# Infinity Paradox: Infinity + 1 = ??? | Joel David Hamkins and Lex Fridman Source: https://www.youtube.com/watch?v=2Nb2GAahJIw Recap page: https://rapidrecap.app/video/2Nb2GAahJIw Generated: 2026-01-02T02:32:34.596+00:00 --- ## Quick Overview The union of two countably infinite sets, such as the set of natural numbers (representing hotel rooms) and the set of passengers on Hilbert's train (represented by pairs of car and seat numbers), remains countably infinite because the union of any two countable sets is countable, demonstrated by mapping the combined set back onto the natural numbers using prime factorization. **Key Points:** - The union of two countably infinite sets (like the natural numbers and the train passengers) is always countably infinite. - One scenario involves one new guest arriving at the already full Hilbert's Hotel (infinite rooms, numbered 0, 1, 2...), requiring every current guest in room 'n' to move to room 'n+1' to free up Room 0 (0:17). - A more complex scenario involves an infinitely long train arriving, where each car 'c' has infinitely many seats 's', requiring a mapping function to assign unique rooms: Room $3^c 5^s$ (1:53:53). - This mapping $f(c, s) = 3^c 5^s$ ensures every new passenger gets a unique room number because prime factorization is unique, and since $3^c 5^s$ is always odd, it maps to the odd-numbered rooms, leaving the even-numbered rooms for the original hotel guests (who move to $2n$) (2:01:14). - The core mathematical principle is that the union of two countable sets is countable, meaning the combined set can still be put into one-to-one correspondence with the natural numbers (4:33). - The speaker notes that this concept, which violates intuitive notions of size and infinity, is a remarkable property of infinity (8:53). ![Screenshot at 1:16: The screen displays the first scenario of Hilbert's Hotel Paradox where one new guest arrives, illustrating that the original guest in room 'n' moves to room 'n+1' to accommodate the new guest in Room 0.](https://ss.rapidrecap.app/screens/2Nb2GAahJIw/00-01-16.jpg) **Context:** This video features a discussion between Lex Fridman and mathematician Joel David Hamkins about the counter-intuitive nature of infinity, specifically exploring paradoxes related to Hilbert's Hotel, an infinitely large hotel that is fully occupied yet can always accommodate more guests. The conversation focuses on demonstrating how infinite sets (like the natural numbers) behave when combined with other infinite sets, such as an infinite number of new guests arriving on a train. ## Detailed Analysis The discussion centers on the properties of countable infinity, using the Grand Hilbert Hotel Paradox as the primary illustration. The first scenario shown is accommodating one new guest in a fully occupied infinite hotel (rooms $0, 1, 2, ext{...}$), solved by moving every existing guest from room $n$ to room $n+1$, freeing room 0 for the new arrival (0:17). The second, more complex scenario involves an infinite number of new guests arriving via an infinitely long train, where each car $c$ has infinitely many seats $s$. To accommodate these new guests while keeping the original guests (who move to even rooms $2n$), a bijective mapping onto the natural numbers is required. The solution presented maps each new guest $(c, s)$ to a unique room number given by $3^c 5^s$ (1:53:53). Since the prime factorization theorem guarantees that every positive integer has a unique factorization into primes (only 3s and 5s in this case), the resulting number $3^c 5^s$ is unique for every pair $(c, s)$. Furthermore, since $3^c 5^s$ is always an odd number, these new rooms are distinct from the even-numbered rooms assigned to the original hotel occupants, demonstrating that the union of two countably infinite sets is still countably infinite (4:01). The speaker emphasizes that this result—that you can add an infinite quantity to an already infinite quantity and still have the same 'size' of infinity—is a remarkable property of infinity that violates intuition (8:53). ### Hilbert's Hotel - Scenario 1 (One New Guest): Initial state is fully occupied (Room 0, 1, 2... have guests) - Solution: Move guest from room $n$ to room $n+1$ - New guest takes Room 0 (0:17) ### Hilbert's Hotel - Scenario 2 (Infinite New Guests via Train): Hotel rooms are numbered 1, 2, 3... (1:53:00) - Train has infinite cars 'c' and infinite seats 's' per car - Original guests move to even rooms ($n o 2n$) (2:01:14) ### Mapping New Guests - New guest $(c, s)$ maps to room $3^c 5^s$ - This mapping is unique due to unique prime factorization - Since $3^c 5^s$ is always odd, it preserves the one-to-one correspondence with the natural numbers (4:01) ### Mathematical Conclusion - The union of two countably infinite sets is countably infinite - This property is counter-intuitive and shows that $\aleph_0 + \aleph_0 = \aleph_0$ (8:53) ![Screenshot at 0:17: Illustration of Scenario 1: One new guest arrives, requiring all existing guests to shift rooms \($n o n+1$\) to open Room 0.](https://ss.rapidrecap.app/screens/2Nb2GAahJIw/00-00-17.jpg) ![Screenshot at 1:16: Visual representation of the Grand Hilbert Hotel Paradox showing the initial fully occupied state and the process for accommodating one new guest.](https://ss.rapidrecap.app/screens/2Nb2GAahJIw/00-01-16.jpg) ![Screenshot at 2:22: Lex Fridman describes the initial state of the hotel as fully occupied, illustrating the paradox setup.](https://ss.rapidrecap.app/screens/2Nb2GAahJIw/00-02-22.jpg) ![Screenshot at 3:52: Visual representation of Scenario 2: Infinite new guests arrive on Hilbert's Train, showing the mapping of old guests to even rooms \($n o 2n$\).](https://ss.rapidrecap.app/screens/2Nb2GAahJIw/00-03-52.jpg) ![Screenshot at 6:01: A complex diagram illustrating the second scenario where an infinite train arrives, showing the mapping function for new guests using prime powers \($3^c 5^s$\).](https://ss.rapidrecap.app/screens/2Nb2GAahJIw/00-06-01.jpg)