Checking Proofs in the Metamathematics of First Order Logic

1 ) I t i s the most t r a d i t i o n a l , i . e . luetanuith ema t i cs , as i t appears in l o g i c books, i s u s u a l l y s t a t e d i n terms o f s t r i n g s . 2) Axioms in terras of a b s t r a c t syn tax are s imply theorems o f the theory expressed in terms of s t r i n g s . Thus the two r e p r e s e n t a t i o n s look sub s t a n t i a l l y the same w i t h respec t t o " h i g h l e v e l " theorems. 3) I l l f o r m e d fo rmulas can be ment ioned. T h i s is o f course imposs ib le in an a x i o m a t i z a t i o n in terms o f the a b s t r a c t syn tax . The p r o p e r t i e s o f w f f s r e l e v a n t to our theory have been d e f i n e d by the p r e d i c a t e s FR, FRN, GEB and SBT, F R ( x , f ) is t r u e i f f the v a r i a b l e x has a t l e a s t one f r e e occur rence i n the wf f f , w h i l e FRN(x ,n , f ) and CEB(x ,n , f ) are r e s p e c t i v e l y t r u e when the v a r i a b l e x occurs f r e e or bound at the p lace n in the fo rmu la f . In a d d i t i o n to these p r e d i c a t e s , some gene ra l i zed s e l e c t o r f u n c t i o n s are d e f i n e d , wh ich eva lua te the f i r s t o r the k t h f r e e occurrence of a v a r i a b l e in a w f f , or the number of i t s f r e e occu r rences . The p r e d i c a t e SBT i s then d e f i n e d . I t ax io tnat izes the n o t i o n o f sub s t i t u t i o n of a term f o r any f r e e occur rence of a v a r i a b l e in a w f f .