Paradoxes arise when one or more convincing truths contradiction either each other, clash with other convincing truths, or violate unshakeable intuitions. They are frustrating, yet beguiling. Many see virtue in avoiding, glossing over, or dismissing them. Instead we should seek them out, if we find one sharpen it, push it to the extreme, and hope that the resolution will reveal itself, for with that resolution will invariably come a dose of Truth.
History is replete with examples, and with failed opportunities. One of my favorites is Olber's paradox. Suppose the universe were filled with an eternal roughly uniform distribution of shining stars. Faraway stars would look dim because they take up a tiny angle on the sky; but within that angle they are as bright as the Sun's surface. Yet in an eternal and infinite (or finite but unbounded) space, every direction would lie within the angle taken up by some star. The sky would be alight like the surface of the sun. Thus, a simple glance at the dark night sky reveals that the universe must be dynamic: expanding, or evolving. Astronomers grappled with this paradox for several centuries, devising unworkable schemes for its resolution. Despite at least one correct view (by Edgar Allen Poe!), the implications never really permeated even the small community of people thinking about the fundamental structure of the universe. And so it was that Einstein, when he went to apply his new theory to the universe, sought an eternal and static model that could never make sense, introduced a term into his equations which he called his greatest blunder, and failed to invent the big-bang theory of cosmology.
Nature appears to contradict itself with the utmost rarity, and so a paradox can be opportunity for us to lay bare our cherished assumptions, and discover which of them we must let go. But a good paradox can take us farther, to reveal that the not just the assumptions but the very modes of thinking we employed in creating the paradox must be replaced. Particles and waves? Not truth, just convenient models. The same number of integers as perfect squares of integers? Not crazy, though you might be if you invent cardinality. This sentence is false. And so, says Godel, might be the foundations of any formal system that can refer to itself. The list goes on.
What next? I've got a few big ones I'm wrestling with. How can thermodynamics' second law arise unless cosmological initial conditions are fine-tuned in a way we would never accept in any other theory or explanation of anything? How do we do science if the universe is infinite, and every outcome of every experiment occurs infinitely many times?
What impossibility is nagging at you?