So the whole idea of an algorithm that operates on or outputs real numbers is nonsensical
You can work with programs over infinite streams in certain situations. For example, you can write a program that divides a real number by 2, taking an infinite stream as input and producing another infinite stream as output. Similarly, you can write a program that compares two unequal real numbers.
True, I forgot about that. I guess that would allow extending the notion of “computable well-ordering” to sets of size 2^aleph_0. Of course that doesn’t necessarily mean any of them would actually be computable, and I expect none of them would. Well, I guess it has to be the case that none of them would, or else we ought to be able to prove “R is well-orderable” without choice, but there ought to be a simpler argument than that—just as computable ordinals in the ordinary sense are downclosed, I would expect that these too ought to be downclosed, which would immediately imply that none of the uncountable ones are computable. Now I guess I should actually check that they are downclosed...
You can work with programs over infinite streams in certain situations. For example, you can write a program that divides a real number by 2, taking an infinite stream as input and producing another infinite stream as output. Similarly, you can write a program that compares two unequal real numbers.
True, I forgot about that. I guess that would allow extending the notion of “computable well-ordering” to sets of size 2^aleph_0. Of course that doesn’t necessarily mean any of them would actually be computable, and I expect none of them would. Well, I guess it has to be the case that none of them would, or else we ought to be able to prove “R is well-orderable” without choice, but there ought to be a simpler argument than that—just as computable ordinals in the ordinary sense are downclosed, I would expect that these too ought to be downclosed, which would immediately imply that none of the uncountable ones are computable. Now I guess I should actually check that they are downclosed...
What does it mean to be downclosed?
I mean closed downwardly—if one is computable, then so is any smaller one. (And so the Church-Kleene ordinal is the set of all computable ordinals.)