Infinity and moreover a hierarchy of infinities is very real and has real consequences. You want your Fourier transform to work, you have to quantify infinities (this was in fact what prompted development of set theory: how to make Fourier analysis precise).
Supertasks are real too. If you take Turing machine something like the lamplighter group pops up at the other end and you have no way to handwave that away.
Unless of course you subscribe to Fictionalism as your philosophy of mathematics, in which case stuff still has consequences just your epistemological stance changes from studying them in favour to some kind of mysticism that absolves you from the effort.
Supertasks are real too. If you take Turing machine something like the lamplighter group pops up at the other end and you have no way to handwave that away.
Unless of course you subscribe to Fictionalism as your philosophy of mathematics, in which case stuff still has consequences just your epistemological stance changes from studying them in favour to some kind of mysticism that absolves you from the effort.