By Masazumi Hanazawa (auth.), Gert H. Müller, Gaisi Takeuti, Tosiyuki Tugué (eds.)

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**Additional info for Logic Symposia Hakone 1979, 1980: Proceedings of Conferences Held in Hakone, Japan March 21–24, 1979 and February 4–7, 1980**

**Example text**

F contains (ii) for a s t r u c t u r e F is c l o s e d (iii) under F is n o r m e d , m, we properties (i)-(v) F is a : R 1 and t h e i r n e g a t i o n s . A, V , 3 S, V S . , say a 2 - c l a s s is in F, t h e r e is a f u n c t i o n o: QC{(x,P) IQ(x,P)})---+ 0 such that c o r r e s p o n d i n g r e l a n t i o n s (x,P) < (x',P')and(x,P) < (x',P') are in F, w h e r e < o" o --o and --o < defined and by (x,P) ~ Q A[(x',P') ~ Q ---+ ~(x,P)< ~(x',P')] (x,P) E Q A,[(x',P') ~ Q ---+ ~(x,P)< s(x',P')] respectively.

Sem. , Vol. 6 (1978), 269-321. [4] Y. Kakuda, On a condition for Cohen extensions which preserve precipitous ideals, Jour. Symbolic Logic, to apDear. [5] K. Kunen, Some application of iterated ultrapowers in set theory, Ann. Math. Logic 1 (1970), 179-227. [6] K. Namba, On the closed unbounded ideals of ordinal numbers, Comment. Math. Univ. St. Pauli, XXII-2 (1973), 33-56. [7] R, M. Solovay, Editor), Real-Valued measurable cardinals, Axiomatic set theory (D. Scott Proceedings of Symposia in Pure Mathematics, Vol.

4. (i) In the topos of G-sets, a terminal object is given by ({*},~(*,*)=~). Let a:l--YV a. We define j(a)EV~ G) by j(a)(x) = yc~V(2) [x=y]Aa(*,y), dom(J(a)) : V (G). (ii) Let a I ..... aner(l,Un). Let JJr I ..... an)Jj be the following open object: V{y) t >i: U n ~y r Prop. 4. (i) Let al, .... anEr(l,Un). Then ..... ant)}. an)Ii = |r ..... J(an))~. (ii) For any u(V (~), there exists acr(1,U n) such that |j(a)=u~ = 1. Hence the a-classes (r(1,Un),ll = If) and (V(~),| = ~) is isomorphic by the ~-class isomorphism given by |j( )= ~.