(Enumerable) set of natural numbers might not be effectively enumerable

Question

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?

Answer 1

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.

Answer 2

Every subset of $\Bbb N$ is enumerable. The "decisive element" that makes a set of integers not effectively enumerable is the absence of an algorithm enumerating it. As Asaf's answer points out, the existence of such sets is by a cardinality argument: there are only countably many algorithms, but uncountably many sets of integers.