The Infinite Hotel Paradox, or Hilbert's Hotel, illustrates the strange properties of infinity. Imagine a hotel with an infinite number of rooms, each occupied by a guest.

Scenarios

1. One More Guest:

  • To accommodate one new guest, move each current guest from room n to room n+1.
  • This frees up room 1 for the new guest.

2. Infinite New Guests:

  • Move each current guest from room n to room 2n.
  • This frees up all odd-numbered rooms for the infinite new guests.

3. Infinite Buses with Infinite Guests:

  • Assign current guests to rooms with numbers that are powers of a prime (e.g., 2, 4, 8, 16).
  • Use a systematic method to assign the new guests to the remaining free rooms.

Key Insights

  • Infinity’s Counterintuitive Nature: Infinite sets can still accommodate more elements, defying our usual understanding.
  • Countable Infinity: The hotel’s rooms are countably infinite, meaning they can be matched one-to-one with the natural numbers.
  • Infinite Subsets: Even infinite subsets can be as large as the set itself.

Conclusion

The Infinite Hotel Paradox reveals how infinity defies intuition and enriches our understanding of mathematical concepts related to infinite sets.