  We analyze $\mathrm{C}^\ast$-algebras, particularly AF-algebras, and their
$K_0$-groups in the context of the infinitary logic $\mathcal{L}_{\omega_1
\omega}$. Given two separable unital AF-algebras $A$ and $B$, and considering
their $K_0$-groups as ordered unital groups, we prove that $K_0(A)
\equiv_{\omega \cdot \alpha} K_0(B)$ implies $A \equiv_\alpha B$, where $M
\equiv_\beta N$ means that $M$ and $N$ agree on all sentences of quantifier
rank at most $\beta$. This implication is proved using techniques from
Elliott's classification of separable AF-algebras, together with an adaptation
of the Ehrenfeucht-Fra\"iss\'e game to the metric setting. We use moreover this
result to build a family $\{ A_\alpha \}_{\alpha < \omega_1}$ of pairwise
non-isomorphic separable simple unital AF-algebras which satisfy $A_\alpha
\equiv_\alpha A_\beta$ for every $\alpha < \beta$. In particular, we obtain a
set of separable simple unital AF-algebras of arbitrarily high Scott rank.
Next, we give a partial converse to the aforementioned implication, showing
that $A \otimes \mathcal{K} \equiv_{\omega + 2 \cdot \alpha +2} B \otimes
\mathcal{K}$ implies $K_0(A) \equiv_\alpha K_0(B)$, for every unital
$\mathrm{C}^\ast$-algebras $A$ and $B$.
