Summary: For any (possibly, non-unital) AF \(C^*\)-algebra \(A\) with comparability of projections, let \(D(A)\) be the Elliott partial monoid of \(A\), and \(G(A)\) the dimension group of \(A\) with scale \(D(A)\). For \(D\subseteq D(A)\) a generating set of \(G(A)\), let \(\mathcal P\) be the set of all formal inequalities \(\alpha_1+\cdots+a_k\leq b_1+\cdots+b_l\) satisfied by \(G(A)\) for any \(a_i,b_j\in D\). By Elliott’s classification, \(\mathcal P\) together with the list of all sums \(a_1+\cdots+a_k\in D(A)\) uniquely determines \(A\). Can \(\mathcal P\) be Gödel incomplete, i.e., effectively enumerable but undecidable? We give a negative answer in the case when \(D\) is finite, and a positive answer in the infinite case. We also show that the range of the map \(A\mapsto D(A)\) precisely consists of all countable partial abelian monoids satisfying the following three conditions: (i) \(a+b=a+ c\Rightarrow b = c\), (ii) \(a+b=0\Rightarrow a= b =0\) and (iii) \(\forall a, b\in E\) \(\exists c\in E\) such that either \(a+c= b\) or \(b+c = a\).