Let $a \in \mathbb{C}$. Ahlfors says we let $a + \infty = \infty$ and $a \cdot \infty = \infty$. But we cannot define $\infty + \infty$ without violating the laws of arithimetic (i.e. field axioms).
I don't see why this is. Don't we have $\infty + \infty = \infty$ by applying the distributive law to $2\cdot \infty$? What am I misunderstanding?