Determine the number of effectively computable methods

Determine how many effectively computable methods exist in total under the notion of effective computability associated with the Church–Turing thesis, or otherwise characterize the size of this set.

Background

In discussing the difficulty of proving the Church–Turing thesis, the authors point to unresolved issues surrounding what counts as an effectively computable method and the lack of knowledge about how many such methods there are. This uncertainty contributes to the challenge of providing a definitive proof of the thesis.

Quantifying or characterizing the set of effectively computable methods would inform the limits of computation and provide clarity about the scope of algorithmic processes considered by the thesis.