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

By Leopoldo Nachbin (Eds.)

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R A % # S)*A 0 + 2 (R*A) (S*A). S)*A . ( R o S)*A = R* S*A - . - D (R o S) = S'l* D R A D ( R o S ) - l = R* D S. R CS The proof is easy. ). F i n a l l y , some o f t h e p r e v i o u s r e s u l t s a r e extended t o g e n e r a l i z e d Boolean o p e r a t i o n s . The p r o o f is l e f t t o t h e reader. 12 (iii) THEOREM SCHEMA, (nx 0"' Xn-1 Let 7 39 be a term and 4 a formula. CT : @ } ) * A C- r-X0". Xn-1 Then IT* A : $1. PROBLEMS ( x ) and ( x i ) , from ( i ) . 1. 10 2. 12. 3. Show t h a t : ( i ) R o ( S n T ) = ( R o S ) n ( R o T ) i s not t r u e i n general, ( i i ) (R = 0 V S = 0 ) - R o ( V x V ) o S = 0, ( i i i ) R n S n T C- R o S - ' o T .

Since W F (R), t h e r e i s a y E (DRu D R - ' ) we have t h a t Then ( D R % U A such t h a t A X I O M A T I C SET THEORY O R ( y ) n ( ( D R u DR-l ) Therefore A) Ax Reg THEOREM, PROOF, B u t OR(y) c - R-'*{yI 0. 23 1 , . 22, v= 5DR. Hence O R ( y ) cA. A. v ). we have W F ( E L ) A EL C D Y. 21, Assume Ax Reg. 49 we g e t , s i n c e . V = But V > A . 22 i s a c t u a l l y e q u i v a l e n t t o W F ( R ) . 23. -. + F i n a l l y , the important well-ordering r e l a t i o n s a r e introduced.

Similarly, A A is the greatest lower bound (glb). L u b X ( z , A ) means that z is the least upper bound of A. Thus P we can express that this least upper bound exists by LubR(J)A,A). Similarly for greatest lower bounds. If P O ( R ) , then the existence of these bounds implies their uniqueness. 16 DEF IN IT ION (i)U L O ( R ) (if) LLO(R) + + + + I PO(R) AWxWy(x,g E DR+ 3 z ( x , y R z AVu(x,gRu+zRu))). PO(R) AWxWq(x,qEDR + 3 z ( z R x , y A b u ( u R x , q + u R z ) ) . AXIOMATIC S E T T H E O R Y - ( i i i ) LO(R) ( i v ) CULO(R) CLLO(R) (v) (vi ) C L O (R) - 47 ULO(R) A LLO(R).

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