Finite sum over uncountable set

Question

Consider the sum $S=\sum_{x\in I}P(x)$, where $P(x)$ are positive real numbers. When the index set $I$ is finite, $S$ is of course finite. When $I$ is countably infinite, it is also possible that $S$ is finite. For example $\sum_{x\in \mathbb{N}}\frac{1}{2^x}$. Is it true that $S$ cannot be finite if $I$ is uncountable?