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		We first exhibit a monotonic scientist that identifies . Let GN be an index for N. Let be an index for . Let Gj be an index for Lj that can be effectively obtained from j. Let denote an index for that can be effectively obtained from j and m. Let . Now let M be a scientist that, for each  s

		.

		It is easy to verify that M is monotonic and identifies .

		Next we show that no weak-monotonic scientist identifies Suppose scientist Mj identifies Consider the language Lj. Clearly, because otherwise and Mj does not identify Lj. Let , hence . Suppose  s  is an extension of Tj[m] such that and Such a  s  exists as and Mj identifies Thus we have , which violates the weak-monotonicity property for identification of by Mj. Thus Mj fails to be weak monotonic.

		The next proposition shows that there are collections of languages that can be identified by weak-monotonic scientists but not identified by any monotonic scientists.

		5.57 Proposition .

		The above proposition (together with Proposition 5.54) implies the following corollary.

		5.58 Corollary

		(a)

		(b) .

		Proof (Proposition 5.57): This argument is based on a technique due to Lange and Zeugmann [119].

		For each m and n, we define: