6 1. ELEMENTAR Y EMBEDDING S For a give n filte r F o n a se t X , w e sa y tha t a se t A C X i s F-positive i f A intersect s eac h membe r o f F. I f I i s a n idea l o n a se t X , I + denote s th e set o f F-positiv e subset s o f X , wher e F i s th e filte r dua l t o I. Equivalently , I+=V{X)\I. 1.1.11 Definition . A filte r F o n a n ordina l K, is normal i f i t i s close d under diagona l intersections , i.e. , i f fo r al l (A a : a K) C F , th e diagona l intersection A{Aa :aK) = {(3K\[3e f] A a } a(3 is als o i n F. Equivalently, F i s norma l i f eac h regressiv e functio n whos e domai n i s F-positive i s constan t o n a n F-positiv e set . A n idea l i s sai d t o b e normal i f its correspondin g filte r i s normal . LEMMA 1.1.12 . Let j : V * M be a nontrivial elementary embedding with critical point K and let U {^ 4 C K \ K G j(A)}. Then U is a nonprincipal normal ultrafilter on K. P R O O F . Th e proo f o f Lemma 1.1. 9 show s tha t U i s a nonprincipa l ultra - filter. T o show that U i s normal, fi x / : n n suc h tha t {a | f{a) a} G U. Then * £ j(ia I f( a ) &}), s o J(f)( K ) K- Le t (3 = j(f)(K) an d le t A = {a K \ f(a) = /?} . Then j{A) = {a J(K) | j(f)(a) = j(f3)}. Sinc e j(0) = 0, K G j(A), s o AeU. It follow s immediatel y fro m th e definitio n o f normalit y tha t i f U i s a nonprincipal norma l ultrafilte r o n a n uncountabl e cardina l K then U i s K- complete, an d s o i n particular , b y Lemm a 1.1.10 , V K /U i s wellfounded . LEMMA 1.1.13 . If U is a nonprincipal normal ultrafilter on K and i is the identity function on n 7 then [i}u K. P R O O F . Fo r eac h a ft, le t f a b e th e constan t functio n fro m K t o {a}. Then fo r eac h pai r a (3 below K, {7 « I fad) 7/3(7) } = « e U, and fo r eac h a K, {7 * I fM i(7) } = « \ ( a + 1 ) G tf. This show s tha t [i]f / K. NO W i f # i s a functio n wit h domai n n an d {7 A C I 5(7) i(7) } G U, then b y th e normalit y o f U ther e i s a n a K such tha t {7 « I 5(7) = /a(7) } ^ which show s tha t [zj^ y = ft.
Previous Page Next Page