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E. intuition- are decidable. 123). For " 'a proves it is decidable structure' free statements, of the connectives), In general we deal with mathematical objects without N(x) for for a notion N(x) for all x' . N(x) and 24 In f o r m a l way systems proofs (note that formal ed m e a n i n g valid). proofs of i n f e r e n c e s , In formal ments in general. connectives, can be e m p l o y e d the a b o v e p r i n c i p l e check whether the p r i n c i p l e holds not 'a proves only In our d e s c r i p t i o n below, in a s p e c i f i c do not n e c e s s a r i l y they systems can even m e c h a n i c a l l y Moreover, are r e p r e s e n t e d the g e n e r a l render the because N(x) are holds, one for all x' but of the m e a n i n g holds intend- they evidently for notions principle symbolic for state- of the once logical it holds for atoms.

Ce(~)_ = x := 3 y ( e ~ y = ×+1). 48 R(~,x). if e a proof number. is a c o n s e q u e n c e Theorem a K. this number n) e * represents in K thus it b e l o n g s m * of the of K 49 Notation: when functionals The convenient we w i l l ~e is d e n o t e d inductive definition by write e(6) for ¢e(<). The class of K*. 5. Lemma. (i) l < ' x E K* (ii) let for then e of type Vx(e (00)0 C K*) ~ e E e x be defined by e x = l ~ ' e ( ~ * ~), K* X (iii) One now K* easily is the least class satisfying (i) a n d (ii).

So it b e l o n g s properties. Vn (Vx D(n • x) ~ Dn) Vn (Vx P ( I m . a ( n • x • m)) ~ P ( I m . a ( n • m)) 53 54 C o n s i d e r b = I m . m). If b0 ~ 0, t h e n h is a c o n s t a n t positive the o t h e r hand, ( P ( l m . b ( x , m)) ~ Pb) holds the c l o s u r e b0 = 0, t h e n Vx properties. N o w BI allows T h e r e f o r e Hyp us to c o n c l u d e function so Pb holds. If, on because of 4 holds. e. P(a). H e n c e we h a v e s h o w n Ka ~ K0a. N o t e t h a t the fact that e r a n g e d o v e r lawless in the p r o o f 701, (of.

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