Talking about what will never be specified

There real line has an uncountable number of elements. Natural language and all our formal languages have at most a countable number of possible expressions. Hence there are real numbers r which we will never be able to define in the sense that r = the x. P(x) for some property P(x). But we are referring to these numbers hence defining them. Is there a contradiction here ? No, because we are defining them as an indefinite plurality. We still cannot separate them or distinguish one of them. No matter what specification we propose to single out an indefinable r it will always escape through our fingers. There will always be more than one indefinable satisfying the specification.
Perhaps this might offer a clue for a new interpretation of quantum theory ? Note that if particles behaved classically and had smooth trajectories specified by a differential equation, then the position and momentum at any definable time would be itself definable (even if not computable). Hence we could postulate that in quantum theory the particle evolves through indefinable states and hence cannot be described by a differential equation - only in the sense of indefinite pluralities as above - that is, a stochastic differential equation or wave-function differential equation. In this way there would be no need of a 'collapse of the wave function'. What are some objections to this interpretation ?