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		(a)

		(b) , and

		(c)

		13-2 Before Fulk's work [72], Kurtz and Smith [116] gave the following, weaker for-malization of Barzdins's * conjecture, which they refuted.

		Conjecture: is r.e. indexable if and only if .

		Proposition 13.5 implies that this conjecture is false, but we can give a direct refutation as follows.

		13.11 Definition is said to be stable if and only if, for all recursive operators  Q  such that ,.

		(a) Suppose f is a limiting-recursive function and . Show that is stable.

		(b) Construct a limiting recursive-function f such that is not r.e. indexable.

		(c) Conclude that the above conjecture is false.