Exercise 417Logic, sets and proofs › Countability and cardinalityHigh schoolAdded 3 Nov 2026

Hilbert's hotel

#Hilbert hotel#infinite#bijection

Statement

A hotel has countably infinite rooms, all occupied. Show that it can accommodate countably many new guests.

Hint

Shift existing guests from room $n$ to room $2n$ , put new guests in odd rooms.

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Shift. Guest in room $n$ moves to $2n$ .

New guests. New guest $k$ gets room $2k-1$ .

Bijection. Every room is filled: evens for old, odds for new.

Conclusion. A countably infinite hotel can accommodate countably many new guests: $\aleph_0+\aleph_0=\aleph_0$ .