There is a widespread misconception that the lack of a real counterpart to the idea N is due to N being infinite. On the contrary, Cantor showed that the much 'larger' infinity 2N is isomorphic as an abstract set with the idealization 'imagine the set of all points in this room.' The latter is an idealization of something that we regard as really there, though of course we can't 'list' all the points in this room by any syntactic process.
F. William Lawvere and Stephen Schanuel, Conceptual Mathematics, (Cambridge 2009) p. 308.
