The Grand Hotel Paradox, also known as Hilbert’s paradox of the Grand Hotel, is a thought experiment proposed by renowned mathematician David Hilbert. It challenges the concept of infinity and the property of infinite sets.
David Hilbert
David Hilbert was a renowned German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. He discovered a range of fundamental mathematical ideas, including commutative algebra, the calculus of variations, spectral theory, and the foundations of geometry.
The Paradox
In the Grand Hotel Paradox, Hilbert imagines a hotel with infinite rooms. Initially, every room in the hotel is occupied by a customer. However, new visitors arrive at the hotel expecting their own room. Hilbert's model imagines the existing guests simultaneously moving rooms when new guests arrive.
With every new guest, the original guest in room 1 moves to room 2, room 2 to room 3, and so on, with the guest originally in room n moving to room n+1. The original hotel is filled with n rooms, and the n+1 room is still within the infinity range. Therefore, the infinite hotel has no final room, so every guest can be accommodated. After this, room 1 is empty, and a new guest can be accommodated. With this pattern, when x guests seek a room, the hotel's infinite property simply expands towards n+k rooms, able to accommodate any new incoming guests in Hilbert's model.
A normal, finite hotel wouldn't be able to accommodate any more incoming guests. However, Hilbert's hypothetical infinite hotel shows that existing guests and incoming guests can be accommodated even though all rooms have initially been filled. This challenges the concept of infinity.
Conclusion
The Grand Hotel Paradox challenges our comprehension of 'infinity' and enhances our understanding of mathematical concepts. It is important to continually question and expand our assumptions and knowledge, pushing the boundaries of mathematics.
Works Cited
OpenLearn from The Open University. “Hilbert’s Infinite Hotel - 60-Second Adventures in Thought (4/6).” YouTube, 3 Oct. 2011, www.youtube.com/watch?v=faQBrAQ87l4. Accessed 6 Feb. 2024.
Stanford Encyclopedia of Philosophy. (2021) "Hilbert's Hotel." In: Stanford Encyclopedia of Philosophy. Available at: https://plato.stanford.edu/entries/paradoxes-contemporary-logic/#HilHot Accessed: February 4, 2024
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