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Note that there are many A's which are both subtle and ~ 21 irides- W. Boos cribable invisible 37 below the least almost ineffable ~. 17 If A is a set of ordinals, Definition. 18 results ( Kunen, ite regular ineffable which about [A] ~ is the set of in- of A. iff f ( ~ ) < x ~ for all ~ in [A~ n for f iff f ~ [ B I n The ineffability-case earliest characterizations and some problems. is constant. of the following, due to Kunen, was among the ineffability. Baumgartner cardinal ). If A is an unbounded K , A is (i) subtle /(2) iff each regressive is (i) of order-type f: [ A ] 2 ~ 3/(2) 9 K subset almost of an infin- ineffable has a homogeneous unbounded in ~ / (~) set X ~ A / (3) stationary in K .

Since any element of the subtle filter on K is stationary, invisible in R(~) is ~-Mahlo for all ~ another way to see that a subtle K Not only are there many invisible given a closed unbounded C ~ many invisible X ~ C well. A (A~L below a subtle ~C) ~ , there are, , with A which are homogeneous K. ~ for ~ for ~EC,

Km] <~ 0 otherwise. language of p o w e r < has a homogeneous > 2 1 , let f~(X) = 1 if An H homogeneous ~f(X)] (~) = for each f~ is homogeneous for f. Proof of (2). Define f: [ K ~ D < ~ the n-type of X = homogeneous { ~ I ~ > S ( the language ~[X] for f is homogeneous Proof of (3). If not, no homogeneous set of type ~. } for n > ~ for ~ for each of ~ An H . K~ let fu : [ y 3 < ~ Y< ) by f(x) = and X ~ E X ] m o Define g: [ K ~ < ~ > 2 have > 2 by g( < x ~ ..... Xn> ) = fxm ( < x 0 . . . Xn_l> ) for n ~ i a n d ~ in [ K ~ ] n+l If H has type ~ and is homogeneous for g, H g ~ y is homogeneous for fy for each W E H.

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10th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics by Alt R., Vignes J.

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