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		(a) Show that infinite scattered sets exist and that each scattered set is either finite or immune.

		(b) Show that there are hyperimmune sets which are not scattered and that there are infinite scattered sets which are not hyperimmune.

		(c) Show that if S is a scattered set and  p  is a recursive isomorphism, then  p (S) is also a scattered set.

		(d) We say M Exscat-identifies f if and only if and is scattered. We also define Exscat(M) and Exscat in the usual way. Show that .

		7-8 Here we show Proposition 7.10: . Let and . Clearly, . Fix n. Show that if , then , a contradiction.

		7-9 Here we develop a proof of Proposition 7.14.

		(a) Show that for each , , and , there is a recursive set R such that and ad(L, R) > a. Hint: Consider the two cases of ad(L, N) > a and .

		(b) Show that for each rational , .

		(c) Show that for rationals q and r with , .

		(d) Prove Proposition 7.14.

		7-10 C.S. Peirce [147] held that one should not expect science to converge on a final theory of a phenomenon X; instead, the expectation should be that science will produce a series of better and better approximations to X. Fulk and Jain [73] define the following identification criteria intended to model Peirce's theory of inference. M Ap-identifies f (written: ) if and only if there is a sequence of subsets of N such that (i) for each n, ; (ii) ; (iii) ; and (iv) for infinitely many n, is infinite. This turns out to be an extremely powerful criteria, as there is an M that Ap-identifies every element of [73, Theorem 1]. We work through a proof of this below.

		(a) For each , let pf denote the least p such that  j p = f. Show that there is an M' that takes a function argument f and a numeric argument m and which is such that, for each f in and each , and  j p' = f.