By R. Chuaqui

ISBN-10: 0080871623

ISBN-13: 9780080871622

ISBN-10: 0444861785

ISBN-13: 9780444861788

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**Extra info for Axiomatic Set Theory: Impredicative Theories of Classes**

**Example text**

Xi, = F*A f o r some A . e. ( x i ) and f x i i ) a r e ' l e f t t o t h e r e a d e r . We a l s o have d i s t r i b u t i v i t y o f i n v e r s e image o f f u n c t i o n s w i t h generalized intersection. 9 THEOREM SCHEMA, L& r be a t m and 6 a 6o/un&. On t h e o t h e r hand, i f x $ implies that f o r a l l such r . e. y Therefore FIX E DEFINITION BA i s t h e & a n 06 F'x = :$I, t h e n f o r a l l Xo... f o r some y nx 0.. x n-1 :$I. e. Xn-l, and hence F'xE r x c ~ - l *n '0' * 'Xn-1 I 6uncLionn w L t h domain 8 and m n g e included i n A.

So suppose C E V . 3, {B,C} E V and, hence, t h e r e i s a t r a n s i t i v e s e t u such t h a t { B , C } E P . We have, B , C E u , so { C } C u. S i n c e - u . Using, now, Ax Sub. we o b t a i n u i s t r a n s i t i v e , B E u. Hence, B u { C ) c BU{C} E Y . 2, PROOF,: 06 B . 4. PROBLEMS 1. Prove i n B (a) A U B , E V - A E V A €3 E V . ( b ) A E V V B E V + A n B E V . (c)A+O 2. A B+O-+(AxBEV++AEVA - B E V ) . Prove t h a t i f we e l i m i n a t e t h e r e s t r i c t i o n which does n o t c o n t a i n A f r e e i n Ax Class, G and, hence, B , become i n c o n s i s t e n t .

Iv) IRWAF(A) =R*A. I ( i ) C l e a r l y i m p l i e s ( i i ) , and ( i i ) i m p l i e s ( i i i ) , because CCyl : y E A l PROOF A = U . The i m p l i c a t i o n o f ( i i i ) t o ( i v ) i s proved, as f o l l o w s : F(A) = u {F(Iy}) : Y E A ) f o r a l l A. R = Suppose D e f i n e R, by [F({x}) : xEV]. 1 and Def. 2, R*CxI = F ( I x 1 ) f o r a l l x E V . 12 ( i v ) , R*A = u {R*{xI : X E A I = u { F ( C x I ) : x E A 1 = F ( A ) . 12 : $ I ) = R*(U{P : $1) = UER* 7 : $j = UCF(r) : $1. e.

### Axiomatic Set Theory: Impredicative Theories of Classes by R. Chuaqui

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