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		Replacing d-conformity by uniform d-conformity in Definition 10.30 produces the following paradigm.

		10.31 Definition Let and .

		(a) A scientist M UapdExa-identifies f (written: ) just in case, for all p such that  j p is uniformly d-conforming with f, and  j M(p,f) =a f.

		(b) .

		Ap and Uap are abbreviations for 'Approximate partial additional information' and 'Uniform approximate partial additional information,' respectively. The Uap-type additional information is intuitively better than the Ap type, and indeed the results to follow will confirm this. Since any Uapd-type additional information is also an Apd-type additional information, we have the following.

		10.32 Proposition For all and all , .

		We next establish that there are collections of functions that can be identified with some Uap-type additional information of nonzero density, but cannot be identified in the presence of the best possible Ap-type additional information even if the the scientist is allowed to converge to a finite variant program.

		10.33 Proposition Suppose . Then .

		Proof: Let n0 = 0 and, for each , let n2i + i + 1 and n2i+2 = n2i+1 · 2i.

		For each j, define

		Also, define

		Finally, let

		We show that in the next two claims and thus establish the proposition.