Comment by shiandow

Well there's no law against it.

Okay I suppose the axiom of choice is somewhat necessary to make it make sense. But only because otherwise such an indexed object may fail to exist.

Anyway arbitrary indexes are useful, you often end up doing stuff like covering a space by finding a covering set for each individual point. And then using compactness to show you only need finitely many to cover the whole space. It is doable without uncountable indices, but it makes it very difficult to write down.