By Saharon Shelah

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T h e s u b s e t of ~ w h i c h a r e i n b u t n o t i n V, b e c o u n t e d when the members V[G] of P , b e i n g in V, c a n c o u n t o n l y s u b s e t s of ~ w h i c h a r e i n V? H e r e we s h a l l m a k e s u r e t h a t V[G] c o n t a i n s c) Is t~1 of V, w h i c h we h a v e m a p p e d n o n e w s u b s e t s of ~. o n t h e s e t of a l l s u b s e t s of G0 i n V[G] a l s o t h e # l of V [ G ] ? H e r e t h e a n s w e r is e a s i l y p o s i t i v e b e c a u s e V and V[G] h a v e the same sets of n a t u r a l n u m b e r s .

Ii) f o r e v e r y Q - n a m e a n d o r d i n a l a t h e r e is a f u n c t i o n H with d o m a i n a, ( H e V) s u c h t h a t [~-Q" if v is a f u n c t i o n f r o m a t o V t h e n 1-(~8) e H(fl) f o r e v e r y ~ < a ' ; a n d [ H ( ~ ) [ < ~ f o r ~ < a. C. 20 R e m a r k : O n c e we p r o v e t h i s l e m m a we k n o w t h a t all t h e c a r d i n a l s i n V here are cardinals also in V[G] a n d t h e r e f o r e k is a c a r d i n a l a l s o i n V[G], a n d if ~ is t h e a - t h i n f i n i t e c a r d i n a l ~a i n V i t is a l s o t h e a - t h i n f i n i t e c a r d i n a l i n v[ a].

9. D is obvio u s l y u p w a r d closed, we shall n o w s h o w t h a t D is d e n s e . 9), l e t p --~ r , q . Then, b y t h e d e f i n i t i o n of ~ , p c ~) a n d we h a v e p r o v e d t h e d e n s i t y of ~). S i n c e ~ is dense G C~D#r q

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