Union of infinite sets and infinity

Question

Note that if $a \in \cup^n_{i=1} A_i$ then clearly $a\in A_j$ for some integer $j$ (This follows from the definition of the union of finite sets). But how can we be so sure that this holds for an infinite union? May someone please explain this to me? Because I know that intuition breaks down while dealing with infinity, so can someone clarify?

Answer 1

Nothing new happens in the case of infinite unions. An element belongs to the union of a family of sets if and only if it belongs to at least one of them.

Answer 2

That's a definition of union (and of symbol $\cup$) - union of family of sets is by definition set that includes any element from any set from this family, and nothing else. Formally, if we have some set $B$ of sets (like $B = \{A_1, \ldots, A_n\}$ in fintie case), then by definition $C = \bigcup_{A \in B} A$ means $\forall x\colon x \in C \leftrightarrow (\exists A \in B\colon x \in A)$.

Answer 3

Actually, a little care is required here. As long as your collection of sets is itself a set, then its union is also a set. That's the Axiom of Union. But if your collection of sets is so "big" that it's not a set (say, the proper class of all sets), then its union will not be a set either and in that sense can be said not to exist.

So, for instance, if you had some collection of sets that was indexed by all ordinals or all cardinals (of arbitrary size), then you would not necessarily know that the "union" of that collection was actually a set.