Achilles and the Tortoise
Most famously, the paradoxes discuss Time as a continuum. If we have already laid out in our minds a notion of classical motion through a continuum, the infinite subdivisibility of Time must follow. But note the introduction here of a separation between object and background space. To the Eleatics, this is the source of the problem, not the mathematical necessity of infinity itself. By placing a fixed finite (relative to the observer) object in a continuum, we have allowed ourselves to ask questions about its motion which are physically unfeasible.
But the resolution comes not from concrete physical axioms about an objective reality, based as they are on the very prejudices that lead to paradoxes in the first place. Rather, it comes from refining the mathematics until its definitions are capable of quantitatively describing the physical problem correctly. We have known this for thousands of years, but do many physicists really appreciate this today?