Note of numbering computable Functions

A set X is effectively denumerable iff there be a bijection f:x→n such that both F and f-1 be effectively comput Able functions.

　　theorem. The set of the URM instructions is effectively denumerable.

To see this, we should define the following functions:

(1) bijectionπ: Nxn→n is defined by

　　　　　　　　　　

(2) Bijectionζ: n+ x n+ x n+→n is defined by

　　　　　　　　　　

(3) bijectionτ: Uk>0nk→n is defined by

　　　　　　　　　　

Then we can construct a bijectionβfrom URM instructions to natural numbers:

　　　　　　　　　　

Hence, the URM instruction set is effectively denumerable.

　　theorem. The set of URM programs is effectively denumerable since we can construct the following bijection:

　　　　　　　　　　　　

where P = I1i2 ... is, and this is also known as the Gödel number of a program.

　　theorem. The set of all n-ary computable functions are denumerable since we can construct an enumeration of those functions without Repetitions.

　　theorem. The set of computable functions is denumerable since it is the union of the sets of the n-ary computable to all functions + and hence its bijection to Ncan is constructed by using functionτ.

Moreover, we can construct a total unary function which is not computable by using Cantor ' s Diagonal Method Decrib Ed in Set theory.

　　　　　　　　　　

　　The s-m-n theorem: for M, n∈n+, there are a total computable (m+1)-ary function SNM (E,x) such that

　　　　　　　　　　　　　　　

References:

1. Cutland, Nigel. Computability:an Introduction to recursive function theory [M]. Cambridge:cambridge University Press, 1980

Note of numbering computable Functions