You have a typical recursively enumerable set S of integers, and a set X of lattice points cut out by a multivariate polynomial. We are talking about S being the projection (onto one axis) of X. Given that S can be "pretty bad", can one say anything except that X is "presumably worse"? None of the interest lies in any finite segment of S.
To put it another way, possibly more interesting to geometers, the analogy with constructible sets fails here. There is some hint in the history that Hilbert was misled by elimination theory into thinking that Godel's incompleteness theorem (or suchlike) couldn't be the case. (I'm not putting this very well, and it is disrespectful in a way to Hilbert.) Anyway algebraic geometers "know" that projection doesn't really innovate that much in the way of geometry, and logicians "know" the precise opposite in terms of logic. So at the very least the intuitions are in tension.
(Maybe the mistake of thinking that "geometry of undecidable diophantine systems" was a kind of impossible object, and therefore all r.e. sets would turn out to be recursive, is a mistake intelligent enough to attribute to Hilbert. As a kind of Fundamental Theorem of Proof Theory.)