Page 228

		Proof: Consider the following collections of functions:

		.

		.

		.

		We argue that . Suppose by way of contradiction that a scientist M Bex*,*-identifies S. Let i0 be a program for , the everywhere 0 function. Let M' be a scientist (which uses no additional information) such that, for each  s ,

		For any , max(f(0), i0) is clearly an upper bound on the minimal program for a finite variant of f, hence M' Ex*-identifies every . This is a contradiction, since it was shown in Theorem 4.25 that . Therefore, .

		Proposition 10.7 limits us to the following weak analog of Proposition 10.5 for Bex-identification.

		10.9 Proposition Suppose . Then, .

		Proof: Let , which is clearly in Exa+1. Suppose by way of contradiction that there is a scientist M theft Bexa,a+1-identifies S. Let M' be a scientist (without any additional information) such that, for all  s ,

		Since for any , f(0) is an upper bound on MinProga+1(f), it follows that for each , Thus, M' Exa-identifies S. This is a contradiction, since it was shown in Proposition 6.5 that .

		Propositions 10.9, 10.7, and 10.8 yield the following two corollaries.

		10.10 Corollary For each , .

		10.11 Corollary.