we obtain a canonical element of C(C), since, i f c ha s value

C(c) = C( i ( a , and, i f c ha shence C(C) = C( j (b . Fromth e e q ua li ty r u le s :


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(4 ) I (A,a ,b) .b e A,B,

(3 ) a(2 ) A(1 ) == or =def. '

P ropos i ti o n al e qu a l i ty

We now tu rn to th e axioms fo r equal i ty . I t i s a(de r i ving i t s or ig in from Pr inc ip ia ' Mathematica) toin predicate logic i den t i t y . However, th e word i den tproper ly used fo r de f i n it i o n a l e qu a li t y, =or =d fe.a bo ve . In f ac t , an . equa l i ty s ta temen t, f or in s tance ,a r i thme t ic , does not mean t ha t th e tw o members a re thmerely tha t they have th e same value. Equa l i ty in prhowever, i s a l so d i f f e r en t ou r equa l i ty a = b Eformer i s a proposi t ion, while th e l a t t e r is a judgep r opo s it i o n al e qu a l it y i s nev er t he l e ss i n d is p ens ab lee q u a li ty I (A ,a , b ), which a ss er ts t ha t a an d b a re e qt he se t A, .bu t on which we ca n operate with th e logi( reca l l tha t e .g . t he n eg a ti on or quant i f ica t ion ofnot make sense ) . In a cer ta in sense, I (A , a , b f i s anof =. We then have f ou r k in ds of equal i ty :

Equali ty between objects i s expres sed in a judgementf ined separate ly fo r each ca tegory , l i ke t h e c a te g oror th e category of elements of a s e t , as in (3); (4 )w he re as ( 1) is a mere s t i pu l a t i on , a r e l a t i on betweeexpressions. Note however t ha t I (A , a, b) t ru e i s a ju

. wi l l turn out to be equ iva len t to a = b ! A (which i

. i

(A ? C) B :::> C) :::> (A Y B ::> C t r ue .

I f , moreover , C(z) does not d ep en d o n z an d A, B ar e proposi t ions aswel l , we have

This , when C(z) i s thought as a propos i t ion , g ives

(A r ) (Ag) (Az)D(z, (x)Ap(f ,x) , (y)Ap(g ,yE (n x E A) C( i (x ) ) - ( (n y ! B) C( j (y ) ) - (n z E A + B) C( z .

("Ix E A)C(iex::> "Iy ! B)C(j(y: :> ( 'Vz ! A + B)C(z t r ue .

Example ( in troductory axioms of d i s junc t i on ) . Assume A s e t ,B s e t an d l e t x ! A. Then i (x) ! A + B by +- in t roduc t ion , an d henceO.x)iex) e A -A + B by A-abs t r ac t ion on x.lf A and B a re proposi-t i ons , we have A ::> A V B t r ue . In th e same way, (Ay) j (y ) e B - - A + B,and hence B A V B t r ue .

Example (e l iminatory axiom of d i s junc t i on ) . Assume' A s e t , B s e t ,C(z) s e t (z Eo A + B) an d le t f ! (Tl x e A)C( i e x , g ! (D yE B)C(j(yand z e A + B. Then, by n-e l imina t i on , from x e A, we haveAp(f ,x ) EC ( i e x , and, . from y e B, we have Ap(g,y) e C( j ( y . So ,using z E A + B, we ca n apply +-el iminat ion to obtainD(z , (x )Ap(f ,x ) , (y )Ap(g .y Eo C(z) , thereby discharging .x e A an dy e B. By A-abs t r ac t ion on z , g, f in t ha t order , we ge t

- 61 _


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a=be:A

(x ! A)

Finally,.mi t's th e

c = r ! I(A , a , b )C E I (A,a ,b )

C I (A,a ,b )

I -e l imina t ion

We would then d ie r ve th e fOllowing rul es wh i ha s p ri m it iv e : ' c we h er

I - equa l i t y

note t ha t I - fo rmat ion i s th e only rUleformation o f f am i li es up to now+ , NNW of s e t s . I f only thn ' , were allowed e opera, we would only .Example ( in tro duc to r g e t constant s e t sx ! Y aXiom of i den t i t y )A. Then x = x ! A Assume A sabst rac t ion on x (A) ' and, by I - i n t roduc t i on , r E l (A,x ,x, x r!( \ {XE .A ) I (Ac an o ni ca l p r oo f of th 1 . ,x ,x) . Therefore (Ae aw o f ident i ty on A.

.v'

b e: AE. AI (A,a ,b ) s e t

A s e t

I-formation

I - in t roduct ion

r e: I (A,a ,b )

We now have to explain how to form canon ica l elements of I (A , a , b ) .The s tandard way to tha t I (A,a ,b ) i s t rue i s to have a =b E. A.Thus th e i n tr o du c ti o n r ul e i s s imply: if a = b e: A, then there i s acanonical proof r of I (A,a ,b) . Here r does no t depend on a , b or A;it does no t matter what canonical element I (A,a ,b} ha s when a = b E. A,as long as it ha s one.

t ha t it has th e same sense ) . (1 ) i s (sameness . of mean-ing) , while (2) , (3 ) and (4 ) are extensional (equa l i ty between ob -j e c t s ) . As fo r Frege, elements a, b may have d i f f e r en t meanings, orbe di f ferent methods, bu t have th e same value. Fo r in s t ance , wecer ta in ly have 22 = 2+2 N, but no t 22 _ 2+2.

ohO

. (Ax) r E (V x E A) I (A , x, x)r E I (A,x ,x )

Example ( e l imina to ry aXiom o f i den t i t y )property Sex) prop (x E A) Given a se t A. over A we c l .correspOnding to L . t ha t th e lawe l b n iz ' s p r in c i p letha t eq u 1 I . of indi scernib i l i tya e ements sa t i s fy th e same prope r t i e s ,

We could now adop t el iminat ion and e qu al it y r u le s fo r I in th e sames ty le as fo r Tl , L. . + , namely in troducing a new el iminatory ope ra to r .

Also, note t ha t th e rule f o r i n tr o du c in g equa l elements of I (A,a ,b )i s th e t r i v i a l one:

a = b E Ar = r E. I (A,a ,b )

- 62 - - -


37/52

(AZ)(AW)w E I(A,x,y) ? (B(x) ? B(y

x e A)qx,y

(x ,y )

(x,y) E (E x e. A)B

(y e B(xpx,y = X " A(px , y , qx , y )

(x e: A)

r E IL: x e. A)B(x) ,(px,y ,q

(p ( (x ,y , qx , y )

(p(c) , q (c = c e (1: x e A)B(x)E(c,(x,y)r ) e IL : x E A)B(x) , (p(c) , q( c . c

r- e I LX e A)B(x) , (px , y , qx , y ) , ( x

E (c , (x , y ) r ) e I( ( L x E A)B( x) , ( p (c ) , q ( c ) )

This example is typical. The I -ru les ar e used systemath e uniqueness of a function , whos e existence is givena ti on r ul e , and whose p rope r ti e s a r e expressed by th eequal i ty rules.

C E (L x. e A) B ( x )

and hence, by I-el imination , (p(c) ,q(c = c e. (L x E

By I - int roduct ion,

i s derivable. I t i s analogue of th e second n -equacould also be derived, provided t he T I- ru le s were formth e same pattern as the other rules . Assume E: A, Y eprOjection laws , px ,y = x 6 A and qx,y = y e BE-int roduct ion ( equa l e lement s form equal pairs) ,

Fo w take t he f am il y in th e rule of I:-ellmination1(0:' x E A)B(x), (p(z) ,q (\z ,z ) . Then we obtain

Y EO A and

(x e. A)B(x) se t

B(y)(x )x = Y 6 A

(z e. I (A,x ,y

w 6 'B(y)EO B(x) ? B(y)

(w 6 B(x

( Vx E A)( v s e A)( I (A, x , y) .::> (B (x)::> B(y) true.

. " . .propertie s is no t possible , and hence th e meaning of ident i:ty . ha s: t. obe de fined in anothe r wa y , wi t h ou t invalidating Leibniz 's

Example (proof of th e converse of th e projection l aws ) ".We cannow prove tha t t he in fe r ence rule

C E: (L x e A)B(x)c = (p(c) ,q (c e (Ex e A)B(x) .

(a = b) == (V X)(X(a) ? X(bfrom whi ch Leibniz 's principle is obvious, since it i s taken to definet he meaning of identity . In the present language , quantif ication over

The same problem (o f justifying Le ibn i z' s p r i nc i p le ) was solvedi n P r in c ip i a by th e us e o f impred ica ti ve second order quantif ication .There one defines

(AX)(Ay)(Az)(AW)We. ("Ix E A)(Vy 6 A)(I(A,x ,y ) ::> (B(x) ? B(y)

hence B(x) = B(y) by subst i tut ion. So, a ssuming w E B(x), by equal i tyof s e t s , we obtain w EO B(y) . Now, by abstraction on w, z , y , x in th a torder , we obtain a pro of o f th e c lai m:

To prove i t , assume x E A, Y e. A and z 6 .I(A ,x,y) . Then x

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38/52

c(mn)d e C

(m = 0 ,1

se tn

cm E C(mn) , (m =0 , ' 1 . . n-1

N -formationn

.N - introductionn

N -el iminationn

Finite se ts

been . seen to be equal to mn and cm e C(mni 'i s asion o ve r t he f in i te se t Nn; it i s a kind of

Here, as usual, t he f am fl y of se ts C(z) se t (z eN ) mnknow th e premissesresul t i s m fo r son '

and n-1. Select th e cor-r-espond t ng element c of. ' . . mexecuting i t . The resul t i s a .c a no nLc e L element

as a property over N : Assuming we. n .as follows: f i rS t execute c, whose

So we "have th e sets' NO with no elements, N,. with th e selement 01, with canonical elements O2, '12 , etc .

Note th a.t , up to 'now, we have no o pe ra ti on s t o bfrom nothing, b ut o nl y operations t o o bt ain new sets f(and from families ' of sets) . We now introduce f in i te sgiven outright; hence t he ir s et formation rules willActually, we have in f in i tely 'many ru les , one group ofn = O. 1,

th e e qu iv al en ce o f t he se tworules. we c an p rove(2 ) th e co rre sponding prop-indexed familY as in ,

(3 x E A)I(B,b(X) ,y ) (y E B) ,

1 , f elements).(properties and indexed 1es 0o f l oo ki ng a t s ub se ts o f a se t B:

p Lx) e B (x E (E y E. B)C(y ,

an indexed family of elements(2 ) a s ub se t o fb Lx ) E B (x e A).

roposi tional (property)(1 ) a subset of B is a P .C(y) (y EO B) ;

ty as in (1 ) th e correspondingand, conversely, given a proper indexed fami ly is

Using th e identityconcepts. Given anerty i s

Exampleare two 'ways

When C (z ) d oe s not depend on z , it i s possiblewe have th eby th e above explanat ion,

- -


39/52

.1. t rue '(B t rue)

A t rueA V B t rue

.1. -el1mina t .Lon

C t rue.1. t rue

Then 0 , i s a ( c an o ni ca l ) p r oo f of ,l r , s ince 0 , E N,t i on . So T' i s t r ue . We now want to prove t ha t 0 , ielement of N" tha t i s , t ha t th e ru le ,

i s derivable . In fact" , from 0 , 6 N" we ge t 0 , : 0 ,r I (N"O"O;> . Now appl t 'N , -e l imina t ion with I (Nfo r the family of se t s C(z) (z E N,> . th e asswe ge t R, (c , r ) E I (N"c ,O , ) , an d hence c : 0 , EN , .

proof (construct ion) no t only in t he c on cl us io n b utpremiss. We then arr ive a t th e l og i ca l i nf er en c e r u

which i s eas i ly seen to be e q ui va ie n t t o th e form aExample (about N,) . We define

' t r a d i t i o n a l l y cal led ex fa lso quodlibet . This ruleordinary ma th ema ti cs , b ut in th e form

(m: 0, ' , . . . , n-1)

N _ in t roduc t ion ) :n

R (m ,c o , , c n _ , )n n

N _equali tyn

The rule becomes s imply :

N _eliminat ionR (c ) E: ct c)

tha t we 'u n de r s t a n d t ha t we sha l l neverf th e ru le i sTh ex p ' lanat ion 0 ' t R (c )e we sha l l never have to execu eO , e lement C6 NO' so t ha tge t executing program of th e formThus th e s e t of ins t ruct ions fo r b ti ' i mi la r t o th e programmi'ng s ta tementR (c ) i s vacuous . It s s

, '2in t roduced by Dijks t ra'2 Se e note 2 .

, in t he c on cl u-, i of m - " ..., n-fo r each ch o ce ' - ,(one such ru le Id b to pos tu la te th e r ul es fo r ns ion) . An a l t e rna t i ve approach wou ' e == N + N e t c . , and ,equal to a n d ' on iy , ',d e f i n e N2 :: 'N, + N" N3 - , 2then de t ive a l l o the r r u l e s . no i n tr o du c ti o n r u le hence noExample (about NO) NO ha s

na tu r a l to putelements; it i s thus

From th e meaning of Rn , givenn ru les (no te t ha t mn 6 Nn by

Consistency_ Co ,.' th e rule of

- 68 -


40/52

a l l th e usual rules o f n e ga ti on .we can easi ly derive

What can .we s ay a bo ut th e consistency of ou r syWe can understand consistency i n two diffe rent ways :( ,) consistency. Then , to prov

th e consistency of theory T, we conside r anothercontains codes f or p r opo si ti on s of th e original theocate Der such tha t Der( 'A') e xp re ss es t he fact thatA with code ' A' is derivable in T. Then we define Co- 'Der( ' l . ' ) =.Der( '.L ' ) ::>.l. and (try to ) prove tha t 'T' . This method is t he o nly one applicable when, l ikegive up thehope 'o f a semant i ca l of thof inference; it could be followed , with success , ali s t i c type t he or y, b ut , since we have been as meticus eman ti cs a s a bo ut i t s syntax , we have no need of i tconvince ourselves . d i r e c t l y of i t s consistency in thminded way.

( 2) S impl e minded consistency . This means simplbe proved, or tha t we ahaLl, never have th e r ight to(which, unl ik e t he p ro pos it io n Cons above , i s no t aproposi tion) . To convince o ur se lv es o f t hi s , we a rg uc e 1- would hold fo r some element (construction) c ,yi.eld a canonical element d e JL ; bu t th is is impossno canonical element by defini ton ( reca l l that we deThus JL true cannot be proved by means of a system oSo , in case we h i t upon a proof of 1-. t rue , we wouldmust be an er ror somewhere in t he p ro of ; and, i f a fJL t rue is found, then a t least one o f th e formal ruis no t correct . Reflecting on th e meaning of each of

true

t f N2is e i therprove tha t any e lemen 0

in th e proposi tional form

Boolean == N2

Thus t he o p er at io n R, canof N,-equality t r iv ia l izes .with .

) We make th e def in i t ion(about N2 .

value as c , .As fo r N, above, we can

O2 or '2 ' bu t obviously only

t ru th values t rue ,Then we can define i f c Co

which means that c yieldsc is true ,

Example (negation) . If we pu t'" A:: - , A == -A == A- NO

which consists ' of the twoth e ty p e used in programmingBoolean is f I - ,false . So we c ou ld p ut true == O2 and a se = 2'R ( c c c ) because, i fc, 2 ' 0 ' ,O2, then R2(C,CO'c,) has th e. d R (c c c) has th e sameC . otherwise c yields '2 an 2 '0 "same value as 0'

and th e rulebe dispensed

Example

t h definit ion R,(C,cO)Conversely, by making eN _elimination ' i s derivable from th e rule, .

c = 0 , e N,

- 11 -

i n t u i t i o n i s t i c type theory, we eventual ly convince ourselves tha t


41/52

N s e t

a '"a ' E

e(x,y) ! C(x')(x e: N, y E C(x

o ! N'

d e C(O)

c as e w it h any o the r i n tr oduc ti owhatever element a i s Th us a Eor a ' wh .1 ' . . er e a 1 ha s value e i thewe reach an element a which hn as

R(c,d , (x,y)e(x,y ! C(c)c c: N

N-introduction

.so far , we have no means of constructing an ith e simplest one, namely th e s e t of

nowby th e rules :

N-formation

N-elimination

Natural numbers

Note t h a t , as is th ei s always canonical,a h as v al ue e i t h e r 0u n t i l , eventual ly,

where C(z) s e t (z e N ) . R(c,d,(x,y)e(x ,yf i r s t execute c, get t ing a canonicalo or a ' fo r some a N. In th e f i r s twhich yields a canonical element

i s explainelement of N, whcase , continue bf E C ( O ) ; b u t , sincet h i s case , f i s 1a so a canonical element o f C(C) = C(

second case, ' sUbsti tute a fOr x and R(a d ( ) ('.' . . " x, e x, y

they are correct; therefore we wil l never find a proof of JL trueusing them.

Final ly , note that , in case , we must rely on t he s impl eminded cons i st ency o f a t l e a s t t he t he or y T' in which Cons i s provedin order to o bt ai n t he simple minded consistency (which i s th e formo f consi s tency we really ciare about) from themetamathematical con-sis tency of th e or iginal theory T.In fact ; once c Cons fo r some ci s proved, one must a rg ue a s follows: i f T were no t consistent, wewould have a proof in T of 1.. t rue, 91' a ! NO fo r some a. By coding,t h i s woul d give 'a ' G Der( ' . l ' ) ; then we would obtain A p ( c , ' a ' )! JL ,i .e . that JL true i s derivable in T '. At t h i s p oi nt , to conclude thatJL true i s no t provabie in T, we must be convinced that JL true i sno t provable in T '.

ou t to be th e same concept when p ropos i ti ons a r e

- -

preceding value) fo r y i n e (x ,y ) so to e(a,R(a,d,(x , y ) e (x ,y) .


42/52

Mathematical induction

a, w

o ! N,= a E N,

b ' E Nb eN

a '

pd (0 )pd (a ' )

. (V X ! N ) ( V yEN) (I (N ,x ' ,y ' ) ::> I (N ,x , y

pdf a ) == R(a,O, (x,y)x) .Example (the predecessor function) . We pu t

a E Npd I a ) e N

This defini tion is just i f ied by computing R(a,O,(x0, t he n pd( a) also y ie ld s 0, and, i f a yields b ' ,th e same value as R(b' ,O,(x,y)x) , which , in t u rn ,

def in i t ion , bu t here these equalities . r e no t defprecisely , we have

which is an i ns ta n ce o f .N-e l imi na t i on , and

which we obtain by N- e qua l i t y .Usi ng p d, we can derive th e third Peano axiom

' v a lue as b. So we have pd(O) = 0 and pd(a ')

Indeed, from a ' = b ' ! N, we obtain pd(a ') = pd(b ')gether with pd(a ') = a e Nand pd(b') = b eN , yiesymmetry and t ransit iv i ty . We can also obtain it in(Vx ,y ) (x ' = s' => x = y) , tha t i s , in the p resen t

e(x,y) E C(x')(x E N, Y E C(x

e(x,y) E C(x')(x EA , y EC(X

C(c) true

(x EN , . C(x) true)C(O) true C(x') true

d eC(O)

d e C(O)

C E N

a e N

unt i l we eventually r eac h t he v al ue 0 . This expLana tLon of th e

R(O,d , (x,y)e(x,y = d E C(O)

N- eq ua l i t y

I f we expl ic i t ly write ou t th e proof (construction) of we s eetha t it is obtained by recurs ion. recursion and induction turn

R(a ' ,d, ( x , y ) e ( x , y = e(a,R(a,d,(x ,y)e(x , y ) E CIa')) - t- a l function (prop-'vident. Thinking of C(z) (z e N as A

er ty) and supp res s ing the proofs ( con s tr uct i on s ) i n th e second andth i rd premisses and in the of th e rule of N- e l i mi n a t i o n ,we arrive a t

elimination r ul e a ls o makes th e e q ua li ty r u le s

Executing it , we ge t a canonical f which, by th e r ight premiss, is inC(a ' ) (and hence i n C(c) since c a ' EN) under th e assumptionR(a ,d ,(x ,y)e(x,y e C(a) . I f a ha s value 0, then R(a,d,(x,y)e(x,yis in C(a) by th e f i r s t case . Otherwise , continue as in th e secondcase ,

t h e le a s t a such tha t Ap(f,b) =In f ac t , assume x E. N, YE N and z EI (N,x ,y ) . By I-e l im ina t ion,


43/52

E N,}dO, f)lj1(a ' , f )

under th e premis se s a E N a nd feN - N2 , an d hence

{if such b ex i s t s ,

r (a , f ) =a, o therwise .

(r(O) = (AOO E ( N -+ N,ljL(a ') (),.f)R2(Ap(f,O) ,0 ,AP(r(a) , f)') e (N -+

r ex ) E (N -. N2) -+ N (x E N) such t ha t

so t ha t we ca n define th e funct ion f (a ,.f) we ar e l opu t t ing r ( a , f ) =Ap(r(a) , f ) . The requirements on rei s f i ed t h r o u g h an ord inary p r imi t ive r e c u r s i on , bu tth i s task i s fu l f i l l ed by th e ru l e of N- e limi na ti on

Su c h a function wi l l be obtained by . solving th e recu

where f :; C\x)Ap( f ,x ' ) i s f sh i f t ed on e s te p to th e.Ap ( f , x) =.Ap(f , x ' ) E N2 (x EN) . In fac t , in c ase thr (O , f ) = E N, i r respect ive o f wha t funct ion f i s .suc :e s so r form , r (a ' , f ) = f (a , f ) ' EN , provided tha'2 E N2 ; otherwise , f (a ' , f ) = eN . Th er ef or e t o coc a n sh i f t f un t i l th e bound i s 0 , bu t checking eacha t i s t rue == 02 or fa lse == '2 ' Even if it admitr ecu r s ive so lu t ion , th e problem i s most so lvet y p e s , as we sha l l n6 w see in de t a i l . We wa nt to f i

b E N(a + b ) ' e N

a e N

bEN

a + b '

bEN

a E N

a E Na +

a E Na +

Usua l proper t ies of th e product a . b ca n then eas i l y be derived .( the bounded We want t o so lv e th e problem:

given a boolean funct ion f on natura l numbers, L e . fE N - .N 2, f indth e l e a s t argument , unde r th e bound a EN , fo r which t he v al ue of f i str u .e . The so lu t i on wil l be a f un ct io n f -( x , f ) E N (x E N, f E N -N 2)sa t i s f ying :

a + b == R(b,a , (x ,y)y ') .

a b == R(b ,O , ( x ,y ) (y + a .

from which we ca n also derive t h e co rr e spond ing axioms of f i r s torder ar i thmet ic , l i k e in t he p re ce di ng example . Note again t ha t th eequal i ty here i s not def in i t ional .

Example (mu l t i p l i c a t i on ) . We define

The meaning o f a + b i s to perform b t imes l he s u c ce s so r op er a ti o non a . Then on e eas i ly de r ives th e ru les :

x ' = y ' E N; hence x = y EN , from which r e I (N,x ,y ) by I - i n t r o -duc t ion . Then , by A-abs t r ac t ion , we obtain t ha t (AX)(Ay)(Az)r i s aproo f ( c on s tr u ct io n ) o f th e claim .

Example ( add i t ion ) . We define

Lis ts


44/52

b e Lis

e(x,y,z) E:

(a.b) E List(A

(x EA, Y eList (

a A

d E: C(nil)

n i l " 'List(A)

C E List(A)

List- introduction

A se t

l i s t rec(c ,d ,(x ,y ,z )e (x ,y ,z E C(c)

List-el iminationwhere we may also use th e nota tion () :: n i l .

List(A) se t

List-formation

where C(z) (z E List(A is a family of sets . The inscute l i s t r ec are: f i r s t execute c , which yields ' eithec as e c on t in u e by executing d and ob t a i n f e C(nil) =with a E A and b E L is t( A) ; i n th is case, executee(a ,b , l i s t r ec (b ,d , (x ,y , z ) e (x ,y , z) which yields a ca

where th e in tu i t ive explanation i s : List(A) is th e se'el ement s of th e se t A ( f in i te sequences of elements o

We can follow , the pa t te rn used nat o i nt ro du ce o ih er defined sets . We s ee h

" of l i s t s '.,'> (f E N-N 2) ,

(y e (N-+ N2) - N) fE N - N2Ap(y,f) EN

which, in t ur n, e qu al s t he value of

Next , Ap( f,O) is evaluated . I f th e value of Ap(f,O) i s true =O2,then th e value of f ( a , f ) is O. I f , on th e other hand, th e value ofAp(f,O) i s false 5 '2 ' then th e value of r (a , f ) equals valueof f (b , ! )"

Observe how t he e v al ua ti on o f , .. (a ,f ) ==Ap(R(a, (Ano, (x,y) (AnR 2(Ap(f ,0) ,0 ,Ap (y , f ) ' , n proceeds. Firs t , ais evaluated. I f th e v alu e o f a is 0, th e value of equals th evalue of ApAf)O, f ) , which is 0. I f , on th e other hand, th e valueof a i s b ! , th e value o f r ( a , f ) e qu al s t he v al ue of

Ap(f,0)EN2 oeN Ap(y,r) 'eNo N R2(AP(f,0),0,AP(y,f)') EN

a N (AnO e (N-N2)-aoN (AnR2(AP(f,0),0,AP(y,f)') E (N-N2)-+N/4(a) = R(a,O.no,(X,y)(AnR/AP(f,O),O,AP(y,f)' e (N-N2 ) -N f E N-N 2

,..(a,n == Ap(p.(a),n e N

W r it te n o ut in tree form th e above proof of r ( a , ,c) E N l oo ks a sfollows:

Wl l o r d e r i n gsl .. .,f & Ca .b)) =,C(c) . i f we pu t g(c) _ l i s t r e b ( c , d , ( x ,y ,z)e(x,y , z) ) ,


45/52

Lis t - e q u a l i t y

( x ," A)B(x) s e ts e t

(Wx E A)B( x) s e t

W- f o rma t i on

Wha t does i t mean fo r c to be an element of (Wx e A)Bt h a t , when calculated , c yields a value of th e fo rm ssome 'a and b , where a E A and b i s a function such thchoice of an element v B(a) , b applied to v yie l dssup(a l ,b l ) , wh er e a l E A and b l i s a function such t hchoice of v l in B(a l ) , b l appl ie d to v l ha s a value su n t i l in any case ( i .e . however t he s uc ce ss iv e choiceeventual ly r ea ch a bottom element of t he form sup(ani s empty , so t hat no choice of an element in B(a ) i snfollowing picture , in whic h we loosely wr it e b( v) f orhelp (look a t i t from bottom to top) :

The concept, of wello'rdering and th e principle ofinduction were f i r s t introduced by Cantor . Once th eymulated in ZF , however , they l os t t h e ir o r i g i n a l compt ent . W can construct ordinals i n t u i t i o n i s t i c a l l y a st r e e s , which means tha t they are no longer t o t a l l y or

e(x,y,z) 'E Cx .y , ( x E A, Y E Li st(A), z E C(y

e( x, y ,z ) E Cx . y

d E C(nil)

(x e A, y & L i s t (A), z E C(y

b 'E List{A)

d E C(nil) ,t i s t r e c ( n i l , d , (x ,y ,z )e (x , i , z = d & C(nil) :

= e( a ,b , l i s t r e c ( b , d , ( x ,y ,z ) e ( x , r , z ) E

, a E. A

Simila r rule s could be gi v en fo r f i n i te tre es and other induc-t ively define d concept s .

then f i s th e v al ue o f e( a ,b ,g(b).

then i t holds fo r sup(a,b) i t s e l f ) , then C(c) holds


46/52

d(x,y,z) e G(sup(x,y

G(c) true

(V x E. A)(Yy e B(x) - - (Wx ! A)B'Iv E B(xG(Ap(y ,v

C E. (WXE A)B(x)T(c,(x,y,z)d(x ,y , z E G(c)

.We l imi na t i o n. (x e A, y e B(x) -+ (Wx e A)B(x), z E. (

c E (Wx E A)B(x)

c E (Wx e A b i t more formally ,

Now we resolve t h i s , th e W-elimination rulpremisses i s tha t G(sup(x,y i s true , provided thatY E B ( x ) - ( W x E A)B(x) and (Y v EB(xG(Ap(y,v id(x,y,z) be th e function which g iv es t he proof of Gterms of x E A, Y E B(x) _ (Wx E A)B( x) and th e proo(V v eB(xC(Ap(y , v , we arrive a t th e rule

where T("c,(x,y,z)d(x,y,z i s executed as follows . Fwhich yields aup Ia j .b ) , where a e A and b e B(a) _ (Wth e components a and b and s u b s t i t u t e them fo r x and

. d ( a , b , z ) . We must now s u b s t i t u t e fo r z th e whole seqf unc t ion va lue s . This sequence i s (AV)T{Ap(b,v),(x,cause Ap(b ,v ) E (Wx E A)B(x) (V E B(a i s th e funcates t he s ub tr ee s ( pr ed ec es so rs ) o f sup(a,b) . Thend(a,b,(Av)T(Ap(b,v), (x ,y ,z)d(x,y,z)}) yields a canoe E G(c) as value under t he assumption t h a tT(Ap(b,v) ,(x,y,z)d(x,y,z E G(Ap(b,v (v e B(a .

c sup(a,b)

b ! B(a) - (Wx e A)B(x)sup(a ,b) e (Wx E A)B(x)

a E AW-introduction

From th e e xp la na ti on o f what an element of (Wx E A)B(x) i s ,. wese e th e correctness of th e elimination rule , which i s a t th e same time t r a n s f i n i t e induction and t r a n s f i n i t e recursion . The appropriateprinciple of t r a n s f i n i t e induction i s : i f t he p ro p er tyG(w) (w E (Wx E A)B(x i s inductive ( L e . if i t holds f'or- a i l pre-decessors Ap(b,v) e (Wx E A)B(x) (v ! B(a of' an element sup La j b ) ,

By th e preceding exp lanat i on , t he fol lowing ru le fo r introducing ca -nonical elements i s j u s t i f i e d :

Think o f s u p( a, b) as the supremum ( l e a s t o rdi nal g r ea ter than a i l ) ofth e o r di na ls b (v ), where v ranges B(a).

We migh t also have a bottom clause, 0 E (Wx E A)B(x) fo r in -stance, bu t we obtain 0 by taking one s e t in B(x) s e t (x EA ) t o beth e empty s e t : if aO E A and B(a o) : . No ' then RO(y) E (Wx E A)B(x)(y e: B(aO so that sup(aO ,(Ay)RO(y E. (Wx e A)B(x) i s a . bo t t om e l -ement .

- 83 _-

We can giv e pictures :

- -

I f we wr i t e f (c) == T(c , (x ,y ,z )d(x ,y ,z , then, when c yields


47/52

is in C' , then we can buil d th e successor oc ' :

(' ) i f

a ' E C;

is a sequence of o rd in a ls i n- 0 , th e-n we can bu:

,( 2 ) if

So o wi l l be inductively defined by th e t hr ee r ul es

d(x ,y ,z ) " C(sup(x,ye B(a) --.. (Wx E: A)B(x)(x E A, y -" B(x) - (Wx "A)B(x) , z e (Ilv E: B(xC(Ap(y ,v)

= d(a,b ,(Av)T(Ap(b ,v ) , (x ,y ,z)d(x ,y , z ) E C(sup(a,b

a e A

Wequa l i ty

" se tCantor gene rated th e second number class from th e i n i t ia l ordinal 0by applying t he fol l owi n g two principles :

" - fo rmat io n

( 1) given 01 E Cl , fo rm th e successor 0( ' e: CJ ;(2) g i ven a sequence of ord inals 0(0' 0(, ,0(2 ' . . . i n 0 , form th el east ord ina l i n Cl grea te r than each element of th e sequence .

i s co r rec t .Example ( the f i r s t number c lass) . Havt"ng access to th e Woper -

a t ion and a family of sets B(x) (x e N2) su ch that B(02) = NO andB('2) = N" we may define th e f i r s t number class as (Wx E N2)B(X)ins tead of taking i t as primitive .Example ( the second number class) . W give here th e rules fo r a

simple set o f o rd in al s , namely th e s e t () of a l l ordinals of th e sec-ond number class , an d show how they a r e o bt ai

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