Let A be a set of partial-recursive unary functions on the domain of natural numbers such that the set is recursively enumerable, where denotes the -th partial-recursive function in a Gödel numbering.
Then for any unary partial-recursive function , we have:
- a finite function such that
In the given statement, a finite function is a function with a finite domain and means that for every it holds that is defined and equal to .
Perspective from Effective Topology
For any finite unary function on integers, let denote the 'frustum' of all partial-recursive functions that are defined, and agree with , on 's domain.
Equip the set of all partial-recursive functions with the topology generated by these frusta as base. Note that for every frustum , is recursively enumerable. More generally it holds for every set of partial-recursive functions:
is recursively enumerable iff is a recursively enumerable union of frusta.
- Cutland, Nigel (1980). Computability: an introduction to recursive function theory. Cambridge University Press.; Theorem 7-2.16.
- Rogers Jr., Hartley (1987). Theory of Recursive Functions and Effective Computability. MIT Press. p. 482. ISBN 0-262-68052-1.
- Odifreddi, Piergiorgio (1989). Classical Recursion Theory. North Holland.
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