The Church-Turing thesis is the claim that any function that can be computed by a definite, mechanical, step-by-step procedure can be computed by a Turing machine, the simple abstract device Alan Turing defined in his 1936 paper “On Computable Numbers, with an Application to the Entscheidungsproblem.” In the same period Alonzo Church arrived at an equivalent notion of effective calculability using his lambda calculus, and the two formulations were shown to define exactly the same class of computable functions. Because several independent attempts to pin down “computable” all landed on the same class, the thesis asserts that this class is the right and complete one.
The thesis is not a theorem to be proved, because it connects an informal idea, what a human or machine could in principle calculate by following rules, to a precise mathematical definition. But its evidence is strong: every reasonable model of computation proposed since, from lambda calculus to register machines to cellular automata, has turned out to be equivalent in power to the Turing machine. That convergence is why we can speak of computation as a single, well-defined notion rather than a property that depends on which machine you use.
For artificial intelligence, the Church-Turing thesis is foundational and double-edged. It underwrites the idea that a general-purpose computer can, in principle, run any effective procedure, which licenses the hope that intelligence, if it is some computable process, could be implemented on a machine. It equally sets limits: Turing’s same paper proved there are well-defined problems no machine can solve. The primary source used here is Turing’s 1936 paper.
For a general reader, this thesis is the reason we can talk about what computers can and cannot do in the abstract, without worrying about the particular hardware in front of us.