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		We next show that .

		Define:

		.

		We leave it to the reader to verify that We argue that . Suppose by way of contradiction that a scientist M that is consistent on identifies . Then by Kleene's recursion theorem (Theorem 2.3), there exists an index e such that, for all x,

		Suppose  j e is total. Then , and for all x > 0, . Hence, , a contradiction. Therefore,  j e is not total. Suppose x is the least number such that  j e(x) is not defined. Then and Let be such that Again, by Kleene's recursion theorem, there exists an index e1 such that, for all x,

		.

		It is easy to verify that However, since M is not consistent on . This is a contradiction. It follows that no such M can exist.

		§5.6.2 Popperian

		As discussed in the context of languages, scientific practice usually demands the testability of hypotheses, a constraint that was labeled "accountability" in Section 5.2.3. We