Thesis The actual infinite is a useful fiction, not something real
Where it stands
Aristotle, in Physics III, allowed only the potential infinite: you can always count one more, but there's never a completed infinite totality. For two thousand years, most mathematicians agreed.
Then Cantor, in the late nineteenth century, treated infinite sets as completed objects and showed, with his diagonal argument (1891), that some infinities are larger than others. Kronecker opposed him fiercely. Hilbert later declared that no one would drive mathematicians out of the paradise Cantor had created.
I teach Cantor's argument and my students love it. But I'm tempted by Aristotle. Nothing in the physical world is actually infinite as far as we know. Infinite sets seem to me a hugely useful idealisation, like a perfect circle, not a description of anything real.