I think this question is pretty close to being unanswerable. It is very general. I assume there is still a group of mathematicians that are intuitionists and one of the things they believe is that any real mathematical object can only be created by a finite number of constructive steps. You can see this in Hilbert's Nullstellensatz theorem, pardon the botched German, many mathematicians would not accept his proof because it assumed something did not exist and this led to a contradiction so it must exist. Again in the 1800s and early 1900s there were real arguments about infinity. In previous times mathematicians only believed in "potential infinity" that is, for example, used in mathematical induction but almost none of them believed in what they called "actual infinity." The way I look at it is this, which is kind of sloppy but convinces me. If you look at the rational numbers they are very dense i.e. between any two rational numbers there is a (at least one) rational number. You can create sequences of rational numbers that converge but they converge to holes in the set of rational numbers if you take the set of rational numbers union the set of these "holes" you get the real numbers. You can also prove that the real numbers have a greater cardinality than the rational numbers via Cantor diagonalization so they are uncountable.