To prove that a segment has the same number of points with half a segment one might say that one can find a bijective function mapping every point from the segment to the half segment. Let' say: $$\forall x\in(0,1), f(x)=x/2 $$ At first, it might seem that this argument holds because of the bijective function. However, the more I think about it, it seems that there is an implicit assumption being made when saying for any x in (0,1), because this might imply a way of choosing every point in the set, which cannot be necessarily taken for granted.
Of course, I understand that one simply assumes that $\forall x \in (0,1)$ is possible, but I can't shake off the feeling this might not be such a trivial assumption. Can you think of any field of mathematics or mathematical logic or philosophy which engages with this assumption?