April 2012
Intermediate to advanced
416 pages
10h 40m
English
Content preview from Theory of ComputationBecome an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
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2.9 THE RECURSION THEOREM
The recursion theorem, by that name, is due to Kleene but owes its existence (and proof) to the work of Gödel (1931) on the incompleteness phenomenon. We look at few versions of this very powerful tool in what follows, and then come back to this comment regarding the connection with Gödel’s work.
2.9.0.26 Theorem. (Kleene’s Recursion Theorem) If 
z
.f (z, n) 
, then for some e, we have 
(n) ⋍ f (e, n), for all n.
Proof. Let 
. Then
by 2.6.0.38
Take e = (a, a). □ ...
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