One of the reasons that these sort of paradoxes can exist is because real numbers can get arbitrarily small. In principle, you could build a horn that very closely matches Gabriel's horn, but it would necessarily be an approximation of the mathematical object because you do not have available particles of arbitrary size with which to build. This last notion hints at an important property of the real numbers: they can be divided into pieces smaller than any size you can specify. They are useful abstractions upon which to build extremely successful models of reality [1], because at those scales, real numbers work very well. However, at extremely small scales, the correspondence between real numbers and reality breaks down. What does it mean to model a horn made of atoms with numbers that are arbitrarily small fractions of the width of an atom? This is not a 'real numbers / axiom of choice bad' rant, but rather I'd like to point out that real numbers exist in our minds, and using our physical intuition about to reason about them at very small scales is bound to lead to paradoxes.

[1] Electromagnetism, classical mechanics, etc