Experiment #64 · Thought experiment

Hilbert's Hotel

A hotel with infinite rooms — always full, always with room

David Hilbert · 1924 (lecture); popularised by Gamow 1947 · Mathematics, philosophy of infinity

First published: D. Hilbert, lecture "Über das Unendliche" (1925); discussed in G. Gamow, *One Two Three… Infinity* (1947).

A hotel with countably infinite rooms, all occupied, can still accommodate one more guest — or infinitely many — by shifting everyone.

Hilbert's hotel dramatises the counterintuitive properties of actual infinity. With every room occupied, a new guest can be accommodated by moving the guest in room *n* to room *n+1* for all *n*; infinitely many new guests can be accommodated by moving each guest from room *n* to room *2n*. The "hotel" is always full and always has room. The case is a vivid pedagogical illustration of Cantor's arithmetic of infinite cardinals — and a frequent reference point in arguments (Craig, Oppy) over whether *actual* infinities can be physically instantiated.

Formulation

Hotel with rooms numbered 1, 2, 3, …; all occupied. New guest arrives: shift n → n+1 for all n, room 1 frees up. Infinitely many new guests arrive: shift n → 2n, all odd rooms free up. The set ℕ has the same cardinality as ℕ ∪ {0} and as 2ℕ.

Compare this experiment

Dimensions Engaged

◉

Matter

Bears on Matter · Extent: are actual infinities physically realisable, or are they at most idealisations?

✦

Space

Bears on Space · Extent: a finite, infinite, or unbounded universe each engages different versions of Hilbert's puzzles.

⧉

Information

Tests Information · Granularity: countable vs uncountable infinities of information states.

Responses — How Schools Engage

Affirms / takes the bait 2

Platonism (Classical)

Actual infinity is mathematically real; Hilbert's hotel correctly describes its properties. The strangeness reflects our finite intuitions, not a defect in the mathematics.

Pythagoreanism

Cantor's transfinite arithmetic vindicates a deep Pythagorean commitment: number governs reality at all scales, including infinite ones.

Denies / rejects the premise 1

Catholic/Thomistic

Aquinas-style finitism: actual infinities outside God are impossible. The hotel is consistent but unrealisable. (Craig's kalam cosmological argument descends from this position.)

Reframes the question 2

Analytic Metaphysics / Logical Atomism

The hotel illustrates that "consistent" and "intuitive" come apart; it does not by itself decide whether actual infinities can be physically instantiated.

Logical Positivism

The puzzle is mathematically legitimate; "actual infinity" outside formal mathematics has no clear empirical content without further specification.

Holds it inconclusive 1

Naturalism

Mathematics admits consistent infinite structures; whether the physical universe contains actual infinities is an empirical question (cosmological topology) that remains open.