Define: 𝜇-Recursive functions and show how all computable functions are 𝜇 -recursive

Sure, here is the explanation:

𝜇-Recursive functions are a class of functions that are defined using certain operations that are guaranteed to be computable. They are named after the mathematician David Hilbert's metamathematical program, which aimed to ground all of mathematics on simple, finite procedures (or algorithms).

The basic operations for defining 𝜇-Recursive functions are:

1. Zero function: Z(n) = 0 for all n in natural numbers.
2. Successor function: S(n) = n + 1 for all n in natural numbers.
3. Projection function: P^n_i(x1, x2, ..., xn) = xi for all i in {1, 2, ..., n} and for all xi in natural numbers.
4. Composition: If g: N^n -> N and h1, h2, ..., hn: N^m -> N are 𝜇-Recursive functions, then f: N^m -> N defined by f(x1, x2, ..., xm) = g(h1(x1, x2, ..., xm), h2(x1, x2, ..., xm), ..., hn(x1, x2, ..., xm)) is also a 𝜇-Recursive function.
5. Primitive recursion: If g: N^m -> N and h: N^(m+2) -> N are 𝜇-Recursive functions, then f: N^(m+1) -> N defined by f(x1, x2, ..., xm, 0) = g(x1, x2, ..., xm) and f(x1, x2, ..., xm, y+1) = h(x1, x2, ..., xm, y, f(x1, x2, ..., xm, y)) is also a 𝜇-Recursive function.
6. 𝜇-Recursion (or minimization): If g: N^(m+1) -> N is a 𝜇-Recursive function, then f: N^m -> N defined by f(x1, x2, ..., xm) = 𝜇y[g(x1, x2, ..., xm, y) = 0] (the least y such that g(x1, x2, ..., xm, y) = 0) is also a 𝜇-Recursive function.

The Church-Turing thesis states that a function is computable if and only if it is computable by a Turing machine. Since all the basic operations for defining 𝜇-Recursive functions are computable by a Turing machine, and since any function that can be obtained from these basic operations using composition, primitive recursion, and 𝜇-recursion is also computable by a Turing machine, it follows that all computable functions are 𝜇-Recursive.