Hilbert's Grand Hotel: Difference between revisions
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Hilbert's Grand Hotel is a famous analogy and paradox used to explain the notion of [[countability]]. One starts off by imagining a hotel, with an infinite amount of rooms, and each is occupied. One's intuition says that it's not possible to fit any more people - however, due to the
Firstly, if there is a single new guest who wants a room, it is possible to accommodate
One can also accommodate countably infinitely many new guests, by requiring that every current guest in
In fact, it's even possible to
However, not every infinite batch of guests can fit in Hilbert's Grand Hotel. If a bus brings infinitely many guests whose names are all infinite strings made up of "a" and "b", and every string has a guest, not all of the guests can fit. In fact, it's possible to pair up each name to a real number, showing that there are more real numbers than natural numbers, even though there are infinitely many of both!
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