MATH.18:5.2 - An enlarged domain changes the available solution

The natural numbers N={0,1,2,...} embed in the integers Z. Inclusion preserves 0, addition and equality of expressions evaluated on natural-number inputs. It also reflects those equalities: two included natural numbers are equal in Z precisely when they were equal in N.

Suppose the next task is to solve x+1=0. The integer solution x=-1 is outside N. The interpretation useful for additive calculations therefore has not supplied a natural-number solution.

To express the original existential question in Z, retain its range: “there exists an integer x with x>=0 and x+1=0.” That statement remains false. To use Z as a calculation space for another natural-number equation, compute there and test the recovered candidates against the source domain.

The revised use keeps the beneficial integer calculations and makes their return condition explicit. A demand to identify all natural-number and integer solution questions would instead fail this comparison.