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		now formulate a similar requirement for the paradigm of function learning.

		5.63 Definition (Case and Ngo-Manguelle [34])

		(a) M is Popperian on f just in case for all n, .

		(b) M is Popperian on just in case M is Popperian on each .

		(c) M is Popperian just in case M is Popperian on .

		(d) .

		(e) .

		An index for a total computable function is a testable hypothesis, since it is possible to test its accuracy against the data provided by a finite sequence. Such testability motivates the terminology "Popperian," since Popper (e.g, [153]) insisted on this aspect of scientific practice. Requiring the testability of hypotheses appears to be a useful requirement. However, like many other seemingly rational constraints, both versions of Popperian scientists pay a price for their rationality, as implied by the next result.

		5.64 Proposition .

		Proof: Clearly, .

		We first show that .

		Consider again the collection of self-describing computable functions Clearly, . Suppose by way of contradiction that M is Popperian (on ) and identifies . Then, by Kleene's recursion theorem (Theorem 2.3), there exists an e such that, for all x,

		.

		Since M is Popperian,  j e is total and thus . Also, for all x > 0, Thus , a contradiction. Hence, .

		Let be as defined in the proof of Proposition 5.62 on page 116. Clearly, . By Exercise 5-22 .

		We describe a nice characterization of [Ex]Popperian First, recall from Definition 5.10 that is said to be r.e. indexable exactly when there is an such that . We define the class

		.