Cabal Seminar 77 – 79: Proceedings, Caltech-UCLA Logic by Claude Dellacherie (auth.), Alexander S. Kechris, Donald A.

By Claude Dellacherie (auth.), Alexander S. Kechris, Donald A. Martin, Yiannis N. Moschovakis (eds.)

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Extra info for Cabal Seminar 77 – 79: Proceedings, Caltech-UCLA Logic Seminar 1977 – 79

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1 that ~ A * $4(~ ) is not wellfounded . 2. each S4 is a tree on Scales for ~ g A, $4(~ ) ~ : ~ ~ ¢, l i.

J SI(~ ) Indeed, let for which is by g s [J]: h = g V 3 i [ h ( i ) < g(i) A ~/j < i(h(j) = g(j))] . But only special trees those of the form h ~ [J], have honest leftmost branches. We shall show that are among them. ~ g A: %(i) = rank< (i). Clearly L g [Sl(C~)]" On t h e other hand if g e [Sl(C~)] , then g : (,, ~ w 1 is such that i < C{ j ~ g(i) < g(j) , thus L(i) = rank< (i) ~ g(i) , and we are done. (b) Define now for ~ ~ A: h(~) = %(i) . e. A (and thus an Wl-scale ). and assuming that for all ~i(~n) = g(i), = lim(~n,~(~n) ) e [Sl] n~ ~ n for all large enough and moreover L(i) Indeed, letting s A n) while ~ n - we conclude = ~i(~) ~ g(i).

The main step in the proof that J is a caliber is a generalization of the classical Souslin theorem on uncountable analytic sets. We state it as a lemma. (1)This is not the classical definition of a semipolar set in potential theory. However it is equivalent although this is difficult to prove. (2)The class of semipolar sets is a ~-ideal of IM-thin sets. 1. (a) C (b) if Let C be a map from A is analytic and compact sets K into [0, i] such that is nondecreasing and going up. Then if map ~(E) A KO, ~ C(A) = 1, such that is analytic and from the Cantor space C(A N K i ) = C(A) = i, [0, i] N into the U G K(~) is compact and contained in (3) for any G 8(HI, H2,...

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