By Leopoldo Nachbin (Eds.)

ISBN-10: 0444861785

ISBN-13: 9780444861788

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

**Example text**

T h u s , we have: - R , R-l, R o S , R o , R1, - R = - R2, ,... C- Vx V ; R-l= (RnVxV)-', ( R n V x V ) , ROS=(R R3 n v x V ) o . ( s n V x V ) . e. these v a r i a b l e s a r e assumed r e l a t i v i z e d t o subclasses o f Y x V . 4 EXAMPLES, The following a r e examples of a p p l i c a t i o n s of these operations: D V IN = - I D , I D = I N ~ I N - ~I ,N ~ E L= E L , = ( - ( E L ~ ( - E L ) - ~ ) ) D- V~ O, D (AxB)'~ = BxA , V (AxB) o (CxU) = 0 ( A x B ) o ( C x U ) = Ax17 if Cn B + O = v x v , if .

Viii) u E d = a = n { a ) . ( i x ) U {a,b) = a u b . 27 AXIOMATIC SET T H E O R Y PROOF, I s h a l l give a proof of ( v ) and ( v i ) , leaving t h e r e s t of t h e theorem t o t h e reader. ( v ) : I f x E U P A , then x E q c A f o r some q , t h e r e f o r e , x E A ; and hence U P A 2 A . On the other hand, l e t x E A ; then { x ) E P A by Def. Hence X E { x } € P A and X E u P A ; i . e . A L u P A . ( v i ) : Let x E A ; then x C - U A. Since x E V , x E P U A We s h a l l c a l l a c l a s s A LttrransiLLve i f U A Z A .

Xo ... 'n- 1 IT Then : $1 = I x : tf T xo be a term, 0 a formula, a n d x , ... w xn-l (4 -+ x E 7)) . In general, i t i s necessary t o i n d i c a t e the v a r i a b l e s which a r e bound by t h e operator. Thus, t h e term I { x , q l : x + q ) without p u t t i n g which varia b l e s a r e bound could be any o f t h e following terms: (1) , I { x , y ) : x + q) = I z : 3 x ( z = Cx,ql A x + q)} . 28 ROLAND0 CHUAQUI These four terms a r e d i f f e r e n t . (1) defines a unary operation t h a t assigns t o each given y the c l a s s of p a i r s { x , y l w i t h x # y .

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

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