By Andrzej Mostowski
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2) ZfO 4 Z(r, a) < 9, then Bcf) = A,(r,u,(fx(r,u),fL(r,u)) n fM(r,u). 1. There exists exactly one transJiniiesequence C, = C / (T ) such that C, = B(Cl5) c, = if Tpl,(r,C)if Z(r, 5) f 10, Z(r, = 10. The terms of this sequence are called constructible from T by B. All the theorems established in this chapter are valid mutatis mutandis or this more general notion cf ccnstrwtibility. CHAPTER IV FUNCTORS AND THEIR DEFINABILITY The general topic of this chapter is as follows. We consider functions which correlate with each set A a set F(A) contained in A.
N l . We shall call H a definition of F. 1. If F and G are strongly definable functors with n+1 and m+ 1 arguments, then so are the functors W , P I , ... YP,) = A - W , P , , ... ,Pfl), 52 IV. ,p,, and permutations of these arguments yield strongly definable functors. +] , ... ,x,+,) is a definition of P and (x,)H is a definition of Q. Identifications and permutations of variables in H yield formulae which define functors arising from F by identifications and permutations of its variables.
If H is (xi)H'. The blanks ... are to be filled here by the description of F(xi),,given on p. 24. We now define G. are ordinals and which has the following properties: for each subformula H' of H the class is a function; if H' is an atomic formula, then K("') = Id = ( ( 7 , 7 ) : y E On}; if H' is either l H * or H * A H * * or (xi)H*, then K,',') is equal to O(K(R*),K\ff**));( 5 , V ) E K(H). , K ( H )is the set of those (ci, ,B) for which (H, ( a , p)) E K. We can easily show that the function G defined above satisfies the conditions (i)-(iv).