I don't see the problem here. It shouldn't be a surprise that listing the computable numbers is itself uncomputable as it is equivalent to the halting problem -

I don't see the problem here. It shouldn't be a surprise that listing the computable numbers is itself uncomputable as it is equivalent to the halting problem - to determine whether an algorithm will either keep on producing new decimals or halt at some point is clearly undecidable. It's when you introduce things as constructible/defineable numbers that things gets hairy.