An Alternative Proof That The Real Numbers Are Uncountable

Summary

In this video, Up and Atom revisits a proof about the uncountability of real numbers, aiming to clear previous confusion. The speaker outlines a more accessible presentation of the proof, which involves demonstrating that any list claiming to enumerate all real numbers is inherently incomplete. This is achieved through a strategic process involving intervals on the real line and alpha-beta series which converge under certain conditions. By covering various scenarios where convergence either does or does not occur, the proof showcases why any attempt to list all real numbers falls short, reinforcing their uncountable nature.

Highlights

• The video revisits a proof about real numbers' uncountability to clear previous confusion 📚.
• Introduces a process of shrinking intervals using alpha-beta series for better comprehension 🔄.
• Explains how different converging scenarios affect the completeness of the real number list 🤔.
• Elaborates on why infinite series' convergence impacts any claim to enumerate all real numbers 🌌.
• A reminder of the abstract nature of real numbers and why they defy complete listing 🔢.

Key Takeaways

• A renewed look at proving real numbers can't be fully listed 🧙‍♂️.
• Simplified understanding of alpha-beta series in proofs ✨.
• Why listing all real numbers is impossible 🚫.
• Discovering gaps in the numbers using intervals 🔍.
• Demystifying infinite steps and their relation to completeness ♾️.

Overview

In the video, Up and Atom tackles a complex mathematical proof demonstrating why real numbers are uncountable. This follow-up video clarifies confusion from a previous attempt by offering a more detailed explanation. By cleverly using alpha and beta series within shrunken intervals, the host unfolds the proof step-by-step.

The proof dives into how an assumed complete list of real numbers is tested against these intervals. By studying where series converge to a point, the video explains why a missing number would indicate an incomplete list. The intricate relationship between intervals and series in the proof is central to understanding real numbers' uncountability.

Key scenarios include cases where intervals converge infinitely or finitely, showing why real numbers can't be perfectly enumerated. Up and Atom concludes by comparing this to rational numbers through a reference to ratio impossibilities, emphasizing real numbers' unique nature. With sources linked for deeper exploration, this video is a thorough exploration of a classic mathematical mystery.

Chapters

• 00:00 - 00:30: Introduction and Apology In this introductory chapter titled 'Introduction and Apology,' the narrator reflects on a previous video they uploaded about the concept of infinity, specifically addressing Cantor's theorem on different sizes of infinity. They acknowledge confusion among viewers due to their attempt to quickly summarize complex proofs on the uncountability of the real numbers, leading to some critical details being missed. The chapter serves as an apology and sets up a deeper dive into the topic to clarify these missing details in future content.
• 00:30 - 01:00: Proof Outline The chapter, titled 'Proof Outline', addresses the feeling of frustration when an understanding of a proof is incomplete. The narrator acknowledges this frustration and aims to rectify it by providing a comprehensive explanation of the proof from start to finish, with the hope of addressing any lingering confusion or questions. The chapter sets up the scenario where someone claims to have listed all real numbers, and the objective is to demonstrate that their list is missing a real number. Before delving into specifics, the chapter promises to outline the overall strategy to avoid further confusion.
• 01:00 - 01:30: Interval Creation and Shrinking This chapter discusses the method of interval creation and shrinking on the real line. Starting with a random interval, the goal is to demonstrate, through interval shrinking, that if this process does not converge to a specific number, all real numbers cannot be listed. If the interval does converge, it must imply either a need for an infinite amount of time to do so, meaning the converged number is not on the list, or it is achieved within a finite timeframe.
• 01:30 - 03:00: Convergence Series Explanation The chapter explores the idea of a convergence series by explaining a method to create and shrink intervals on the real line. It starts with selecting two numbers, alpha 1 and beta 1, and then consistently finding new pairs of numbers, alpha n and beta n, from the list that fall between the previously selected numbers. This iteration is aimed at demonstrating how a sequence or series converges by gradually narrowing down the interval to highlight missing real numbers from the list.
• 03:00 - 04:00: Finite Steps Convergence Scenario The chapter discusses the concept of convergence in finite steps by examining the sequences alpha and beta. If no number is found between alpha n and beta n, the process halts. Discovering only one number between the two sequences results in equal subsequent terms. The chapter emphasizes the convergence limits of the alpha and beta sequences, called alpha infinity, illustrating the end goal of this mathematical process.
• 04:00 - 05:00: Infinite Steps Convergence Scenario The chapter discusses the concept of infinite steps and convergence scenarios, focusing on mathematical series. It points out that to prove convergence, one must demonstrate that the series converge, but for this discussion, the proof is assumed. The text highlights that the limits of these series (denoted as alpha infinity and beta infinity) must be the same; otherwise, there would be additional real numbers in between. It concludes that the interval must converge to a single point, meaning both alpha infinity and beta infinity represent this point.
• 05:00 - 08:00: Recap of Critical Cases In this chapter, the discussion focuses on 'eta,' a unique number. Two scenarios are presented: one where a series converges to eta in a finite number of steps, and another where it converges in an infinite number of steps. It's shown that in both scenarios, some real numbers are missing from the list, serving as a key point. For the scenario where convergence takes a finite number of steps, the specific step count is labeled as 'm'. The alpha and beta terms are considered prior to reaching eta, indicating a comparison or calculation approach.
• 08:00 - 09:00: Self-Referential Argument Explanation In this chapter, the concept of self-referential arguments is explored through the lens of real numbers. The discussion begins with the properties of real numbers, emphasizing that an infinite number of real numbers exists between any two given real numbers, like alpha m-1 and beta m-1. The narrative then posits a scenario where a specific real number, eta, is identified, suggesting that this could imply a disregard for other real numbers between alpha m-1 and beta m-1, apart from eta. It further explores the idea of series converging to eta within a finite number of steps and the implications of eta being on a particular list, hinting at a deeper philosophical or mathematical question about completeness and enumeration of real numbers.
• 09:00 - 10:00: Conclusion and References This chapter concludes with a discussion on the convergence process, focusing on a hypothetical scenario involving infinite steps required for convergence to a point labeled as eta. It poses a question regarding the placement of eta on a theoretical list of numbers, emphasizing the paradox of infinity in practical terms. The transcript highlights a key point about infinite sequences and convergence, questioning the idea of assigning a natural number to eta and prompting readers to consider the limitations of finitely definable processes.

An Alternative Proof That The Real Numbers Are Uncountable Transcription

• 00:00 - 00:30 Hey guys, how's it going? I uploaded a video last week called "my infinity is bigger than your infinity" all about the different sizes of infinity and Cantor sets. In it I went through two proofs about why the real numbers are not infinitely countable or enumerable and there was a lot of confusion about the first one. So the reason for the confusion was my fault. I tried to sketch out the proof but I also didn't want the video to be too long so I tried to go through it quite quickly but in doing that I left out some major details and
• 00:30 - 01:00 basically gave you an incomplete proof. I know there's nothing more frustrating than feeling like you didn't understand something, especially when it's not your fault, so I decided to make this video going through the proof again from start to finish and hopefully that will clear any of the confusion and questions that you have. So here we go. Suppose someone says they have a list of all the real numbers. You want to step on their dreams and show them that their list is missing a real number. Before we get into the details of how we're going to do this, to avoid confusion I want to give the overall strategy of what we're doing.
• 01:00 - 01:30 Starting from an arbitrary interval on the real line the goal will be to shrink that interval by using the real list in such a way that we can show that if the process of shrinking doesn't converge to a number then all the reals can't be on the list. Then we'll show that if it does converge to a number then either it must take an infinite amount of time to do so, meaning that the number the interval converges to can't be on the list, or it takes a finite amount of time to
• 01:30 - 02:00 converge, meaning that the list must be missing some other real numbers. If that made absolutely no sense, don't worry, it will, I hope. All right let's get started. Here we're defining a process for creating and shrinking an interval on the real line. Start with two real numbers from the list, alpha 1 and a larger number beta 1. Now find the first two numbers from the list that are between alpha 1 and beta 1, alpha 2 and beta 2. Keep going with this pattern. For any alpha and beta n find the first two distinct numbers from the list that are
• 02:00 - 02:30 between them, alpha and beta n plus 1. If you don't find a number the process stops. If you only find one number between alpha n and beta n, alpha n plus one is equal to beta n plus one. Now we have two series, the alpha series and the beta series. We can take the limit of those series, in other words what the alpha and the beta series converge to, and call them alpha infinity
• 02:30 - 03:00 and beta infinity. In a full mathematical proof we would have to show that the series do in fact converge, but just take my word for it. Now these limits have to be the same number. If there were a gap between alpha infinity and beta infinity then there would be reals in between, for example alpha infinity plus beta infinity divided by 2. So then the interval must converge to a point, which means alpha infinity is equal to beta infinity. We'll call this number
• 03:00 - 03:30 eta, the funny-looking n. Now we're going to show two scenarios, one where the series' converge to eta in a finite number of steps, and one where it takes an infinite number of steps. And then show that both scenarios mean there are some reals missing from the list. Suppose this real number eta is on the list, which means it takes a finite number of steps to converge to it. We'll label this finite number of steps m. Now look at the alpha and beta terms just before eta,
• 03:30 - 04:00 alpha m-1 and beta m-1. According to how the reals are defined there should be an infinite amount of real numbers between them, like any weighted average between the two. So then if we do find the real number eta, that means we're missing all the real numbers between alpha m-1 and beta m-1 except eta, so in the case where the series' converge to eta in a finite number of steps and eta is on the list, there are real numbers
• 04:00 - 04:30 that aren't. So then what if it takes an infinite amount of steps to converge to eta? Well that means there are an infinite amount of alphas and betas that the process picked before it. Because of how we defined our process they must have come before it on the list. To put it another way, say you asked your friend "What natural number does eta correspond to on the list?", what would he say? Infinity isn't an answer and if he gives any natural number like 300 billion and 49, you can
• 04:30 - 05:00 say, "Well no it's not because there are an infinite number of alphas and betas before it". Contrasting this to any enumerable set like the rationals, for any number, your friend would be able to tell you the natural number that it corresponds to. In the previous video we showed how every element in an infinitely enumerable list could be mapped to the set of natural numbers. For the set of reals we've shown this can't hold, so the real number eta cannot be
• 05:00 - 05:30 on the list, therefore the list is never complete and the real numbers are non enumerable. So just to recap what we've done. There are three critical cases we analyzed to show that the argument works. Case 1: the alpha series and the beta series don't converge to the same limit, but this means we're missing any real number in between the alpha and beta limit, so in this case the list is missing reals. Case 2: the series' converge to the same limit eta in a finite number of steps m, but this means that the alpha and beta
• 05:30 - 06:00 before eta alpha m-1 and beta m-1, which must be distinct, have an infinite number of reals in between them which can't be on the list, otherwise the process wouldn't have converged to the limit eta yet. So in this case there are also reals that aren't on the list. Case 3: it takes an infinite amount of steps to converge to the limit eta, but then there needs to be an infinite amount of alphas and betas on the list before it, so eta itself cannot be on the
• 06:00 - 06:30 list. But why doesn't this argument work for enumerable steps like the rationals? Well we're generating this number eta in a pretty complicated way. We're taking the limit of two infinite series' and saying this number is what they converge to, but the way the real numbers are defined means we could prove that eta is real. But for the argument to work with rationals we would have to show that eta is a rational number, but it turns out to be impossible to show that eta can be expressed as a ratio of two integers. We need a more expressive number system to
• 06:30 - 07:00 include this abstractly defined number. What makes this argument so powerful is that it's a self referential argument, which basically uses the list to generate a number that can't be on it. So yeah that's the complete proof. If you still have questions I've linked a bunch of sources in the description. Again sorry for the initial confusion and thank you for your patience. See you in the next video. Bye!
• 07:00 - 07:30