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

By Leopoldo Nachbin (Eds.)

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R i s an 3 z(xRz A z R y ) ) . e. i f R = R - l and 2 R C R. There a r e s e v e r a l o t h e r ways o f s a y i n g t h a t R i s an equivalence relation: (i)says t h a t i f R i s symmetric and t r a n s i t i v e , then i t i s r e f l e x i v e . N o t i c e t h a t t h i s i s n o t t r u e f o r r e f l e x i v e i n A. The usual n o t i o n o f e q u i v a l e n c e r e l a t i o n i s eqLLiuaLence treeation i n A ( i . e . symmetric, t r a n s i t i v e , and r e f l e x i v e i n A ) .

Suppose t h a t A x B = C x D A A f 0 A 8 f 0. 4) aEC and b E D . T h i s shows t h a t (l)c+o A D#O Suppose now, t h a t x E A ; t h e n ( x , b ) -C. therefore, A C E Thus x e C AxB = CxD. S We need, besides t h e ordered p a i r o f two s e t s u , b y t h e concept o f an that ordered p a i r o f two a r b i t r a r y c l a s s e s A, B. Namely an o b j e c t [ A , B ] satisfies: (1') [A, 81 i s a c l a s s , f o r any A, 8 , ( 2 ' ) [ A , 81 = [ C , D]-A = C A B = D . 5 I n o r d e r t o l i f t these l a s t l i m i t a t i o n s , t h e f o l l o w i n g d e f i n i t i o n i s i n t r o duced.

S i m i l a r l y ( 2 ) i s a l s o a unary operation assigning t o each x the c l a s s o f p a i r s { x , q l w i t h + y. ( 3 ) i s the special c l a s s formed by a l l p a i r s Cx,yl with x f q. ( 4 ) i s a binary operation t h a t f o r x , q with x f y gives { { x , ~ } } a n d f o r x,y with x = q gives 0. , I s h a l l omit t h e variables Xo Xn-,, when t h e s e a r e a l l t h e f r e e variable i n 7 . Thus, { { x , q l : x # y l stands f o r {{x,ql : x f y l . XY With t h i s newly introduced notation we can reformulate t h e d e f i n i t i o n of t h e Cartesian product: A X B = { z :3 x 3 q ( z = ( x , y ) A x E A A x E 8 ) ) = {( x , y ) : x E A A y E B } .