[tshaddox]

Yes, I chose one arbitrary method of enumeration. That’s not important to the point, which is that algorithms are enumerable and thus you don’t need the axiom of choice to select one out of the set of all algorithms.

[shadowlight]

Yes I know that's your point. I'm saying you can't enumerate algorithms without selecting an algorithm.

One way of selecting an algorithm is to select a way to encode the algorithms.

It's easy to see that this is true. You chose an arbitrary example above. Try to do the same without choosing anything arbitrary. You can't.