A formalist might say that mathematics has no infinities. Even infinite sets are just definitions of finite length, even if expanded all the way to ZFC axioms. The problem is your intuitive understanding.
A realist might say that mathematics exists within this universe, therefore it is equally subject to its constraints just like anything else. Problems arise via misapplication or misinterpretation of the math.
A model theorist might point out that even in models of math with only a countable collection of objects, it's still true that the reals are uncountable. The problem isn't with infinitie, it's that the finitary objects have subtle interactions.
A logician might object that ZFC already does start of with an intuitive notion of infinity, i.e. the counting numbers is a natural collection. The problem is that this inevitably has unintended consequences.
A historian might gently point out that this debate already occurred vigorously about 150 years ago, and the verdict was that we just gotta live with the weirdness of infinities. The problem is that we lose too much useful math by trying to throw them out.
Etc.