Which hypothesis states that any effectively calculable function is Turing machine computable?

The hypothesis that every effectively calculable function is computable by a Turing machine is the Church–Turing thesis.

It connects an informal idea—something that can be calculated by a definite mechanical procedure—with formal mathematical models of computation. In the 1930s, Alonzo Church studied computability through lambda calculus and recursive functions, while Alan Turing developed his abstract machine model. Their approaches were shown to describe the same class of computable functions.

The thesis says that a function is effectively calculable if and only if a Turing machine can compute it. The word “thesis” matters: effective calculability is an informal concept, so the claim is not a theorem that can be proved in the ordinary mathematical sense. Its extraordinary acceptance comes from the equivalence of many independent models of computation.

Church’s theorem is a different result, associated with undecidability in formal logic. Turing completeness describes whether a particular system can simulate a universal Turing machine; it is related, but it is not the name of this hypothesis. The Church–Turing thesis also says nothing about how efficiently a function can be computed.