The word used for the property referred to in the Wolfram article really should be dense, not continuous. The set of rationals is dense, but incomplete and totally disconnected.
The main property lacking is exactly what I stated earlier: for some perfectly reasonable questions, rationals only allow you to work with approximations that you can prove are always wrong. That’s mathematically very undesirable. It’s much better to have a theory in which you can prove that there exists a correct result, and then if you only care about rational approximations you can just find a nearby rational and accept the error.
The word used for the property referred to in the Wolfram article really should be dense, not continuous. The set of rationals is dense, but incomplete and totally disconnected.
The main property lacking is exactly what I stated earlier: for some perfectly reasonable questions, rationals only allow you to work with approximations that you can prove are always wrong. That’s mathematically very undesirable. It’s much better to have a theory in which you can prove that there exists a correct result, and then if you only care about rational approximations you can just find a nearby rational and accept the error.