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By J. Barwise

ISBN-10: 3662110350

ISBN-13: 9783662110355

ISBN-10: 3662110377

ISBN-13: 9783662110379

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Proof. We need to show Ca is one-one and that ca(x)Eciy) implies xEy. We prove both of these by proving 'v'x'v'yP(x,y) where P(x,y) is the conjunction of: x,yEa /\ cix)=ciy)--+x = y, X,YEa /\ ca(x) E ca(y)--+x E y, and X,YEa /\ ciy) E ca(x)--+ y EX. Given an Xo we can assume, by induction on E, (1) 'v'XEXo 'v'y P(x,y) in our proof of 'v'y P(xo,Y). Given an arbitrary Yo we can assume in our proof of P(Xo, Yo), again using E-induction. Thus, suppose xo,YoEa. Case 1. cixo) = ciyo). Suppose Xo # Yo' We see that both xo, Yo must be sets since ca(p) = p.

O. We can introduce a L\ predicate R by definition so that the following are provable in the resulting KPU: (i) R(Xl, ... , xn,p)+-+P(x 1 , ••• , xn,p); (ii) R(Xl, ... , xn,a)+-+Q(xl' ... , xn,a, {bETC(a) I R(Xl, ... , xn,b)}). Proof. Introduce the characteristic functions G, H of P,Q respectively. Use Recursion to define the characteristic function F of R and then note that R(x 1 , ••• , ~ xn,y)+-+F(x 1 , ••• , x n,y)=1 +-+ F(Xl, ... , x",y)#O, so that R is shown to be L\. 0 In Table 4 we give some examples of operations defined by recursion.

3 Theorem. Let S be a relation on natural numbers. e. iff S is Ll on IHF. (ii) S is recursive iff S is ~1 on IHF. 3 that are just as easy to prove. 1 is admissible. 48 II. 3 we assume familiarity with the elements of ordinary recursion theory. 3 (=». e. Nevertheless, first we prove the (=» part of (ii) to help us prove the corresponding half of (i). It clearly suffices to show that every recursive total function on the integers f can be extended to a ~l function j on lHF by the definition: j(X) = f(x), =0, for XEW for x¢w.

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Admissible Sets and Structures: An Approach to Definability Theory by J. Barwise

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