Page 111

		account for some data; a weak-monotonic scientist must keep generalizing until the current hypothesis fails to account for some data. A nontrivial proof shows that weak-monotonic scientists and conservative scientists identify the same collections of languages. (See Exercise 5-21.)

		We now consider the interplay between the three generalization strategies defined above. It is easy to verify the following proposition.

		5.54 Proposition

		(a)

		(b)

		(c)

		(d)

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

		5.55 Proposition

		Before proving this proposition, we note the following corollary that follows from the above proposition and Proposition 5.54.

		5.56 Corollary

		(a)

		(b)

		Proof (Proposition 5.55): Let M0, M1, . . . be an enumeration of total computable scientists along the lines of Corollary 4.16. For each j and m, let and . For each j, let Tj denote the text and let

		Note that Sj is recursive. Now define the collection of languages:

		We show that .