The Infinity Problem that BROKE Mathematics

Summary

Highlights

• Cantor's idea that infinite sets have different sizes was groundbreaking and controversial. 🌌
• Gödel proved you can't disprove the continuum hypothesis, while Cohen proved you can't prove it within ZFC. 🧠
• This issue leads to philosophical questions about whether mathematics is discovered or invented. 🎓
• Hilbert's dream of a complete and consistent math system was shattered. 💥
• The mathematical community remains divided on the interpretation of mathematical truths. 🔍

Key Takeaways

• Gayo Cantor revealed that not all infinite sets are equal; there are different sizes of infinity. 🎢
• Kurt Gödel and Paul Cohen proved that the continuum hypothesis can neither be proven nor disproven within ZFC theory. 🤯
• The continuum hypothesis sparked debates on whether math is invented or discovered. 🤔
• Mathematicians confront the idea that logic and universal truths might have inherent limits. 🚧
• The mathematical community is divided between Platonism and Formalism regarding mathematical truth. ⚖️

Overview

Chapters

• 00:00 - 01:00: Introduction and The Infinity Problem The chapter "Introduction and The Infinity Problem" delves into a fundamental question in mathematics: determining which of two containers holds more items. This seemingly simple question leads to one of mathematics's most significant open problems, challenging our understanding of concepts like infinity, logic, and truth. It compels mathematicians to ponder the existence of universal truths and question if logic has limitations that prevent us from uncovering certain truths. The chapter sets the stage for readers to explore the profound journey mathematicians embarked on to tackle this problem.
• 01:01 - 04:00: One-to-One Mapping and Countable Infinities The chapter delves into the concept of one-to-one mapping and countable infinities, presenting a complex problem that challenged and broadened the understanding of mathematical reality.
• 04:01 - 07:00: Cantor's Discovery: Different Sizes of Infinity The chapter explores Cantor's discovery of different sizes of infinity. It begins by explaining the concept of a one-to-one mapping, which is used to determine equal quantities in mathematics. The analogy of pairing apples in containers is used to illustrate this idea: if each apple in one container can be paired with an apple in another container with none left over, then the two containers have an equal number of apples. However, if a one-to-one mapping cannot be established and some apples are left unpaired, it suggests that one container has more apples than the other. This foundational concept leads into discussions about varying sizes of infinity in Cantor's work.
• 07:01 - 10:00: The Continuum Hypothesis The chapter discusses the simplicity of certain mathematical concepts compared to more complex ones. It also introduces the natural numbers (1, 2, 3, 4, 5, etc.) as an example of a simple mathematical set, emphasizing that there are infinitely many natural numbers. The concept of sets is briefly introduced, describing them as collections containing objects like numbers.
• 10:01 - 15:00: Zermelo-Fraenkel Set Theory (ZFC) In this chapter on Zermelo-Fraenkel Set Theory (ZFC), the discussion explores the concept of comparing the size of infinite sets, specifically between the set of natural numbers and the set of even natural numbers. It challenges the intuitive assumption that the set of natural numbers might be larger, as the set of even numbers excludes all odd numbers. However, through the method of one-to-one mapping (pairing each natural number with an even number, e.g., 1 with 2, 2 with 4, etc.), it is demonstrated that both sets actually possess the same cardinality.
• 15:01 - 20:00: Gödel's Contribution and Incompleteness This chapter discusses Gödel's contribution to mathematics, particularly focusing on his incompleteness theorems. It starts with the concept of pairing natural numbers with even numbers, illustrating Cantor's discovery of the cardinality of infinite sets. The chapter delves into Gödel's theories, demonstrating that for any system capable of doing arithmetic, there are true mathematical statements that cannot be proven within the system itself. This highlights the limitations and boundaries of mathematical systems.
• 20:01 - 25:00: Paul Cohen and Forcing The chapter discusses the concept of infinity in mathematics and introduces Paul Cohen's work on set theory and forcing. It mentions a one-to-one mapping with the set of natural numbers and clever ways to map negative and positive integers, as well as rational numbers, to the natural numbers. It raises the question of whether all sets of infinite numbers are the same size, prompting further exploration into the nature of infinity.
• 25:01 - 30:00: Independence of the Continuum Hypothesis The chapter discusses the concept of the real number line, which is continuous and uninterrupted, contrasting it with the natural numbers that are discrete and have gaps between them. It explains the process of including various types of numbers like irrationals, algebraics, and transcendentals to eliminate any gaps, extending the line to encompass all numbers from negative infinity to positive infinity. The emphasis is on the continuum nature of the real number line, highlighting its difference from other sets of numbers.
• 30:01 - 40:00: Philosophical Interpretations: Platonism vs Formalism In this chapter, the discussion revolves around the philosophical interpretations of mathematics, specifically contrasting Platonism and Formalism. The transcript highlights the concept of countably infinite sets as introduced by Cantor, which can be visualized by counting or creating a one-to-one mapping, despite the infinite nature of these sets. Cantor's relentless efforts to understand these sets are emphasized.
• 40:01 - 45:00: Conclusion and Nebula Documentary In the concluding chapter of the documentary on Nebula, the focus is on Cantor's groundbreaking work on set theory. The narrator explains Cantor's diagonalization argument, demonstrating that a one-to-one mapping between natural and real numbers is impossible, thus establishing that real numbers are uncountably infinite. It is concluded that the set of real numbers is larger than the set of natural numbers, showcasing Cantor's revelation of different sizes of infinity.

The Infinity Problem that BROKE Mathematics Transcription

• 00:00 - 00:30 i have a question for you which one of these containers has more things in it the simple question sparked one of mathematics's most important open problems shaking our understanding of logic infinity and the nature of truth it forced mathematicians to confront uncomfortable ideas like whether universal truth even exists and whether logic has hidden boundaries we'll never be able to cross in this video we'll explore the remarkable journey mathematicians took to solve it a
• 00:30 - 01:00 journey that ended in an unexpected resolution leaving us with even more questions than answers it's bizarre we'll explore how this single problem turned people's understanding of mathematical reality upside down it's really opening the door to this other world which is mathematically enlightening and enriching it just shows us that that math is richer and more interesting and messier than we thought but first how could such a simple question cause all of
• 01:00 - 01:30 this to understand first what does it mean for there to be an equal number of things in each container well if we can pair up apples from each container with none left over there was an equal amount of apples in each one in math talk this is called a one:one mapping how do we know if one container had more apples well if we can't do a onetoone mapping if there are some apples left
• 01:30 - 02:00 over that's so easy how did mathematicians get so tripped up by that well now let's try the same thing with numbers let's start with the natural numbers 1 2 3 4 5 etc there are infinitely many natural numbers no matter how high you count you can always add more in math things that contain objects like numbers are called sets this is the set of all the natural numbers over here is the set of all the
• 02:00 - 02:30 even natural numbers 2 4 6 8 etc which set has more numbers in it are there more natural numbers or more even natural numbers your hunch might be the set of natural numbers is bigger after all the set of even natural numbers is missing all the odd numbers let's test out our hunch with onetoone mapping we can pair one with two two with four three with 6 four with 8 5 with 10 hang
• 02:30 - 03:00 on for any natural number we can always pair it with an even natural number so does that mean there are as many even natural numbers as natural numbers is the set of natural numbers the same size as the set of even natural numbers yes and a man named Gayo Cantor was one of the first to discover this fascinating fact he also found that the set of all square numbers can be placed
• 03:00 - 03:30 in a onetoone mapping with the set of all natural numbers he even came up with a clever way to map the set of all negative and positive integers to the set of all natural numbers as well as the set of all the rationals all of these infinite sets are the same size so then are all sets of infinite numbers the same size is infinity a onesizefits-all kind of thing well we're still missing a lot of numbers let's add
• 03:30 - 04:00 the irrationals the algebraics the transcendentals let's add every number until there are no more gaps between negative infinity and positive infinity this is the real number line now the funny thing about lines is that they're continuous uninterrupted you can point to anywhere on the real number line and there will always be a real number there this is different to the natural numbers which are discrete there exist spaces between them they're
• 04:00 - 04:30 better represented as dots we can also count them even if we would never stop counting we can at least start counting same for all of these other sets actually this one:1 mapping idea can be kind of thought of as counting as we're saying this is the first element in the list this is the second one this is the third one this is the fourth one and so on so Cantor called these sets countably infinite canour tried and tried but he
• 04:30 - 05:00 couldn't come up with a onetoone mapping between the natural numbers and the real numbers in fact through his now famous diagonalization argument he proved that there is no way to map the natural numbers to the real numbers the real numbers are uncountably infinite now remember if we can't do a onetoone mapping between two sets it means that one set is bigger than the other so Kant concluded that the set of real numbers must be bigger than the set of natural
• 05:00 - 05:30 numbers there are different sizes of infinity he later discovered there are infinitely many sizes of infinity but for the purposes of this video we're only concerned with these two the size of infinity of the natural numbers is often called a left null and is represented by this fancy looking n with a zero it's the smallest size of infinity and can hypothesized that the set of real numbers was the next biggest size of infinity alf he believed that there is no size of infinity between
• 05:30 - 06:00 that of the natural numbers and that of the reals kanto spent decades wrestling with this question but despite his best efforts he could never prove it so it was called the continuum hypothesis little did he know it would just about break mathematics many of his colleagues rejected his ideas entirely claiming a theory of infinity wasn't real mathematics one even accused him of disrupting the youth but one of the most influential mathematicians at the time
• 06:00 - 06:30 David Hilbert thought Cantor's ideas were revolutionary not just his theory of infinity but also his ideas about sets he famously said "No one shall expel us from the paradise that Kant has created for us." If you don't know this guy he was a great mathematician but his main role in math history seems to have been getting his dreams destroyed over and over again and this story is no exception hilbert made a list of what he thought were the 23 most important problems in math and presented them at a
• 06:30 - 07:00 conference for mathematicians the continuum hypothesis was number one on the list now you might be wondering really number one on the list of most important math problems why is an abstract question about infinities so important well to understand why the continuum hypothesis was such a big deal we need to know why Hilbert called this conference in the first place math was going through a pretty big problem at the time and it was in desperate need of
• 07:00 - 07:30 some fixing there were all these new theories that were vastly outside of our physical realm math was becoming more and more confusing and counterintuitive paradoxes and inconsistencies were starting to pop up all over the place the most famous being Russell's paradox which stated simply is does the set of all sets that do not contain themselves contain itself either answer leads to a contradiction mathematics is constrained
• 07:30 - 08:00 only by logical con inconsistency like you can do whatever you want just don't write down a contradiction hilbert's dream was to repair mathematics he wanted a system powerful enough to prove or disprove every mathematical statement clearly and decisively no more mysteries no more paradoxes everything would be either provably true or provably false this doesn't seem like a terribly unreasonable ask after all most of us
• 08:00 - 08:30 think of math as this objective thing a lot of the time when I ask people why they like math they say because there's a definite yes or no answer hilbert believed we just needed the right set of foundational rules the answer to his dream or so he thought came in the form of Zumelo Frankle set theory with the axiom of choice or ZFC for short that was a lot of words let's break it down zfc is basically the official rule book of mathematics it consists of a small
• 08:30 - 09:00 number of carefully chosen axioms that define what sets are how you can combine them and how you can build new mathematical objects safely by rules of deduction think of these axioms as statements that are so obvious they don't need a proof like if two sets have exactly the same elements they're identical or you can always collect together elements to form new sets the brilliance of ZFC is that it provides a framework where mathematical statements can be rigorously proven each proof becomes a careful logical progression it
• 09:00 - 09:30 seemed like the perfect fix for all these inconsistencies and paradoxes that were going on by starting from basic axioms that we all agree are true and following precise logical rules you'd end up with theorems that are simply undeniable truths to prove something true you start with some axioms and through a logical progression prove that the statement follows logically from the theory to prove something false you need to show that the negation of the statement follows logically from the
• 09:30 - 10:00 theory zfc remains the standard foundation of math today powerful enough for essentially all mathematics we do and because ZFC is fundamentally about sets the continuum hypothesis became one of its first big questions the continuum hypothesis wasn't just a weird question about infinity it had deep implications for the power of axiomatic systems some of the greatest minds were in the room and many tackled it but years
• 10:00 - 10:30 turned into decades and no progress was made on solving Cantor's continuum hypothesis until this guy came along kurt Girdle you may know him from incompleteness fame or you may not but either way he was a super smart dude who had a knack for thinking outside the box you could say he was eccentric he actually later died because he was so worried about his food being poisoned that he refused to eat and starved to death anyway Girdle was the first person to make any progress on the continuum
• 10:30 - 11:00 hypothesis in decades but there was a slight problem with his solution and that was it wasn't really a solution at all what Girdle found was more surprising and profound than anyone could have guessed he proved that you cannot disprove the continuum hypothesis now when I first heard this my response was "What how can you prove that you
• 11:00 - 11:30 can't disprove something and why isn't that just a proof that it's true?" So to answer these questions take this analogy imagine you go to a new country and the only law is all buildings must be rectangular prisms so you travel around and you find a city and all buildings are rectangular prisms cool a mathematician would say that this city satisfies the law you also notice that all the buildings have this property that the number of corners or vertices
• 11:30 - 12:00 minus the number of edges plus the number of faces always equals two hm that's interesting you think but then you're like well actually it's not that interesting this fact follows logically from the law that all buildings must be rectangular prisms as all rectangular prisms have eight vertices 12 edges and six faces it would be impossible to have a city where buildings were both rectangular prisms and this number was anything other than two you keep doing
• 12:00 - 12:30 tourism and you notice that all the buildings in the city are red so you wonder does the fact that all buildings are red also follow logically from the law that all buildings must be rectangular prisms you're not sure but one thing you are sure of is buildings being red is logically consistent with the law that buildings must be rectangular prisms to clarify about the word consistent because it's an important word that we're going to keep using mathematically
• 12:30 - 13:00 it just means that if two properties are consistent they can exist together without causing a contradiction there is nothing logical that says they can't go together if two properties are inconsistent they cannot exist together without causing a contradiction they just do not make sense together so again you haven't shown that buildings being red follows logically from buildings being rectangular prisms but you definitely know that two properties can exist together without causing any problems you know you can have a city
• 13:00 - 13:30 where both things are true i mean you're in it girdle did something equivalent with math and the continuum hypothesis zfc theory are the laws of mathematics girdle built a mathematical universe that satisfied the laws of ZFC and where the continuum hypothesis was true he showed that having the set of reals being the next biggest size of infinity after the set of naturals was consistent with ZFC theory however he
• 13:30 - 14:00 didn't prove that this follows logically from the ZFC axioms he didn't show it was a theorem he just showed that the continuum hypothesis is consistent with ZFC in the same way red is consistent with being a rectangular prism even though this isn't a proof of the continuum hypothesis it shows that we can't disprove it remember that to disprove something from a theory you need to show that its negation follows logically from the axioms but how could
• 14:00 - 14:30 that be when here was a place where ZFC and the continuum hypothesis lived together in harmony it'd be like trying to prove that being red and being a rectangular prism was impossible while you're looking at the red rectangular prism city he showed that the negation is not provable because there's a place where the CH is true so how exactly did he do this he took the laws of ZFC and figured out what is that construct and he started constructing and he took everything he
• 14:30 - 15:00 constructed called out the constructible universe and started figuring out what its properties are the constructible universe is the bare necessities universe at every stage everything in L has to exist girdle basically just kept putting in everything he was absolutely contractually obligated to build and then he counted how much stuff that was and what do you know you get exactly all of one real numbers by necessity this
• 15:00 - 15:30 was an entirely new kind of proof and it left us in an awkward position girdle proved that you can't disprove the continuum hypothesis but he didn't prove that the continuum hypothesis was true it might be true and the obvious next question for mathematicians was well can we prove that the continuum hypothesis is true nobody really had any ideas on how to do this and progress stalled for over 20 years until in 1963 a guy named Paul
• 15:30 - 16:00 Cohen came along and turned everything on its head again he proved that the continuum hypothesis cannot be proven from ZFC theory cohen was trained as an analyst not a set theorist but he wanted to get famous and for some reason thought logic was the best way he asked a mathematician at Berkeley "What's the biggest open problem in mathematical logic?" The mathematician replied "The continuum
• 16:00 - 16:30 hypothesis but you shouldn't work on that because there are no leads on how to solve it." Cohen didn't listen and went on to win the Fields Medal for his result the Fields Medal is the highest achievement in mathematics kind of like the Nobel Prize of math so I guess his plan worked he did become famous in a lot of ways Cohen did the opposite of what Girdle did he built a mathematical universe that was consistent with ZFC and where the continuum hypothesis was false if we go back to our city analogy
• 16:30 - 17:00 it's like he forced in a bunch of non- red buildings into our red city thereby showing that not red is also consistent with the law that all buildings must be rectangular prisms you'd be forced to conclude that the property of being red must not follow logically from the property of being a rectangular prism however once again he didn't show that the property of not red follows logically from the law cohen started
• 17:00 - 17:30 with Girdle's constructible universe and forced in a bunch of stuff until the continuum hypothesis was no longer true how did he do this he invented an ingenious technique called forcing forcing is pretty complicated so before we go over what it is directly let's start with a helpful analogy an analogy that I've often used in talking to other mathematicians um is a lot of times we like to think about fields and a field is a place where you can add and subtract multiply and divide and it all
• 17:30 - 18:00 works out really nicely so the rational numbers or the real numbers or the complex numbers these are all examples of fields a thing that we do a whole lot in math is build field extensions so we start with some field and we make it bigger this idea of extending an object we do it all the time in math so you could start with the rational numbers you cannot solve the polomial x^2 minus
• 18:00 - 18:30 2 inside of the rational numbers right the root square of two is not a rational number but you could take the rational numbers and you can build a bigger field that has roo<unk>2 and then there are a couple of things that just have to come along for the ride if you add roo<unk>2 then you have to add roo<unk>2 plus roo<unk>2 because you have to be able to add things and then you have to add like 1 over<unk>2 so that's what I mean by there are some things that have to come along for the ride just logically must so that's kind
• 18:30 - 19:00 of how forcing works you have your model of ZFC you add your new thing that's not going to change ZFC but it can change other stuff so I guess the next question is what did he add cohen added more real numbers to Girdle's constructible universe but how the constructible universe was already meant to encapsulate all of mathematics with the fields the rationals and the reals are a small part of the universe of mathematics it's easy to grab things
• 19:00 - 19:30 from outside of them but this was meant to be the entire universe of mathematics where was he going to pull more numbers from how do you expand an entire universe of mathematics so it turned out the only way we could begin to tackle this problem was to imagine what a random number would look like we can start bookkeeping our expectations about what a larger universe would look like the way we do this is to imagine we
• 19:30 - 20:00 choose a real number at random between zero and one and we one at a time assign probabilities to every question we can ask about it 50% chance is between 0.2 and 0.7 100% chance of being irrational and 100% chance of being different from every wheel we already have forcing is the language that Cohen developed to keep track of all these questions and
• 20:00 - 20:30 assign them an abstract likelihood of being true in doing this and doing this al too many times to randomly add that many more new numbers to our universe he developed a consistent language for describing a universe that with probability one satisfies every individual axiom of ZFC but not the continuing hypothesis one expert I talked to literally said that it's not so easy to explain forcing even to a
• 20:30 - 21:00 professional mathematician so if you didn't get all that don't worry what you need to know is that when Cohen was done there were alf null many natural numbers alf one many real numbers from girdle's constructible universe and alf too many real numbers from Cohen's universe which were considered to be all the real numbers remember that the continuum hypothesis was the assertion that there is no size of infinity between that of all the natural numbers and that of all the real numbers but now we have the
• 21:00 - 21:30 size of the constructible real numbers in between the two thereby showing that the continuum hypothesis is false so here's where we're at girdle constructed a mathematical universe that satisfies ZFC where the continuum hypothesis is true and Cohen constructed a mathematical universe of ZFC where the continuum hypothesis is false so where does that leave us a statement is independent of a theory
• 21:30 - 22:00 just means that you can't prove the statement from the theory and you also can't refute the statement in other words you can't prove the negation of the statement let's just say Hilbert's not happy let's take a moment to really appreciate why cuz it's a bit subtle and the nuances definitely confused me for a while so something that's been implicit in this video is the idea of mathematical models like Girdle built a mathematical universe and then Cohen extended his mathematical universe another word that mathematicians use for
• 22:00 - 22:30 mathematical universe is mathematical model but what does built a mathematical model mean isn't there just one math like the math well that's actually not how set theorists think about things and it's largely due to results like the continuum hypothesis set theorists think in terms of mathematical models we have a theory like ZFC and rather than there just being one math it describes it can describe multiple models that we fit to the theory like Girdle's model and
• 22:30 - 23:00 Cohen's model both fit to ZFC theory this is a pretty different way of looking at math than what we're used to and I bring all of this up because it has a very important consequence for a statement to be provable from the theory it needs to be true in all models for example remember that relation we had between vertices edges and faces earlier on we agreed that that followed logically from the law that all buildings were rectangular prisms right notice how this relation still holds in
• 23:00 - 23:30 both cities we can see that in any city that's made up of rectangular prism buildings this relation will be true however as we've known the entire video the property of red or not red does not follow logically from the law that all buildings must be rectangular prisms it can change from model to model the color of the buildings is independent of the law that all buildings must be rectangular prisms the law simply says nothing about the color of the buildings
• 23:30 - 24:00 drawing a parallel with ZFC and the continuum hypothesis here we have two models of math one where the continuum hypothesis is true and one where it's false therefore it must not follow logically from the axioms of ZFC we cannot arrive to it by starting with the axioms and logically following the rules of deduction the theory simply doesn't say anything about the continuum hypothesis if you produce by some brute force algorithm every single possible
• 24:00 - 24:30 theorem from ZFC the proof of the continuum hypothesis and the proof of the negation of the continuum hypothesis will never show up considering the continuum hypothesis is one of the first questions you would ask when building a theory of sets it raises a lot of questions about the state of mathematics is the continuum hypothesis considered solved is its independence a problem that we need to fix is it okay that there are some questions our current math just can't answer i mean it gets
• 24:30 - 25:00 into contentious philosophical stuff there are generally two main philosophical camps when it comes to answering these questions the first is Ploninism plonism is the idea that there is some kind of objective realm in which all mathematical structures exist there is one true mathematical universe and there is a true answer to all of our questions we all agree from these mathematical facts that there is more
• 25:00 - 25:30 than one universe of set theory the position of a set theory place in this is usually that amongst these universes one is the biggest among them all and that everything else is just a subuniverse of that and under this position when we are asking questions like the contin hypothesis what we are trying to figure out is is it true or false in this biggest of this universes to one who believes that there is this
• 25:30 - 26:00 ultimate universe then there is still hope for us to resolve the hypothesis some set theorists are currently working on building new axioms of CFC in hopes of being able to solve the continuum hypothesis at the opposite end of the spectrum to Pltonism is formalism formalism is the mathematical philosophy that there's nothing more to math than shuffling symbols and rules on a piece of paper math is the consequence of our starting axioms and we get to choose whatever
• 26:00 - 26:30 axioms we want as long as they're consistent different starting rules lead to different mathematics to the formalist this is the end of the story we have our full answer the continuum hypothesis is true or false depending on what model you're in there's a mathematical multiverse and all universes are equally valid i don't know if I think there is a real mathematical universe to me it's more like we're trying to understand what happens if you assume this rather than what's the real
• 26:30 - 27:00 truth of the matter i think it just tells us ZFC doesn't settle all of our questions and that means we get to ask more questions this contrast in how mathematicians view mathematics raises an even deeper question are we discovering mathematical truths that have always existed or are we inventing mathematics as we go along are mathematical axioms like those of ZFC something humanity created or something we uncovered this question of whether math is invented or discovered comes up
• 27:00 - 27:30 again and again when you study mathematics it always fascinated me but I could never find a satisfying answer so I decided to dive deeper so deep that I made a featurelength documentary exploring this question on Nebula nebula is a platform that's quickly becoming the go-to space for the best educational content on the internet it's built and run by creators with the mission to be the best place for us to make work that we couldn't make anywhere else there are
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