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		5.25 Proposition (Scháfer-Richter [167], Fulk [69])

		Proof: For each j, define and, for each j and n, define . Let M0, M1, M2, . . . denote an enumeration of all scientists. For each j, let

		Let . It is easy to see that .

		Suppose by way of contradiction that Mj is a set-driven scientist that identifies . Mj thus identifies the text . So there must be an and an index i for Lj such that Mj( s j,n) = i. In particular, there must be a least such that Mj( s j,n) = i and . But then Mj does not identify content( s j,n), since on the text must conjecture i in the limit, since Mj is set driven. Thus Mj fails to identify , a contradiction.

		§5.3.2 Rearrangement-independent Scientists

		Set-driven scientists ignore the order in which data arrive and base their conjectures only on the content of the data seen at any given time. The following definition introduces a less stringent variation on set-drivenness.

		5.26 Definition (Schäfer-Richter [167], Fulk [69, 71])

		(a) M is rearrangement-independent just in case for every  s , with and , we have .

		(b) .

		Unlike set-drivenness, rearrangement independence turns out not to be restrictive; we delay the proof of this fact to the next section.

		§5.4 Constraint on Convergence

		Suppose that a scientist M identifies . Then for each and each text T for L, M must converge to some index for L. This M does not necessarily converge to the same