(Enumerable) set of natural numbers might not be effectively enumerable It is well known that a set of natural numbers, although trivially enumerable, might not be effectively enumerable. I am trying to understand this fact intuitively. What is the decisive element in the definition of effectively enumerable, which involves algorithm, that makes some enumerable set not effectively enumerable?

The easiest and least constructive way of seeing this is to remember that there are only countably many algorithms. So only a countable collections can be effectively enumerable.

But there are uncountably many subsets... so most sets of natural numbers cannot be effectively enumerated.