Hilbert's paradox

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The paradox stated by Hilbert illuminating the nature of infinite, but countable sets, where the number of rational numbers between zero and one is equal to the total number of rational numbers. The paradox is this: if a hotel with infinitely many rooms is full and another guest arrives, that guest can be accommodated by each existing guest moving from their current room to the room with the next highest number, leaving room 1 free for the new arrival. If an infinite number of extra guests arrived, they could be accommodated efficiently by each existing guest moving to the room whose number is twice their existing room number, leaving an infinite number of odd numbered rooms available for the new arrivals.

Subjects: Mathematics.

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