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		Suppose scientist M is given. We describe a scientist M' such that and M' is decisive. Let be a recursive function that satisfies the following equation.

		For each M and each , define

		Now let M' be a scientist such that, for all , M'( s ) = h(M( s [m]),  s [m]), where m = convpoint(M,  s ). We claim that and M' is decisive.

		Note that if , then M'(f) converges to h(i, f[m]), where i = M(f) and m is the convergence point for M on f. Moreover,  j h(i,f[m]) =  j i. Thus .

		To see that M' is decisive, suppose f and l < m < n are given such that . Let n' = convpoint(M,f[n]) and l' = convpoint(M,f[l]). Clearly, and . Now M'(f[n]) = h(M(f[n']), f[n']) and thus (by construction of h). However, M'(f[l]) = h(M(f[l']), f[l']). Thus, , since, in the construction of h(M(f[l']), f[l']), either a mind change would be observed before input m or  s M'(f[l']) is convergently different from f[m]. Therefore, M' is decisive.

		§5.7 Bibliographic Notes

		The question of consistency has been independently considered by Barzdins * [13] and by Blum and Blum [18] in the context of functions. The reader can find additional material on consistent strategies in Wiehagen and Liepe [202], Jantke and Beick [100], Zeugmann [206], and Fulk [70]. Consistency in the context of languages has been addressed by Angluin [6]. Wiehagen and Zeugmann [203] present an interesting perspective on the power of inconsistent strategies in the context of more practical domains.

		The notion of set-driven scientists was introduced by Wexler and Culicover [194]. Schäfer-Richter [167], and later Fulk [69], independently established that set-drivenness is a restriction on computable scientists. The notion of rearrangement-independence was introduced as a weakening of set-drivenness by Schäfer-Richter [167] and by Fulk [69].