If you read any part of it, read Section 9, his justification of why the Turing Machine model captures computation, an argument that still resonates today.
No attempt has yet been made to show that the “computable” numbers include all numbers which would naturally be regarded as computable. All arguments which can be given are bound to be, fundamentally, appeals to intuition, and for this reason rather unsatisfactory mathematically. The real question at issue is “What are the possible processes which can be carried out in computing a number?”
The arguments which I shall use are of three kinds.
- A direct appeal to intuition.
- A proof of the equivalence of two definitions (in case the new definition has a greater intuitive appeal).
- Giving examples of large classes of numbers which are computable.
Once it is granted that computable numbers are all “computable” several other propositions of the same character follow. In particular, it follows that, if there is a general process for determining whether a formula of the Hilbert function calculus is provable, then the determination can be carried out by a machine.