[fasterik]

Having constructible Cauchy sequences doesn't guarantee that we can construct unbounded operators. I'm no expert, but the little searching I've done suggests this is an open research question.

I don't see the benefit of being able to write something down "in principle." A number can only ever be computed to a finite number of digits in practice. If we're talking about finite approximations, then the standard approach using numerical solutions to the Schrödinger equation handles this just fine, no alternative mathematics needed. If we're talking about theories, then we should choose whatever abstraction is most convenient for expressing the theory.

Personally, I don't believe numbers "exist." The physical universe exists, and numbers are abstractions that we invent to describe it. In that sense, uncomputable numbers are just as "real" as computable ones.