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Hilbert’s Grand Hotel Paradox

On a dark desert highway a tired driver passes one more hotel with a “No Vacancy” sign. But this time the hotel looks exceedingly large and so he goes in to see if there might nonetheless be a room for him:
The clerk said, “No problem. Here’s what can be done –
We’ll move those in a room to the next higher one.
That will free up the first room and that’s where you can stay.

The mathematical paradox about infinite sets associated with Hilbert’s name envisages a hotel with a countable infinity of rooms, that is, rooms that can be placed in a one-to-one correspondence with the natural numbers. All rooms in the hotel are occupied. Now suppose that a new guest arrives – will it be possible to find a free room for him or her Surprisingly, the answer is yes. He (or she) may be accommodated in room 1, while the guest in this room is moved to room 2, the guest in room 2 moves to room 3, and so on. Since there is no last room, the newcomer can be accommodated without any of the guests having to leave the hotel. Hilbert’s remarkable hotel can even accommodate a countable infinity of new guests without anyone leaving it. The guests in rooms with the number n only have to change to rooms 2n, which will leave an infinite number of odd-numbered rooms available for the infinite number of new guests. What the parable tells us is that the statement “all rooms are occupied” does not imply that “there is no more space for new guests.” This is strange indeed, although it is not, strictly speaking, a paradox in the logical sense of the term. Yet it is so counter-intuitive that it suggests that countable actual infinities do not belong to the real world we live in.

Hilbert, Cantor, and the infinite.

The discussion of Hilbert’s hotel relates to the old question of whether an actual, as opposed to a potential, infinity is possible. According to Georg Cantor’s theory of transfinite numbers, dating from the 1880s, this is indeed the case, namely in the sense that the concept of the actual infinite is logically consistent and operationally useful [Dauben 1979; Cantor 1962]. But one thing is mathematical consistency, another and more crucial question is whether an actual infinite can be instantiated in the real world as examined by the physicists and astronomers. From Cantor’s standpoint, which can be characterized as essentially Platonic, numbers and other mathematical constructs had a permanent existence and were as real as nay, were more real than – the ephemeral sense impressions on which the existence of physical objects and phenomena are based. From this position it was largely irrelevant whether or not the physical universe contains an infinite number of stars. Contrary to some other contemporary mathematicians, including Leopold Kronecker and Henri Poincaré, Hilbert was greatly impressed by Cantor’s set theory. This he made clear in a semi popular lecture course he gave in Göttingen during the winter semester 1924-1925 and in which he dealt at length with the infinite in mathematics, physics, and astronomy. A few months later he repeated the message in a wide-ranging lecture on the infinite given in Münster on 4 June 1925. The occasion was a session organized by the Westphalian Mathematical Society to celebrate the mathematical work of Karl Weierstrass. Hilbert’s Münster address drew extensively on his previous lecture course, except that it was more technical and omitted many examples. One of them was the infinite hotel. “No one shall expel us from the paradise which Cantor has created for us,” Hilbert famously declared in his Münster address. On the other hand, he did not believe that the actual infinities defined by Cantor had anything to do with the real world.

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