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		7.4 Definition Suppose and  a ,

		(a) The asymptotic agreement between A and B (written: an(A, B)) is .

		(b) The asymptotic agreement between  a  and  b  (written: aa( a ,  b ) is

		(c) The asymptotic disagreement between A and B (written: ad(A, B)) is 1 - aa(A, B).

		(d) The asymptotic disagreement between  a  and  b  (written: ad( a ,  b )) is 1 - aa( a ,  b ).

		Note that . So, ad( a ,  b ) is an asymptotic upper bound on the amount of disagreement of  a  and  b  on sufficiently large initial segments of N. The next proposition says that there is no finite function  b , partial recursive  a , and such that the "a-radius ball around  a ," i.e., includes all recursive extensions of  b .

		7.5 Proposition For each a , finite function  b , and partial recursive  a , there is a recursive f such that and ad( b , f) > a.

		Proof: Fix a,  b , and  a . Let If ad( a , f0 > a, we are done. Suppose . We construct from f0 an f as required. Let . Since , we have . Choose some . So, den(A) > b > 0. As A is r.e., we have, by Lemma 7.3, that there is a recursive with . Without loss of generality we assume . Let . By definitions of f and f0 and our assumption that , . Since B is recursive and , also. Observe that . Therefore,

		(by Lemma 7.2(b))
		(by the choice of B)
	< 1 a	(by the choice of b).

		Therefore, by Definition 7.4(d), ad( a , f) > a, as desired.

		§7.3 Approximate Explanatory Identification

		We now make use of the density notions of the previous section to develop a series of identification criteria based on approximations. (We follow the pattern of Chapter 6 by