Hilbert's Hotel Paradox — Infinite Set Calculator
Hilbert's Hotel is a fully occupied hotel with infinitely many rooms — yet it can always fit more guests. Enter your room number and the number of new arrivals to see the exact room-shifting mechanism and explore what 'infinity + n = infinity' means concretely.
You move from room 7 to room 10 — shifting up by 3 to free room 1 for a new guest
- 1
Shift amount (new guests)
k = 3 = 3Every existing guest shifts up by k rooms, freeing rooms 1 through k - 2
Your new room number
7 + 3 = 10
How does this calculator work?
Hilbert's Hotel: a full infinite hotel can always take more guests. k new guests → every current guest in room n moves to n + k, freeing rooms 1..k. An infinite bus → current guests move to 2n, new guests take odd rooms. Both leave the hotel 'full' with the same infinite cardinality ℵ₀.
Formula
How this is calculated
Hilbert's Hotel, conceived by mathematician David Hilbert to illustrate the counterintuitive properties of infinite sets, is a thought experiment: imagine a hotel with infinitely many rooms numbered 1, 2, 3, … all fully occupied. Can it accommodate a new guest? Yes — the manager simply asks every current guest to move from room n to room n + 1. Room 1 is now vacant for the newcomer, and every existing guest still has a room. No guest is turned away; the hotel still has infinitely many occupied rooms.
With k new guests, every current guest shifts to room n + k, freeing rooms 1 through k simultaneously. This works for any finite k, however large. This demonstrates that adding a finite number to an infinite cardinal (aleph-null, ℵ₀) leaves the cardinality unchanged: ℵ₀ + k = ℵ₀.
More strikingly, even a countably infinite busload of new guests can be accommodated: move every current guest from room n to room 2n (even rooms), then assign new guests to odd rooms 1, 3, 5, 7, … The natural numbers can always be put in a one-to-one correspondence with a proper subset of themselves — which is in fact the defining property of an infinite set. This is why mathematicians say the set of natural numbers, even numbers, odd numbers, and prime numbers all have exactly the same infinite cardinality ℵ₀.
Frequently asked questions
The thought experiment is attributed to German mathematician David Hilbert, who used it in lectures around 1924 to illustrate properties of infinite sets. It was first published in 1947 by George Gamow in his popular science book 'One Two Three… Infinity'.
It demonstrates that a countably infinite set (one that can be put in one-to-one correspondence with the natural numbers) always has 'room' for finitely many or even countably infinitely many more elements. However, a hotel with uncountably infinitely many guests (the cardinality of the real numbers, ℵ₁) cannot be accommodated — that is the content of Cantor's theorem.
ℵ₀ (aleph-null or aleph-zero) is the cardinality of the natural numbers — the 'size' of a countably infinite set. It satisfies ℵ₀ + k = ℵ₀, ℵ₀ + ℵ₀ = ℵ₀, and even ℵ₀ × ℵ₀ = ℵ₀. The next larger infinity is ℵ₁, which equals the cardinality of the real numbers (the continuum).
Also known as
TG we-Calculate Editorial Team. (2026). Hilbert's Hotel Paradox — Infinite Set Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/hilberts-hotel-paradox-calculator
TG we-Calculate Editorial Team. "Hilbert's Hotel Paradox — Infinite Set Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/hilberts-hotel-paradox-calculator.
TG we-Calculate Editorial Team, "Hilbert's Hotel Paradox — Infinite Set Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/hilberts-hotel-paradox-calculator
@misc{wecalculate_hilberts_hotel_paradox_calculator, title = {Hilbert's Hotel Paradox — Infinite Set Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/hilberts-hotel-paradox-calculator}}, year = {2026}, note = {TG we-Calculate} }
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