An algorithm, we said, is a sequential step-by-step procedure which can be fully specified in advance of being applied to any particular input. Every minimal step is to be ‘small’ in the sense that it is readily executable by a calculator with limited cognitive resources. The rules for moving from one step to the next must be entirely determinate and self-contained. And an algorithmic procedure is to deliver its output, if at all, after a finite number of computational steps. The Church–Turing Thesis, as we are interpreting it, is then the claim that a numerical function is effectively computable by such an algorithm iff it is μ- recursive/ Turing-computable (note, we continue to focus throughout on total functions).

The Thesis, to repeat, is not a claim about what computing ‘machines’ can or can't do. Perhaps there can, at least in principle, be ‘machines’ that out-compute Turing machines – but if so, such hypercomputing set-ups will not be finitely executing algorithms (see Section 44.4).

And as we also stressed, it is enough for our wider purposes that we accept the Thesis's link between effective computability by an algorithm and μ- recursiveness/Turing computability; we don't have to take a particular stance on the status of the Thesis. But all the same, it is very instructive to see how we might go about following Turing (and perhaps Gödel) in defending a bolder stance by trying to give an informal proof that the intuitive and formal concepts are indeed coextensive. So in this chapter I attempt such a demonstration.