TITLE: Terminology: arbitrary vs. finite
QUESTION [7 upvotes]: How is arbitrary different from finite in mathematics? An arbitrary union of subsets and a finite union of subsets look the same to me! Is an infinite union of subsets arbitrary?

REPLY [8 votes]: The word "arbitrary" does not specify a property of the thing it applies to. It is merely a rhetorical way to call attention to the fact that what you say is not restricted to things with any particular property.
It's still left to the reader to figure out from context what the thing you're calling attention to not saying is, but that is okay, because the word "arbitrary" has no formal effect.
So saying that "such-and-such is true for arbitrary unions" formally means neither more nor less than saying that "such-and-such is true for unions", period.
When we say "let $A$ be an arbitrary widget", we're not formally saying anything different from "let $A$ be a widget" -- just making sure that the reader notices that the only thing we know about $A$ is its widgetness. We can follow up with "remember that $A$ was arbitrary ..." which again just mean "remember that we didn't assume anything in particular about $A$".