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		If the enumeration of We fails to progress past some odd stage, then We is finite and fails to identify some text for We.

		If the construction progresses past all stages, then We = N and fails to converge on some text for We and so fails to identify We (in this case e and p are equivalent indexes, both for N).

		Thus, in all cases fails to identify {Wp, We} = [h(e)].

		In conjunction with Exercise 4-1, Proposition 4.36 implies:

		4.37 Corollary is not performable on [·].

		Thus, no parameterized scientist can adapt herself to the identification of an arbitrary, identifiable on the basis of an [·] description of . In this sense there is no universal learning machine for [·] and TxtEx.

		The proof of Proposition 4.36 rests on the possible presence of equivalent indexes in the description of a class of languages. Disallowing such equivalences in fact makes performability easier. The next definition helps to make this claim precise.

		4.38 Definition .

		So, if then Wi contains no two equivalent indices. (The symbol ''" may be read as "no equivalences.")

		4.39 Proposition is performable on [·].

		Proof: Exercise 4-15.

		Is the finiteness qualification in Proposition 4.39 essential? The next proposition shows that it is.

		4.40 Proposition is not perform able on [·].

		Thus, no parameterized scientist can adapt itself to the identification of an arbitrary, identifiable even on the basis of an [·] description of that is purged of equivalent indices. A proof of 4.40 is given in Osherson, Stob, and Weinstein [141, Proposition 3C] and omitted here.