I can say arbitrary stuff like the set of all sets with positive numbers but when I say the set of all algorithms written in English and C++ suddenly I'm getting too specific. Where is the line drawn?
Your assuming all algorithms are defined in terms of English that's how you can order them alphabetically. English is an arbitrary language that comes from human culture. Same with a programming language. You are defining a set in terms of concepts that are cultural.
Algorithms themselves have no specific order. In order to define enumeration you must first start off by *selecting* which algorithm gets the first enumeration. This is a completely arbitrary choice.
No, I'm not assuming any particular encoding of an algorithm. I'm just assuming by "algorithm" you mean a computable function, and we know there are only countably many computable functions. This is not a cultural notion.
And yes, which particular encoding you decide to use is arbitrary, but the point is that you can enumerate the set of all algorithms, and thus you can select one without needing the axiom of choice.
Your own example used the word "alphabetical." So your example is false because it uses a "particular" encoding.
Try to select an algorithm out of the set of all algorithms without using an encoding. If you must use an encoding, please ensure that it's not a "particular" encoding.
You can't.
The point is all encodings in the known universe are "particular."
Additionally, to even use an encoding you have to *select* and encoding from the set of all encodings.
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.
