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If  is ever found, then:
 set  ,
set ys+1 to the least number of the form  that is not in the finite set  ,
set  ,
set s s+1 be the least extension of s s such that content( s s+1) = Ss + 1,
set  , and
go to stage s + 1. |
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To show that M fails to accountably identify  , we consider two cases. |
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Case 1: All stages of the construction halt. Let L = We. It is easy to see that  and  . Let  . Observe that T is a text for L. But for each i we have  . Thus M fails to identify L. |
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Case 2: Some stage s starts but does not halt. By Kleene's recursion theorem there exists an e' > e such that  . Clearly,  . However,  . Thus, M fails to be accountable on We'. |
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The global version of accountability turns out to be more restrictive than the class version. To show this, let us first recall from Chapter 4 that  and introduce the following. |
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5.17 Definition  . |
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5.18 Proposition  . |
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Proof: We first show that  . By the S-m-n Theorem (Theorem 2.1), there is a recursive g such that, for all i,  . Let GN be an index for N. Consider the scientist M that behaves as follows on each s : |
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It is easy to verify that M is accountable on  and that is identifies  . |
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Now suppose by way of contradiction that an accountable scientist M identifies  . Through an implicit use of Kleene's recursion theorem (Theorem 2.3), we exhibit a  which M fails to identify. We construct a text for this We in stages. |
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Stage 0: s 0 =  . |
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