34 - The Church–Turing Thesis

Summary

Right back in Chapter 2 we stated Turing's Thesis: a numerical (total) function is effectively computable by some algorithmic routine if and only if it is computable by a Turing machine. Of course, we initially gave almost no explanation of the Thesis. It was only very much later, in Chapter 31, that we developed the idea of a Turing machine and saw the roots of Turing's Thesis in his general analysis of the fundamental constituents of any computation.

Meanwhile, in Chapter 29, we had already introduced the idea of a µ-recursive function and noted the initial plausibility of Church's Thesis: a numerical (total) function is effectively computable by an algorithmic routine if and only if it is µ-recursive.

Then finally, in Chapter 32, we outlined the proof that a total function is Turing computable if and only if it is µ-recursive. Our two Theses are therefore equivalent.

Given that equivalence, we can now talk of

The Church–Turing Thesis The effectively computable total numerical functions are the µ-recursive/Turing computable functions.

Crucially, this Thesis links what would otherwise be merely technical results about µ-recursiveness/Turing computability with intuitive claims about effective computability; and similarly it links claims about recursive decidability with intuitive claims about effective decidability. For example: it is a technical result that PA is not a recursively decidable theory. But what makes that theorem really significant is that – via the Thesis – we can conclude that there is no intuitively effective procedure for deciding what's a PA theorem.