Hilbert's Hotel — infinity in practice

What just happened?

The hotel has infinitely many rooms (numbered $1, 2, 3, \dots$) and they are all occupied. In a finite hotel “full” means “no room”. But here it is enough to ask every guest to move one room further: the guest in room $n$ goes to room $n+1$. No one is left without a room — and room 1 has just been freed for the new guest.

The arithmetic of infinity:

Adding one element (or even infinitely many) to a countable infinity still gives the same countable infinity — denoted $\aleph_0$ (“aleph-null”, the smallest infinite cardinal number).

And if infinitely many new guests arrive?

They still fit. Every existing guest moves from room $n$ to room $2n$ — occupying only the even-numbered rooms. All the odd-numbered rooms ($1, 3, 5, \dots$, and there are infinitely many of them) now stand empty, ready for infinitely many new guests:

Where does this idea come from?

This thought experiment was devised by David Hilbert (lecture “Über das Unendliche”, around 1924) to illustrate the counterintuitive properties of infinite sets. It is a practical picture of what Georg Cantor had earlier proved formally: the set of natural numbers, of the even numbers alone, or of the natural numbers plus one element — all have the same cardinality $\aleph_0$ (countable infinity).

Why this matters for your exam:

Hilbert's Hotel is the clearest way to understand that infinity does not behave like an ordinary number — key to the notion of the cardinality of a set, countable sets, and the difference between $\mathbb{N}$ and $\mathbb{R}$.

Remember: in a hotel with countably infinitely many rooms room can always be made — because $\aleph_0$ “plus one”, or even “plus infinity”, is still $\aleph_0$.