Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2010 / B / 4 / f / i / Solution

No. In the standard successor structure, every element other than the named zero has a predecessor:The proposed structure fails this first-order sentence at the integer $-1$. Its nonnegative elements map to positive elements, while every negative element maps to an integer at most $-2$; hence no element maps to $-1$. This gives a direct first-order distinction, without treating the negative branch as a two-sided successor chain.