The Hilbert Hotel, also known as Hilbert’s Paradox of the Grand Hotel, is a thought experiment in mathematics proposed by the German mathematician David Hilbert. It is used to illustrate some counterintuitive properties of infinite sets, particularly involving the concept of “countably infinite” versus “uncountably infinite” sets.
The scenario involves a hotel with an infinite number of rooms, each room numbered with the natural numbers (1, 2, 3, …). Even if all the rooms are occupied, the hotel can still accommodate more guests.
Here are some key points of the thought experiment:
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Countably Infinite Sets: The set of natural numbers (1, 2, 3, …) is countably infinite. This means that the elements of the set can be put into one-to-one correspondence with the natural numbers.
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Room Shuffling: Suppose every guest currently staying in the hotel moves to the room with double their current room number (e.g., the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, and so on). After this shuffling, all the odd-numbered rooms are vacant, so an infinite number of new guests can be accommodated, even though the hotel was previously full.
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Adding More Guests: If an infinite number of new guests arrives, each guest can be assigned a room number equal to twice the current room number of the guest occupying that room. For example, the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, the guest in room 3 moves to room 6, and so on. This process ensures that all the previous guests have even-numbered rooms, leaving all the odd-numbered rooms vacant for the new guests.
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Uncountably Infinite Sets: Despite accommodating an infinite number of new guests, the hotel remains “full” in the sense that it never runs out of rooms. This illustrates the concept of an uncountably infinite set, where the number of elements in the set is greater than the number of natural numbers.
The Hilbert Hotel is a famous example used in discussions about infinity, set theory, and the philosophy of mathematics. It highlights the counterintuitive nature of infinite sets and challenges our common-sense understanding of mathematical concepts.
