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		5.68 Claim Suppose M is recursive-reliable. Then for all , there exists a such that and . Moreover, there is a recursive function t such that, given  s , as above.

		Proof (Claim 5.68): If such a does not exist then M converges to M( s ) on all functions consistent with  s . However, M( s ) can be an index for at most one function consistent with  s . This contradicts the fact that M is recursive-reliable. The moreover clause is obvious.

		Recall that . Fix a recursive-reliable scientist M. We show that M fails to identify . By Kleene's recursion theorem (Theorem 2.3), there exists an e such that  j e may be defined as follows: let  s 0 be such that , and . Clearly, . Also, . Thus, .

		§5.6.4 Decisiveness

		In the course of identification a scientist may conjecture a hypothesis, abandon it after seeing more data, and revert to the abandoned hypothesis once again. A sensible strategy would be to consider only those scientists that never go back to an abandoned conjecture.

		5.69 Definition

		(a) M is decisive on f just in case, for all l, m, with l < m < n and , we have .

		(b) M is decisive on just in case M is decisive on each .

		(c) M is decisive just in case M is decisive on .

		(d)

		(e)

		We shall now see that scientists can adopt a decisive strategy without loss of inductive competence.

		5.70 Proposition (Schäfer-Richter [167]) [Ex]decisive = [Ex]class-decisive = Ex.

		Proof: Clearly, . We show that