Category: mathematics
Associated thinker(s): David Hilbert; George Gamow
Original publication or approximate period: Hilbert’s lectures (1924–25); popularized by Gamow (1947)
Difficulty: Beginner
Type: thought experiment
Themes: infinity; cardinality; bijections
Related experiments: st-petersburg-paradox; bertrands-paradox
A hotel has rooms numbered by all positive integers: 1, 2, 3, and so on, with no last room. Every room is occupied. A new guest arrives. Ask each existing guest in room n to move to room n + 1. Room 1 is now vacant and no existing guest lacks a room.
A bus carrying countably infinitely many guests can also be accommodated. Move each existing guest from n to 2n and give the arriving guest numbered n room 2n − 1. Each old guest has a distinct even-numbered room, and each newcomer has a distinct odd-numbered one.
Why can an infinite collection have the same cardinality as a proper part of itself?
The scenario makes a definition visible. Two sets have the same cardinality when their elements can be paired one-to-one, even if one set is a proper subset of the other. Finite intuitions about “full” combine different ideas: every place is occupied, and no rearrangement can create a vacancy. Those ideas come apart for infinite sets.
Set-theoretic interpretation: the room maps are explicit bijections or injections demonstrating countable cardinality. There is no contradiction in standard infinite-set mathematics. Physical interpretation: whether an actual universe could contain such a hotel involves additional claims about matter, space, time, and possibility. Mathematical coherence alone neither constructs a real hotel nor proves every proposed physical infinity possible.
There is no final guest who falls off the end. Moving guests successively is not needed for the mathematical mapping to be well-defined. The example does not show that every infinity is the same size; some sets cannot be listed by positive integers. It also does not make ordinary subtraction of infinite quantities behave like subtraction of finite numbers.
Countably many buses can be handled by enumerating pairs of positive integers. Cantor’s diagonal argument shows why a hotel indexed by positive integers cannot house every member of an uncountable set.
The IAS article documents Hilbert’s lectures and Gamow’s later popularization. The algebraic maps are original explanatory calculations. No claim about the physical impossibility or actuality of infinite populations is inferred from the hotel alone.
Verified on 2026-10-30. Source coverage: [1] Hilbert’s 1924–25 lectures, Gamow’s popularization, room reassignment proofs. Reflection questions and illustrative numerical examples are original editorial material.