An Effective Procedure for Computing "Uncomputable" Functions
  We give an effective procedure that produces a natural number in its output
from any natural number in its input, that is, it computes a total function.
The elementary operations of the procedure are Turing-computable. The procedure
has a second input which can contain the Goedel number of any Turing-computable
total function whose range is a subset of the set of the Goedel numbers of all
Turing-computable total functions. We prove that the second input cannot be set
to the Goedel number of any Turing-computable function that computes the output
from any natural number in its first input. In this sense, there is no Turing
program that computes the output from its first input. The procedure is used to
define creative procedures which compute functions that are not
Turing-computable. We argue that creative procedures model an aspect of
reasoning that cannot be modeled by Turing machines.
