TITLE: From an axiomatic set theoric approach why can we take uncountable unions?
QUESTION [11 upvotes]: From an axiomatic set theoric approach why can we take uncountable unions in the first place? Sure if $A$ and $B$ are sets then the thing denoted by $A\cup B$ exists and is a set, but if we want to extend this to an uncountable union shouldn't we assume some form of transfinite induction (and hence $\sf AC$)?

REPLY [4 votes]: No. All we do is assume the
Axiom of Union. If $A$ is a set then there exists a set $U$ such that
$$ \forall x\colon (x\in U\leftrightarrow \exists y\in A\colon x\in y).$$