Your question has a logical formulation that leads to an open problem.
A Discretely Ordered Ring is an ordered ring in which the inequality
$x<y<x+1$ has no solutions, or equivalently, an ordered ring containing no element between 0 and 1. There is a simple finite set of axioms for the class of discretely ordered rings. Hilbert's Tenth Problem for discretely ordered rings asks:
Is it decidable whether given a system of polynomial equations with integer coefficients, there is some (at least one) discretely ordered ring in which that system of equations is solvable?
This is equivalent to asking if the set of unsolvable polynomial systems whose unsolvability follows from the axioms for discretely ordered rings can be effectively listed.
The best result so far, due to van den Dries, is that the answer is positive for a single polynomial equation in two variables. The proof uses basic facts about algebraic function fields, especially the Riemann-Roch Theorem. See "Which Curves over Z have Points in a Discretely Ordered Ring?", Transactions of the AMS. March 1981