Let $M$ be an $R$-module which is equal to a direct sum of simple $R$-submodules:
$M=\bigoplus_{i \in I} S_{i}$ where each $S_{i}$ is simple.
Then if $M$ is Artinian then $M$ is Noetherian. I'm reading the proof and there is a part which says note that the index set must be finite, otherwise if $M=S_{1} \oplus S_{2} \oplus S_{3} ...$ then we can find a descending chain which does not terminate.
Question: isn't this assuming that if $I$ is infinite then $I$ is countable? by writing $M$ as $M=S_{1} \oplus S_{2} \oplus S_{3} ...$ doesn't this implies that $I$ is countable? Why we can assume this? we can always take a countable subset of an infinite set but how can we guarantee that $M$ can still be written as the direct sum of the simple modules which belong to this countable subset?