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Extra info for Computability and Incompleteness

Example text

F (x, y − 1) are defined; if there is no such y, µy f (x, y) is undefined. If R(x, y) is a relation, µy R(x, y) is defined to be the least y such that R(x, y) is true; in other words, the least y such that one minus the characteristic function of R is equal to zero at x, y. We have seen that if f (x) is a partial function on the natural numbers computed by Turing machine M , then for each x, we have f (x) output(µs CompSeq(M, x, s)). If f is a total function, is equivalent to =, and one minus the characteristic function CompSeq(M, x, s) is regular; so we have shown that f is “general recursive” as well.

In other words, K is the term λyx. y. Looking at it differently, for every M , K(M ) is a constant function that returns M on any input. Now define D(x, y, z) by D(x, y, z) = z(K(y))x. Then we have D(M, N, 0) 0(K(N ))M M and D(M, N, 1) 1(K(N ))M K(N )M N, as required. The idea is that D(M, N ) represents the pair M, N , and if P is assumed to represent such a pair, P (0) and P (1) represent the left and right projections, (P )0 and (P )1 . For clarity, I will use the latter notations. Now, let us remember where we stand.

Note that according to axioms of set theory, G is still a function; there is no paradox here, just a clarification. I have found that talk of partial functions, computable functions, partial computable functions, and so on can be confusing. The set of all partial functions from N to N is a big collection of objects. Some of them are total, some of them are computable, some are both total and computable, and some are neither. Keep in mind that when we say “function,” by default, we mean a total function.