A set like for instance the ordered set with all even numbers first, followed by all odd numbers like ${\{0,2,4,...,1,3,5,...\}}$ is a computably enumerable set?
From wikipedia:
In computability theory, a set S of natural numbers is called computably enumerable >(c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: There is an algorithm such that the set of input numbers for which the algorithm halts is exactly S.
Or, equivalently,
There is an algorithm that enumerates the members of S. That means that its output is simply a list of all the members of S: s1, s2, s3, ... . If S is infinite, this algorithm will run forever.
From a cardinal perspective is enumerable, in the sense there is a bijection between natural numbers and this particular set, so this particular bijection is a computation that enumerate their elements.
But can I provide a computation that outputs the numbers in the order I want? It seems that is not posible because the Turing machine would output even numbers forever, before having the opportunity to write an odd one...
Any countable ordinal is computably enumerable as a set?