By Neil D. Jones

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**Additional info for Computability Theory: An Introduction**

**Example text**

1 states the set of functions fin S which are effectively computable is countable. 1). | 1. Implications of the Concept of Effectiveness 41 More concretely, the following proposition shows how we can obtain an uncomputable function from any enumeration of all the computable total functions. PROPOSITION II. 8 Let U(k, x) be any function which enumerates the set T, where T = {/: N^> N\f is a computable total function}. Then U cannot be an effectively computable function. PROOF Clearly C/must be total.

2 A set S £ U is effectively enumerable iff S = 0 or S is the range of an effectively computable total function/: N-* U. PROOF Clearly no problems arise if S = 0 , so suppose S φ 0 . For the " if" part, let S be the range of/. Then the flow chart of Fig. II. 1 defines an effective procedure to enumerate S. For the converse, suppose the effective process P will list the elements of S. We may certainly assume that P will produce an infinite series of 35 1. l Procedure to enumerate the range of/. outputs, even if S is finite (by repeating the same element).

In addition, each step must itself be purely finitary, so a single step cannot depend upon an infinite amount of data for its completion, or the completion of an infinite process. An effective process could, in principle, be performed on a purely mechanical basis. That means that a machine (for example, a digital computer) could be constructed or programmed or instructed in advance to perform the process without need for human decision-making once the machine has begun operation. The process is required to terminate eventually if applied to any data which is within its domain of applicability.