But equality over the constructible reals is not decidable either. A constructible real number is essentially a suspendable computer program that outputs the n

But equality over the constructible reals is not decidable either. A constructible real number is essentially a suspendable computer program that outputs the next digit of the number whenever you ask for it, in finite time. With some work you can define operations like addition and multiplication of such objects. But if you try to define an equality test, you will find that the algorithm will correctly differentiate distinct numbers in finite time, but will not terminate on equal numbers (how many digits have to agree before you know they are the same?). Deciding equality here requires a halting oracle.