# Some infinities are bigger than others - simple proof | Joel David Hamkins and Lex Fridman Source: https://www.youtube.com/watch?v=nsTmrTKfjP0 Recap page: https://rapidrecap.app/video/nsTmrTKfjP0 Generated: 2026-01-03T17:33:24.528+00:00 --- ## Quick Overview The discussion confirms Cantor's diagonal argument proves that the set of real numbers is uncountably infinite, meaning it is strictly larger than the set of natural numbers, using a proof method that constructs a real number guaranteed not to be on any assumed enumeration list by differing in at least one decimal digit. **Key Points:** - Cantor's diagonal argument proves that the set of real numbers (R) is uncountably infinite, which is strictly larger than the set of natural numbers (N). - The proof relies on assuming an enumeration (a list) of all real numbers exists, $r_1, r_2, r_3, \dots$, and then constructing a new real number, $z$, that cannot be on that list. - The constructed number $z$ is made by ensuring its $n$-th decimal digit after the decimal point differs from the $n$-th digit of $r_n$ (e.g., if the digit is 4, $z$'s digit becomes 5, and if it is 9, $z$'s digit becomes 0). - This construction guarantees $z$ is different from every number on the list, leading to a contradiction, thus disproving the initial assumption that all real numbers could be listed. - The discussion notes that while some real numbers like $\sqrt{2}$ are irrational but algebraic, transcendental numbers like $\pi$ and $e$ are not algebraic. - The guest mentions that most real numbers are transcendental, and also points out the existence of multiple sizes of infinity, contrasting the countable infinity of rationals/integers with the uncountable infinity of reals. - The guest humorously notes that while the proof is elegant, the concept of the uncountability of reals is not controversial in modern mathematics, unlike some other complex topics. ![Screenshot at 09:23: The screen displays the core of Cantor's diagonal argument, showing a hypothetical enumeration of real numbers \($r\_1 = \\pi$, $r\_2 = e$, etc.\) and the construction of a new number $z$ by taking the diagonal digits, illustrating how $z$ is guaranteed to differ from every listed number.](https://ss.rapidrecap.app/screens/nsTmrTKfjP0/00-09-23.jpg) **Context:** This segment features Lex Fridman interviewing a guest, likely a mathematician (implied to be Joel David Hamkins from the title context, though not explicitly named in the provided clip), discussing the mathematical proof of the uncountability of the real numbers using Cantor's diagonal argument. The conversation delves into the comparison between countable sets (like natural numbers) and uncountable sets (like real numbers) and the nature of transcendental numbers. ## Detailed Analysis The discussion focuses on Cantor's diagonal argument, which proves that the set of real numbers is uncountably infinite, meaning its cardinality is strictly greater than the set of natural numbers. The guest explains the proof by contradiction: assume one can create a complete list of all real numbers, $r_1, r_2, r_3, \dots$. Then, construct a new real number, $z$, such that its $n$-th decimal digit differs from the $n$-th decimal digit of $r_n$. The construction rule mentioned is to change the digit (e.g., if it's 4, make it 5; if it's 9, make it 0). Because $z$ differs from every $r_n$ in at least one decimal place, $z$ cannot be on the list, contradicting the initial assumption that the list contained all real numbers. The guest notes that this proof method is highly influential and has been extremely fruitful in set theory. They also touch upon the classification of real numbers, noting that while $\sqrt{2}$ is irrational, it is algebraic, distinguishing it from transcendental numbers like $\pi$ and $e$, which are not roots of polynomial equations with rational coefficients, and stating that most real numbers are transcendental. The conversation concludes with an affirmation that this proof solidifies the fact that there are different sizes of infinity. ### Cantor's Diagonal Argument - Assuming an enumeration of all real numbers ($r_1, r_2, \dots$) exists - Constructing a new real number $z$ that differs from $r_n$ at the $n$-th decimal place - Concluding that $z$ is a real number not on the list, proving the initial list was incomplete and thus R is uncountable ### Number Classification - Real numbers include rationals (P/Q) and irrationals - Algebraic numbers (like $\sqrt{2}$) are roots of polynomial equations with integer coefficients - Transcendental numbers (like $\pi$ and $e$) are not algebraic, and most reals are transcendental ### Implications of Uncountability - The set of real numbers is strictly larger than the set of natural numbers - The proof method is fundamental to modern set theory and mathematical logic - The concept implies there are different 'sizes' of infinity ![Screenshot at 00:02: Lex Fridman Podcast intro screen showing the host against a space background.](https://ss.rapidrecap.app/screens/nsTmrTKfjP0/00-00-02.jpg) ![Screenshot at 00:04: Lex Fridman initiating the question about uncountable infinities.](https://ss.rapidrecap.app/screens/nsTmrTKfjP0/00-00-04.jpg) ![Screenshot at 00:09: The guest begins explaining the concept of counting the real numbers.](https://ss.rapidrecap.app/screens/nsTmrTKfjP0/00-00-09.jpg) ![Screenshot at 03:19: A slide appears explicitly showing the formula for the Liouville Constant, demonstrating a concrete transcendental number.](https://ss.rapidrecap.app/screens/nsTmrTKfjP0/00-03-19.jpg) ![Screenshot at 08:56: A slide illustrating Cantor's diagonal argument, listing example real numbers \($\\pi, e, \\sqrt{2}, \\ln\(2\), \\sqrt{29}$\) and highlighting the diagonal digits used to construct the new number $z$.](https://ss.rapidrecap.app/screens/nsTmrTKfjP0/00-08-56.jpg)