The Mathematics of Infinity – Hilbert’s Hotel

David Hilbert’s paradox of the Grand Hotel has infinitely many rooms, all occupied. A new guest arrives – move each existing guest from room n to room n+1, freeing room 1. Even an infinite bus of new guests? Move each from n to room 2n, freeing all odd rooms. An infinite number of infinite buses? Use prime powers. The hotel accommodates countably infinite guests despite being full. This illustrates that infinite sets have different cardinalities: the natural numbers are countable; the real numbers are uncountable (Cantor’s diagonal argument).