Technically, though, there aren't any paradoxes here.
The real reason why it is a paradox doesn't have much to do with notion of infinity per se, but with problem of infinite decision aka Supertask.
Which reminds me of an article on physics of supertasks that among other things somewhat outlines how classical theory is more paradoxical before than after quantization http://th-www.if.uj.edu.pl/acta/vol36/pdf/v36p2887.pdf (43pp but easy reading).