## Axiomatic Set Theory: Theory Impredicative Theories of by Leopoldo Nachbin (Eds.)

By Leopoldo Nachbin (Eds.)

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IS A SUBTHEORY OF B . G I n t h i s s e c t i o n , we assume B as o u r t h e o r y and deduce t h e axioms o f G from t h o s e o f B. Since Ax Class and Ax E x t a r e common t o b o t h t h e o r i e s , i t i s enough t o show Ax Em and Ax Num as theorems o f B. A l l theorems i n t h i s s e c t i o n a r e theorems o f B. 3 depend o n l y on Ax Class and Ax Ext. 1 a l s o be used f o r B. 4 @' f o r formulas $ w r i t t e n i n t h e p r i m i t i v e n o t a t i o n . I s h a l l l a t e r extend t h i s r e l a t i v i z a t i o n t o d e f i n e d concepts b u t we s h a l l n o t need i t i n t h i s s e c t i o n .

2 . 2 i s more general, because i t does n o t r e s t r i c t t h e domain t o those X w i t h 53 AXIOMATIC S E T T H E O R Y But, when p o s s i b l e , t h e p r e s e n t r e p r e s e n t a t i o n by f u n c t i o n s i s E V. more convenient. F(x) The f o l l o w i n g d e f i n i t i o n f o r m a l i z e s t h e i n t r o d u c t i o n o f such f u n c t i o n s . 6 D E F I N I T I O N SCHEMA, ( 7 X = Let T :q5) = ( ( 7 , x ) : be a t e r m and q5 a formula. Then GI. For i n s t a n c e , we have when F i s a unary o p e r a t i o n , { ( F ( x ) , x ) : F(x) E V ) .

THEOREM SCHEMA, Let Suppose Hence x r z . R n '0 0 . xRq A yRz. Therefore, le,t r be a tehm and 4 a jjotvnuh. Then, { r : @ I and assume t h e hypotheses o f t h e Xn-1 Then f o r a l l Xo,... Xnml, such t h a t 4 , x r q A x R z. 43 AXIOMATIC SET THEORY The c o m p o s i t i o n o f two t r a n s i t i v e r e l a t i o n s i s n o t , i n genera1,trans i t i v e . 3. PROOF, (RoS) THEOREM, Assume R R2 -R C A S2 C S A R - 2 C _ R A S2 C -S A R o S oS = S OR = SoR. e. + (R oS)2 C - R oS. Then, 2 = R o ( S o R ) o S = ( R o R ) o ( S o S ) = R 2 O S2 R i s den6e, i f R C R -+ 5 R o S .