Unexpected difficulties in a practical measurement problem challenged fundamental conceptions of length and led to a new theory of dimension, while revealing monsters lurking at small scales.
What does it mean for two sets to have the same size? What does it mean, exactly, to say that a set is finite or that it is infinite? Can we provide a precise mathematical definition of finiteness?
The hotel is full, yet nevertheless can accommodate new guests, even infinitely many new guests. Thus the parable of Hilbert's Grand Hotel illuminates the nature of countable infinity.