Unless I am missing something, as far as the work of Turing and Church is concerned, their answers are actually quite clear on this question of "can", namely th

Unless I am missing something, as far as the work of Turing and Church is concerned, their answers are actually quite clear on this question of "can", namely the class of partial recursive functions. This is precisely what a Turing machine can compute. It is also possible to build up to these functions in a concrete way using the simpler class of primitive recursive functions. Rogers book on recursive functions and computability is a great reference on this topic.