By Andrzej Mostowski

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Additional resources for Constructible Sets with Applications

Sample text

This function is again strictly increasing and continuous; its critical numbers are critical for F H , and for G. Let F&) = x > 0 and f E A:'(H). If k A H [ f ] ,then for every a in A we have k A H " { ( j , a>>u f l , 2. If non b A H [ f ] then , there is an element a in A such that non F A H ' [ { ( i , a ) } uf]. Hence we can find an a of this kind already in A Q ( f )Since . f E A:'(H), there is a E < x such that f E ArcH) and hence eCT) < 4 5 ) < G ( t ) < &(5) < FH(X)= x . Thus there is an a in A, such that non k A H ' [ { ( i ,a>}ufl, whence by the inductive assumption we obtain non b A x H ' [ { ( i ,a ) } ufl and non EA,HLfl.

3 are determined by 5. Thus if B is given, then each sequence a determines uniquely a sequence of constructible sets. Terms of a are among the constructible sets and they appear there in the same order in which they appeared in a : the set a, occupies the place C,(a) where E is the ath ordinal such that ZE = 10. The whole sequence a is not necessarily a set constructible in a. 3. Properties of constructible sets From now on we shall assume that (1) < lh(a). Thus elements of a, are some a,, with T,I < 5 and in particular a, a, c a15 for each 5 = 0.

H[f]}, SH(a,f) = {xE a : ((0, x>>ufE D H ( a ) } . We call &(a) the diagram of H in a. This set consists of all finite sequences with terms belonging to a which satisfy H in a. The set S H ( a , f )is called a section of &(a) determined byf; f is here a sequence whose domain is Fr(H)--0). We can describe the section as follows: If 0 4 Fr(H), then the section SH(a,f ) is void. e. satisfy the equations g j =fj f o r j # 0). The set of all go's is the section S,(a, f). We call a class A predicatively closed if conditions a E A ,f E am(")imply SH(a,f) E A for all formulae H.