By Neil D. Jones

ISBN-10: 1483205053

ISBN-13: 9781483205052

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**Example text**

Clearly g is a total unary function, so that g must be in S. This means that for some k, g(x) = u(k, x) for all xe N. But this is clearly impossible, for if we set x to k we obtain u(k,k)+ 1 =g(k) = u{k,k)\ that is, 1 =0. Thus the assumption that S is countable must be false. | This is another example of proof by diagonalization. Note that the argument above would fail if S were taken to be the set of all unary partial functions, since the equation u(k, k) + 1 = u(k, k) is not a contradiction if u(k, k) is undefined.

Show that if a set is effectively enumerable, then it is effectively enumerable without repetitions. 3. Show that a set S φ 0 can be effectively enumerated in increasing order if and only if S is decidable. 4. Show that if S is the range of an effectively computable partial function, then S is effectively enumerable. 5. In what order does the algorithm of Proposition II. 6 enumerate the domain of a partial function gl 6. Show that if S and T are effectively enumerable (effectively decidable), then S u T and S n T are of the same type.

Is defined, or For example, the function/(x, y) = x + y (where x, y,and x + y are expressed in decimal notation) is effectively computable by the algorithm for digit-by-digit addition which is given in elementary schools. It might seem that the definition of an effectively computable function depends on the notation used to represent the arguments (for example, decimal, binary, prime power representation). However, this makes no difference as long as there is an effective way of translating from one notation to another and back.

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