[atomack]

One point regarding whether or not this is unphysical is that in physical systems, conceptually we work in finite systems and at the only very end of our calculations we take the infinite limit while holding physically observable quantities fixed. This is the essence of the thermodynamic limit.

In these paradoxes, infinities are present from the outset and I think it's this that leads to the unphysical outcome. They're not wrong. They are mathematical paradoxes. But it's not a problem they are unphysical (from a physics point of view) because physics uses mechanisms, like the thermodynamic limit, to handle infinite limits sensitively. Then the paradox goes away.

For instance, the 'physics' version of the Hilbert hotel problem would say there are two hotels, one with N rooms and the other with M rooms. Then do all the renumbering you like, the paradoxical situation of filling both hotels and then putting all guests from both hotels into one of them is no longer possible. Finally, if you want to think about hotels with an infinite number of rooms take N and M to inifinity keeping N/M fixed

Edit: add physicified version of Hilbert hotel problem