# The smallest infinity in mathematics | Joel David Hamkins and Lex Fridman Source: https://www.youtube.com/watch?v=V4lhxOtqwjw Recap page: https://rapidrecap.app/video/V4lhxOtqwjw Generated: 2026-01-03T21:32:43.394+00:00 --- ## Quick Overview The discussion between Lex Fridman and Joel David Hamkins centers on the mathematical concept of countability within the context of different infinite sets, particularly demonstrating how the set of all pairs of natural numbers (the integer lattice) is countable, unlike the set of all real numbers, which is uncountably infinite. **Key Points:** - The discussion uses a visual zigzag path on a grid (the integer lattice) to prove that the set of pairs of natural numbers, $\mathbb{N} \times \mathbb{N}$, is countable. - The path starts at (0,0) and snakes through the grid, assigning a unique natural number to every pair (row, column), thereby establishing a one-to-one correspondence with $\mathbb{N}$. - Hamkins argues that this method provides a direct and intuitive way to count points in the lattice, contrasting with potentially overly arithmetic or convoluted counting methods. - The concept of countability is contrasted with uncountability, exemplified by the set of real numbers, which Cantor proved cannot be put into a one-to-one correspondence with the natural numbers. - The visual demonstration shows the path hitting every grid point in the upper-right quadrant, suggesting that even though the lattice is two-dimensional, it can be mapped onto a one-dimensional count (natural numbers). - Lex Fridman acknowledges the visual method as a very nice way to think about assigning room numbers to all points on the grid, thereby confirming their countability. ![Screenshot at 00:39: The graphic displays a coordinate grid with a green zigzag path starting at \(0,0\) and moving up and to the right, illustrating the method used to prove that the integer lattice points are countable by establishing a one-to-one correspondence with the natural numbers.](https://ss.rapidrecap.app/screens/V4lhxOtqwjw/00-00-39.jpg) **Context:** This segment is an excerpt from the Lex Fridman Podcast, featuring mathematician Joel David Hamkins. The conversation delves into foundational concepts of set theory and infinity, specifically focusing on the mathematical proof that the set of all ordered pairs of natural numbers (the integer lattice) is countable, meaning its elements can be enumerated, which is a crucial distinction from uncountable sets like the real numbers. ## Detailed Analysis Lex Fridman and Joel David Hamkins discuss the countability of the integer lattice, $\mathbb{N} \times \mathbb{N}$. Hamkins explains that one can establish a correspondence between the points on the grid and the natural numbers by following a specific path. He illustrates this path visually, showing a zigzagging line starting at the origin (0,0), moving diagonally up and right, then down and right, and repeating this pattern to cover every point on the grid. This systematic path ensures that every pair $(r, c)$ of natural numbers is eventually visited, meaning the set is countable. Hamkins notes that this method is superior to overly arithmetic approaches because it is a direct, visual way to enumerate the points, akin to assigning train cars sequentially. This concept is contrasted with the uncountability of the real numbers, which cannot be enumerated this way. Fridman agrees that this visual approach effectively demonstrates the countability of the lattice, allowing one to assign a unique 'room number' to every point. ### Introduction to Countability - Lex Fridman asks if it's possible to internalize a good intuition about the 'countably infinite' set of pairs of natural numbers. ### The Integer Lattice Proof - Hamkins explains that taking pairs of natural numbers defines the integer lattice, and he shows a graphic illustrating how to count these points. ### Visualizing the Path - The visual aid shows a zigzag path starting at (0,0) and moving diagonally up and right, then down and right, covering every grid point in the upper-right quadrant. ### Interpretation of the Path - Hamkins explains that this path ensures every grid point (every pair of natural numbers) is hit, giving a correspondence between the lattice points and the natural numbers, proving countability. ### Contrast with Uncountability - The discussion touches upon the fact that this method does not work for the real numbers, which form an uncountably infinite set. ### Conclusion on Method - Fridman praises the visual method as a 'really nice visual way to think about it' for including everyone in the enumeration process. ![Screenshot at 00:03: Lex Fridman in the podcast studio, wearing a black suit and tie, speaking into a microphone.](https://ss.rapidrecap.app/screens/V4lhxOtqwjw/00-00-03.jpg) ![Screenshot at 00:19: The guest, Joel David Hamkins, in a light grey suit and bow tie, begins to speak about the concept of infinite sets.](https://ss.rapidrecap.app/screens/V4lhxOtqwjw/00-00-19.jpg) ![Screenshot at 00:39: A graphic comparing an empty 5x5 grid with a second grid showing a green zigzag path, illustrating the method for counting integer lattice points.](https://ss.rapidrecap.app/screens/V4lhxOtqwjw/00-00-39.jpg) ![Screenshot at 01:29: Close-up on the graphic showing the 'Start' point at \(0,0\) and the ongoing green path traversing the grid diagonally.](https://ss.rapidrecap.app/screens/V4lhxOtqwjw/00-01-29.jpg) ![Screenshot at 02:31: Lex Fridman using hand gestures while discussing the visual zigzag method for enumerating the lattice points.](https://ss.rapidrecap.app/screens/V4lhxOtqwjw/00-02-31.jpg)