# Download e-book for kindle: Axiomatic Set Theory: Theory Impredicative Theories of by Leopoldo Nachbin (Eds.)

By Leopoldo Nachbin (Eds.)

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Additional info for Axiomatic Set Theory: Theory Impredicative Theories of Classes

Example text

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 } . (Def. 4) Also, the Cartesian product i s r e l a t e d t o t h e generalized union by: AxB = = I f we take have , 7 U X , { A x C x } : xEB} uy{{q} x . B : yEA3 = X in ( i i ) and ( i i i ) we get U {X : u {X : \$1 n {x : \$1 and n {X : 61. We :3 X(qEX A @ ) I { q : W x(6 + y E X I > . = {y @ l= Suppose, now, t h a t 4 has only X f r e e .

I t should s a t i s f y : a ) ( a , b , c ) i s a c l a s s , i f a, b, c € V b) a, b, c E Y + ( a , b , c ) Y E c ) ( a , b , c ) = ( d , e , 6 ) t--) a = dA b = e A c = 6. Show t h a t , ( i ) EIO,a), {1,61, I2,cI) and { E d , Ea,bl, {a,b,c)l do n o t s a t i s f y ( a ) - ( c ) , and ( i i ) EC O,a), ( l,b), ( 2 , ~ 1) and ( ( a , b ) , c), s a t i s f y ( a ) - ( c ) . 5. Otion 0 6 Rhe &emem2 0 6 a g i v e n A, and t h e powen Ceans 0 6 a CAUA. 1 D E F I N I T I O N (OPERATIONS). (i) u A = {x : 3 y ( x E y € A ) l .

3 = {0,1,21 ( S P E C I A L CLASSES). 1,2,3 1= IO), 2 = IO, 1) , E V . 2 We need an o b j e c t ( i . e . a c l a s s ) r e p r e s e n t i n g an o r d e r e d p a i r (a,b ) o f a r b i t r a r y s e t s a, b. That i s , a c l a s s ( a , b ) w i t h t h e f o l l o w i n g properties. (1) ( 4 6 ) (2) (a,b ) = (3) - i s a class, whenenver (c,d) a, 6 E Y + ( a , b ) a, 6 a = c A b = d. E V . E V. ROLAND0 C H U A Q U I 24 The f i r s t d e f i n i t i o n chronologically was given by Hausdorff in 1912 : {{O,al, {1,6}1.