By Bierutowice, Poland, 1975. Marek, Wiktor. ; Srebrny, Marian, ; Zarach, Andrzej, Conference on Set Theory and Hierarchy Theory

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Additional info for Set Theory and Hierarchy Theory A Memorial Tribute to Andrzej Mostowski: Bierutowice, Poland 1975

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Namba's main theorem is identical with that of [I] (see also [2]). Namba was inte, rested in the independence of (~o,~)-distributive laws in complete Boolean algebras and his theorem is formulated in Boolean terminology. After the important paper of R. Jensen [5], appeared the theorem of [I] and [~] has become interesting since it shows that the Jensen's result is the best possible. That's why I decided to publish the paper [I] **). This paper should be considered as an improved version of [I].

A functional which is (partial) recursive in [2E, 3E, 3E@ ] is called hyper-arithmetic (HA) , hyperanaliti~ (HArt) , hyperprojective (HP)] . The sets of type ~ 2 semirecursive 3 sets nE(~) SR for in F in 2E 2E turn out to be exactly the are also If P . 4, we obtain the following. 6. (a) ~ is type 2 Let P and be a positive H~, then inductive operator: CI(P) is H~ . 58 (b) If F is type 3 and SHAh , then (c) If F is type 3 and SHP , then (d) If in F is type n+3 n+2 E(@) . [] and SR in CI(F) CI(P) is is n+2E(~) SHAh .

Amd b e am increasing continuous sequence of cardinal ~(~, X o) ~ V x B(~,x) (of [4]). 28 Proof. By induction Case I. X = Case 2. k > ~ ~o VT . In order to prove k = v+ . it easily follows sequence : ~ s k~ = ~ ~ < k and for all < ~ ~ > ~ such that B(~,~) that there exists of cardinal numbers ~ E A , T~ > ~ ~ <~(~) ~ ~(~+~ T~ ~ ~ . ~ < ~ continuous <~(~) ~(~) n ~(~)) I , . e. N o w take into T ~ ~(k ; ~ ~ ~> . T h e m < w~(~+~) such functions, ~ if A~ ~ A 2 x v) into ~ Let . ~ C(~,X) = ~ ~ < k , The set ~ be the following =of

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