Peit8Cb. f. Bd. 1, 9.

math. Lopik und 89-90 (1956)

Orundlagen a. Math.,

A GODEL T H E O R E M F O R AN I N F I N I T E - V A L U E D ERWE ITERTE R AUS SA G E N K A L K U L

By ALAN ROSE in Nottingham, England

It will be shown, by methods similar to those used by GODEL for elementary number-theory, that if the usual rules of procedure are used no plausible and complete formalization of a certain infinite-valued Erweiterter Aussagenkalkul exists. The truth-values are all real numbers satisfying 0 5 x 5 1 and 1 is the designated truth-value. The primitives are the LTJKASIEWICZ implication f unc- tion C , a ternary function G and the universal quantifier 17 . If when p , , . . . , pn take the truth-values x l , . . . , x, respectively the g. 1. b. of the truth-value of @(pl, . . . , pn) as xi varies is p(z$, . . . , x ~ . - . ~ , xi+1, . . . , x,J then this is the truth-value of 17 pi @(pl, . . . , pa) (i = 1 , 2 , . . . , n) . If G p q r takes the truth- value g ( x , y , z ) when p , q , r take the truth-values x , y , z respectively then

g(1, 1 , a - w ) = 1 (w = 0 ) 1 , . . .), g( l ,O, 2) = 1 (2 =I- 1) , g(2-9, 2-w, 2-u.w.) = 1 (v = 1 , 2 , . . .; w = 0, 1 , 2 , . 1 . ) A * ) )

where the wth sequence in an enumeration of all finite sequences of non-negative integers is u,o, u , ~ , . . . , uvA,. In all other cases g ( x , y , z ) = 0 . Thus G bears a considerable similarity to GODEL'S p-function.

Let the truth-values of the formulae H p , Dp, Epq be h ( 5 ) , d ( x ) , e ( 2 , y ) respecti- vely where x , y are the truth-values of p , q respectively and h ( x ) = z, d(1) = 0, d ( x ) = 1 ( x =I= 1) ) e ( x , x ) = 1 , e ( z , y) = 0 ( x ?= y ) . Then we can define H , D , E . the logical constants 0, 1 and the LUKASIEWICZ functions N , A , K , 2 by

A P Q =df C C P Q Q , 0 =df.JI ~ p , 1 = d f coo, N P = d f CPO, KPQ = d f N A N P N Q , DPdf = G l O P , H P = , f n p A N C P p p (where p does not occur in P ) ,

EPQ =df K D D C P Q D D C Q P , Z p P = d f N n p N P .

We first show that if p ( x l , . . . , xn) is a primitive recursive function then there exists a formula Y ( p l , . . . , p,,, q) such that if Y ( p l , . . . , p9&, q ) takes the truth- value y (a l , . . . , an, /3) when p l , . . . , p a , q take the truth-values al, . . . , an, p respectively then y(2-"1, . . . , 2-%, F") = 1 if w = p ( x l , . . . , xrL) and in all other cases y ( q , . . . , afi, p ) = 0.

Case Case 11. If p ( x l , . . . , xa) =k, then Y (pl,. . . , p f i , q ) is KnEq H' 1G 11 pl . . . G 11 pn. Case 111. If p ( x l ,... , x J = x d , thenY(pl ,..., p a , q ) is, KnEqpiG1lp ,... Gllp,.

I. If p ( x ) = x', then Y ( p , q ) is KEH'pqGllp.

7 Ztschr. f. hiath. Logik

ALAN ROSE 90

Case IV. If q~ (xl , . . . , qJ=[ (xl(xl,. . . , x,), . . . , xnL(x1, . . . , xn)) and ?Pi are the (n + 1) th degree formulae corresponding to the respective functions xi (i= 1,. . . , m) arid SZ is the (m + 1)th degree formula corresponding to the function c then P(P1,. . ., P a , n) is

2 91 * . 2 qn&" Y1(~1, * * * , pn, 41) * - - u l , ( ~ 1 , . . - 9 Pn, 9,) Q(q1, . - , ~ t n , 4). CaseV. If

~ ( 0 , 22, * * * , z,)=T (22, * * 9 x,), P; (y', ~ 2 , * * 9 3 xn)=x(y, ~ ( y , 5 2 , * xn) x2, * * * ,'x,) and Q ( p 2 , . . . , p n , q ) , A ( p l , . . . , P , + ~ , q ) are the formulae corresponding to

2 r K D r K K 2 s KG r1sQ (p2 , . . . , pa, s ) 17 t C KG 11 t D D C p1 H t Z s Z u KKG rH t s G r t u

It then follows as in the proof of GODEL'S theorem that if P ( x l , . . . , x,J is a primitive recursive predicate then there exists a formula Y ( p l , . . . , pn) of our system which takes the truth-value 1 when pl, . . . , p , take the truth-values 2--"1, . . . , 2-'n respectively if P ( x l , . . . , xn) is true and which takes the truth- value 0 otherwise. It also follows, as in GODEL'S proof, that we can attach GODEL- numbers to formulae and proofs of any formalization of our system which uses t h e usual rules of procedure, in such a way that the predicate ,,a i s the GODEL- number of a formula Yu(p) and b i s the Gtimbnurnber of a proof of the formula P,(P 1)" is primitive recursive. Let the formula of our system corresponding to this predicate be !P(p , q ) . Let the GODEL-number of the formula n q D Y ( p , q ) be x.

Let us now consider the formula 1 7 q D Y ( H " 1 , q ) , i. e. Y"(H" 1). Now Y ( H 5 1 , q ) takes the truth-value 1 if and only if q takes the truth-value 2-v where y is the GaDEL-number of a proof of the formula Y,(Hx 1). Hence h'qDu/(H" 1, q ) takes the truth-value 1 if and only if YT(H" 1) is unprovable, i. e. Y z ( H S 1) takes the truth-value 1 if and only if Yx(H5 1) is unprovable. If u/ , (Hx 1) were provable then Y z ( H " 1) would not take the truth-value 1, contrary to our assumption that the formalization is plausible. Hence Y % ( H " 1) is unprovable and takes the truth-value 1. Thus the theorem is proved.

the functions 5, x respectively then Y ( p l , . . . > P n ? 4) is

A @ , u , P ~ , . . . , P,,, 8) G r p l q .

(Eingegangen am 1. September 1954)