The Infinite Hotel Paradox is a thought experiment that explores the concept of infinity and challenges our intuition about how infinite sets behave. It was first introduced by mathematician David Hilbert in the early 20th century.

The paradox starts with an imaginary scenario where there is a hotel with an infinite number of rooms, each of which is occupied by a guest. Now, suppose a new guest arrives and wants to check-in. Surprisingly, the hotel manager claims that they can accommodate the new guest without evicting any of the current guests, even though all rooms are occupied.

To achieve this, the hotel manager asks each guest to move to the next room number. For example, the guest in room 1 moves to room 2, the guest in room 2 moves to room 3, and so on. This way, the guest in room n moves to room n+1. Now, room 1 becomes available for the new guest to occupy.

This process can be extended to accommodate any finite number of new guests. For instance, if three new guests arrive, the manager can ask each guest to move two rooms down. So, the guest in room 1 moves to room 4, the guest in room 2 moves to room 5, and so on. Now, rooms 1, 2, and 3 are available for the new guests.

This paradoxical situation arises because infinite sets, such as the set of natural numbers (1, 2, 3, ...), do not behave in the same way as finite sets. In a finite set, if all elements are occupied, there is no room for additional elements. However, in an infinite set, even if all elements are occupied, there are still "infinitely more" elements available.

The Infinite Hotel Paradox has been used to illustrate various mathematical concepts, such as the nature of infinity, the difference between countable and uncountable infinities, and the concept of bijection (a one-to-one correspondence between two sets). It highlights the counterintuitive properties of infinite sets and challenges our everyday understanding of numbers and sets.

While the Infinite Hotel Paradox is a thought experiment, it has real-world applications in mathematics, particularly in set theory and the study of infinite cardinalities. It has also inspired philosophical discussions about the nature of infinity and its implications for our understanding of reality.

References:

1. Hilbert, D. (1924). On the Infinite. In Philosophy of Mathematics: Selected Readings (pp. 151-182). Cambridge University Press.
2. Rucker, R. (2005). Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton University Press.
3. Sierpiński, W. (1965). Cardinal and Ordinal Numbers. PWN-Polish Scientific Publishers.