Infinity Paradox: Infinity + 1 = ??? | Joel David Hamkins and Lex Fridman

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).

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.