By R.B. Jensen, A. Prestel
Read or Download Set Theory and Model Theory: Proceedings of an Informal Symposium Held at Bonn, June 1-3, 1979 PDF
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Extra info for Set Theory and Model Theory: Proceedings of an Informal Symposium Held at Bonn, June 1-3, 1979
M~ 1 $ < y ~ y' < y l ..... Yn (i) we I. N o w see that set ~yy, embeddings n (7 ' y , 'Yl ' " " "'Yn ) ' yj 6 I 6 (~Z and is O Z - d e f i n a b l e y7' = nE~ ~_] ~ YY' n U s i n g (2) we get in n+3 . y It is c l e a r that J for on y in < ~ y , Uy > is o b v i o u s . and ~yy, (y) = y'. N o w d e f i n e ~ y y 6 ~yT, (X) = is n o r m a l and < ~ y Normality ~ y = id yy' X 6 U7 (4) Uy f~ ~,, , Uy > is a m e n a b l e To p r o v e y < y' . amenability set U n = U • T It s u f f i c e s to sh o w t h a t U n 6 ~ But this T Y" O Z - d e f i n a b l e from ~yy, and U nY 6 0 ~ y ÷ .
Qv "v be defined as in § I. For is a cardinal"~. v~ S+-Card We note that ~(v) can also be found as follows. +2 and of v* guaran- . f : ~(V) ----* ~(v). t. f* =L~. ---*2 Lv* rng f*. An easy argument d e t e r m i n e d by f : v ===~ v f~ shows and f*(q~) = qv" that H~v (E I) Let f = ~==> f(~(~,i)) = ~ v, v. if Then = ~(~f(i)) f* is u n i q u e l y So, by slight abuse of notation, V,v. we do not d i s t i n g u i s h Note that So ~v. f and We also set We clearly have f(k~) = k v for f*. i(_k~v . and given f(~*)=v*.
Gives the r e l a t i o n on the right side. qued. Note that we could also treat finite (. be the l e x i c o g r a p h i c a l o r d e r i n g on k~s ~2. above. Finally, For g=~ let the f o l l o w i n g result was p r o v e d b y D e v l i n in . P r o p o s i t i o n 5: ir rich. Let Let M = F~(~) [gB]Bg F~. t. ~, ]X[ F regular, there is some by (. t. ~. t. IXI = K + and X ls wellordered ~,). This follows immediately from the proof of Proposition 3. qued. References [I] KoJ. Devlin, Aspects of constructibllity, Springer Lecture Notes in Mathematics 354 (1973)  KoJ.