Hilbert's Paradox of the Grand Hotel

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David Hilbert mengilustrasikan sifat aneh ketidakhinggaan: Grand Hotel dengan kamar 1, 2, 3, ... semuanya terisi. Tamu baru datang: minta tamu kamar 1 pindah ke 2, tamu kamar 2 ke 3, dst. Kamar 1 kosong untuk tamu baru. Tak terbatas bus berisi tak terbatas tamu datang: minta tamu kamar n pindah ke kamar 2n. Semua kamar bernomor ganjil kosong.

Ini menunjukkan bahwa ∞ + 1 = ∞ dan bahkan ∞ + ∞ = ∞ untuk kardinalitas countable. Secara formal, himpunan tak terhingga countable bisa dipasangkan satu-satu dengan subset sejatinya — properti yang menjadi definisi himpunan tak terhingga Dedekind.

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