By Neil D. Jones

ISBN-10: 1483205053

ISBN-13: 9781483205052

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

**Example text**

2. Suppose IH(&) is true. To show. \W{k + 1) must also be true. Let expression e(x) have height k + 1. Then, by definition of A, there must exist expressions p(x) and q(x) of height less than k + 1 such that either e(x) = p(x) + q(x), or e(x) = p(x) · q(x). 22 / Mathematical Basis By IH(/c), there are polynomials such that p(x) = a0 + axx + · · · + amxm and q(x) = b0 + · · · + bnxn. In the first case, let if 1 < / < min(/7, m), if n *
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*X | x is even and x > 4 and x # 10}. 4. Let £/ = {/|/ : TV -> TV}, and define T by (i) The functions e(x) = x and S(x) = JC + 1 are in Γ. 24 / Mathematical Basis (ii) If f and g are in Γ, then so is /z, where h(x) =f(g(x)>) for all xe N. Note: h(x) denotes a function in one variable x, not the value of h as applied to a specific value of x. (iii) Extremal clause. In this case T consists of all functions of the form/(x) = x + a, where a e N. x;)) = (x + a) + 1 = x + (a + 1) is in Γ. Thus T contains all functions f(x) = x + a. *

However, the proof above shows that the set of all unary partial functions contains an uncountable subset, and so it is itself uncountable (recall that every total function is also partial). Exercises 1. Construct an enumerating function for the set containing the functions: x + 1, 4x + 3, 9x + 5, \6x + 7, — 2. Construct an enumerating function f(k, x) the collection of all functions of the form ax + b, for a, be N. 1. 3. Consider the set of functions of Exercise 1. 5 is applied to its enumerating function, what function g do you obtain ?

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