I am just guessing here, but I think the problem is that even if you redefine a real number as "that which can be written by a program", you cannot write a prog

I am just guessing here, but I think the problem is that even if you redefine a real number as "that which can be written by a program", you cannot write a program that enumerates all such numbers. That's kinda because different real numbers require programs of different complexity (for any specific complexity, e.g. the number of bytes, there is only a finite number of programs with that complexity), so listing all the real numbers would require infinite complexity, i.e. not a (finite) program. So there is a countable set of integer, and a set of reals (by the new definition) which is countable (by the usual definition) but cannot be enumerated, so it is technically "uncountable" (by the new definition). The trick is that if you are changing definitions, you may accidentally also change the definition of "countable". Because it is defined as "that which can be mapped to integers", but the problem is, which mappings are considered legal and which are not. In the world of "only things written by programs exist", only mappings which can be generated by programs exist; which is not the same as all possible mappings, so a few previously countable things now become "uncountable".