But such a program can't visit every real number in the interval, because there are uncountably many, but the program will only run for countably infinitely many steps.
You're right, I actually hadn't grasped this, I realised later on but it was too late to edit my comment. And to be honest I still don't completely get it.
Since you can count the cycles your computer takes, then at any point in time it's outputting a countable digit on a list of numbers that is also countable. Accelerating it to infinity lets you finish both of those tasks, but it doesn't break you into another realm.
If you get a nondeterministic computer, where every digit it splits into 10 identical computers that each picked one of the 10 options, then when you run that for countably infinite cycles you'll find that you have uncountably infinite computers and you have finally calculated every real number.
The cardinality of the real numbers is 2 to the power of the countable numbers.
