Ranking Paradoxes, From Least to Most Paradoxical

0:00 People love and hate paradoxes. Paradoxes are places of confusion — of conflicting ideas. They pinpoint the stickiest parts of our knowledge and understanding. And they are often used as a testing ground for our scientific and mathematical assumptions and methods. But, our reactions to different paradoxes vary wildly — some feel truly perplexing, while others are just annoying. So, here are 24 paradoxes, ranked from least to most paradoxical by my 12 year old former students. They — and everyone — have individual perspectives and backgrounds, which informs their reactions to the paradoxes. I think that’s very interesting.

0:46 So, please feel free to — politely — disagree. First off: What is a paradox? “a statement claiming something which goes beyond (or even against) ‘common opinion’ (what is usually believed or held).” “one (such as a person, situation, or action) having seemingly contradictory qualities or phases” “any conclusion that at first sounds absurd but that has an argument to sustain it” The 24 paradoxes on this list were selected using a very simple criterion: they are commonly called paradoxes. I have loosely — and very informally — lumped them into categories. We don’t have time to go into the details of the

1:31 paradoxes. That video would be very, very, very long. But, don’t worry: I left plenty of links in the description. Many of the paradoxes at the bottom of my students’ list are what the philosopher Quine called “veridical” paradoxes — the word comes from Latin and means “truthful.” These paradoxes are surprising or counterintuitive, but they are “true”, in the sense that they follow from some accepted mathematics or logic. One famous example is the birthday paradox. The more people you have in a room, the more likely it is that two of those people share a birthday. How many people do you need to have in one room before there is at least a 50% chance that two of those people share a birthday?

2:17 Most people guess a pretty big number, since there are 365 — or 366 — days in a year. But, it’s 23. Once 23 people are in a room, there is a more than 50% chance that two of them share a birthday. And you can figure this out using math that is taught in an introduction to probability course. Also using introductory probability, you could solve Bertrand’s Box. You have three boxes: one with two gold coins, one with two silver coins, and one with a gold and a silver coin. You randomly pick a coin without knowing what box it came from. You look and the coin is gold. What is the probability that the other coin in the same box is gold?

3:05 Most people say ½. Here’s their logic: you’re equally likely to have pulled from the first or third box — the first box gives you a gold coin and the third box gives you a silver coin. But this logic is flawed — you are not equally likely to have pulled from the first or the third box. There are two gold coins in the first box and only one in the third box. So, if you pick a coin, and you see that it’s gold, there was actually a 2/3rds chance that it came from the first box. Therefore, there’s a 2/3rds chance that the other coin in the box is gold. This did not impress my students and they put it at the bottom of the ranking. They were slightly more impressed with the similar and much-more-famous Monty Hall problem — the one with the doors and the goats and the car.

3:51 I’ll leave the details to the links in the description. These three paradoxes all point to the fact that calculating probabilities can be counterintuitive. It’s a problem in statistics and applied sciences, where computing the likelihood of certain outcomes — where a hurricane will make landfall or whether a patient has cancer — can have life or death consequences. Speaking of scientific predictions, my students were not moved by the Fermi paradox. That’s the one about aliens. The reasoning goes: There is a boatload of stuff *out there* — stars that are similar to our sun and planets that are similar to Earth. There are so many planets that it’s very very likely one of them would

4:36 have developed intelligent life and that we would have some contact with these aliens. But, of course, that hasn’t happened — that’s the paradox, the seeming contradiction. The Fermi paradox is like many of the other paradoxes in this section of the video, which is roughly about philosophy and science. These paradoxes tend to have a lot of context. They’re often part of a long background conversation — one that is harder to summarize than a few mathematical axioms. That background perspective — the assumptions — is so critical to how we react to paradoxes. For example, you may not have spent a lot of time considering the word “heap” — that’s a synonym for “pile”; as in “a heap of sand” —

5:22 but some philosophers have. What makes a heap? If you have one grain of sand, that’s not a heap. Adding a single grain of sand is not going to change that. So, two grains of sand is not a heap. And if two grains of sand is not a heap, then — adding one more won’t change it — so three grains of sand is not a heap. And so on. It follows that any number of grains of sand is not a heap. That’s the paradox of the heap — it hinges on the vagueness of the word “heap” and our intuition that a single grain of sand can’t change something from a non-heap to a heap. But then, how does it ever become a heap? You can contemplate the definition of “heap”

6:09 the next time you’re sitting in traffic — which is the subject of our next paradox. Let’s say there are two different routes connecting downtown to the suburbs. In the afternoon, there are lots and lots of people driving from downtown to the suburbs — so we’ll only look at traffic flowing in that direction. Let’s say there are 2000 total cars. Some parts of the route take a specific amount of time — for example, it’s 30 minutes to drive from *here* to *here*. But, on other segments, the travel time is a function of the total number of drivers. For example, the number of minutes to get from *here* to *here* could be five plus the number of drivers divided by 100 — the more drivers, the slower the traffic. Each driver takes the route that is best for them individually — they’re not collectively planning.

6:58 Half the drivers will end up going one way and half the other way — no one wants to switch, because then, the other route would have more cars and thus take longer. So — doing the arithmetic — everyone has a 45 minute commute. The city wants to decrease the commute time. So, they add a bridge at the midpoint — drivers can switch which route they are taking, if they want — which would, hopefully, improve the traffic situation. But — on average — drivers spend more time getting home. That’s Braess’ paradox — counterintuitively, the extra road can cause extra traffic. In our example, the drivers will all end up changing to the same route — first on the lower path, then the bridge, then the upper path. That makes the commute 50 minutes — 5 minutes longer than before.

7:46 Remember, each driver is calculating the best route individually — and each driver alone has no incentive to switch routes — the other routes would take longer. But, of course, if everyone collectively agreed to stop using the bridge, they’d all save 5 minutes. Our next paradox — or set of paradoxes — is particularly famous in the history of philosophy and mathematics: Zeno’s paradoxes. They all have to do with motion, and time, and space, and infinity. One of them is a funny version of the tortoise and the hare story. The hare can never catch up to the tortoise because, by the time the hare reaches the tortoise’s former position, the tortoise will have gone further — always keeping a tiny lead.

8:31 That continues the whole race, so the tortoise wins. Quine called this a “falsidical” paradox — as in, “false”. My experience is that, when people first encounter Zeno’s paradoxes — there are several of them — they either love them or hate them — and my students kind of loved them. Almost as much as they loved the ship of Theseus. Theseus — a character in Greek mythology — had a ship. As the wooden planks rotted and decayed, he replaced them with new ones. At some point, every plank has been replaced. Is it still *the* ship of Theseus? It sort of feels like yes and no — that’s the paradox. The story is used to discuss identity — what makes an object the same object throughout time? — and composition — how is an object related to its constituent parts?

9:22 Next up: the omnipotence paradox. It’s about an omnipotent being — omnipotent, meaning “all powerful.” In the best-known form, the paradox asks “Can an omnipotent being create a stone so heavy that it cannot lift it?” Omnipotence implies that the being has the power to create any stone. But, it also implies that the being should be able to lift any stone. Hence, the paradox. Earlier, I mentioned three vertical paradoxes. Those were all about probability, but they don’t have to be. For example, I recently made a short video about the potato paradox — which uses basic algebra to produce a counterintuitive answer.

10:09 My students were not impressed. They much preferred 1 = 0.99999… In my experience, people have one of two reactions to this. The first reaction — from people who have never seen this before — is confusion or disbelief. There are lots of precise proofs of this fact, but here’s a simple argument: Starting from the fact that ⅓ = 0.3333…. , we can add ⅓ + ⅓ + ⅓. This is obviously 1. And 0.3333… + 0.3333… + 0.3333… = 0.9999…. Therefore, 1=0.9999…. The second common reaction — from people who have already learned this fact — is annoyance.

10:56 This is everyone who is yelling at the screen, “That’s not a paradox! It’s a fact!” Yes. Many paradoxes are things that we call “facts.” Those are the veridical paradoxes, which, as Quine writes, “we can get used to, thereby gradually sapping its quality of paradox.” You may be used to the fact that 1 = 0.9999… But for many people — at least, at some point in their math education — it seems counterintuitive… surprising… paradoxical. Moving on. Let’s say you’re getting married and want a very tall wedding cake. Each layer should be 1 meter tall — which is ridiculously tall. The bottom layer also has a radius of 1 meter. Then, each subsequent layer narrows.

11:42 The second layer has a radius of one-half of a meter. The third layer has a radius of one-third of a meter. The Nth layer has a radius of 1/Nth of a meter. You tell the baker, “I would like infinitely many layers please.” The baker replies, “I don’t have infinitely much cake batter.” But, you do a quick calculation. The volume of the n-th layer is the radius squared times pi times the height. That’s “1 over n” squared times pi times 1. Adding up all the layers, you get a famous infinite series, which Euler proved is equal to exactly “pi squared over 6”. Multiplying by the pi in front, the volume of this infinitely tall cake — known as Gabriel’s wedding cake — is “pi cubed over 6” cubic meters.

12:27 That’s a little more than 5 cubic meters. That’s not that big! The Guinness World Record for largest wedding cake is roughly 15,000 pounds or 6800 kilos. I don’t know what the volume of that cake was, but my guess is that it’s bigger than Gabriel’s wedding cake. The wedding cake paradox was resolved using a famous infinite series. And, in general, infinite series can be quite tricky. Consider, for example, the sum 1-1+1-1+1-1+1… alternating forever. If we put parentheses like *this*, then it seems like the answer is zero. If we put parentheses like *this*, then it seems like the answer is one. Maybe we should just average those answers and

13:14 get ½ — which is an answer with several other prominent, historical arguments. But, if your calculus teacher asks you the answer, it’s none of the above; it’s a “divergent series” — which kind of means that there is no answer. This is the consensus in modern mathematics following the standard rules for infinite series. Maybe if you’ve learned about those rules, these three different solutions don’t seem very paradoxical — just wrong. But, if you haven’t learned these rules, or if you lived before those rules were invented — or discovered — or agreed upon — then it could be pretty perplexing that there are three different plausible answers to this summation.

14:01 Thompson’s lamp is, in some ways, a physical version of the alternating sum paradox — you’re turning a light on and off instead of adding and subtracting one. You enter a room and turn the light on. After one minute, you turn the light off. Thirty seconds later, you turn it on. Fifteen seconds later, you turn it off again. Seven-and-a-half seconds later, you turn it back on. And so on — always halving the time before you flip the switch. If you do this infinitely many times, it will take exactly two minutes. At that two-minute mark, is the light on or off? That question is the paradox. It seems like it should have an answer — after all, lights are either on or off — but, as you approach two-minutes, you’re turning it arbitrarily quickly on and off,

14:50 and the status of the light at two minutes is very unclear. The Ross-Littlewood paradox is similar to Thompson’s lamp — it’s also something called a “supertask,” which means you have to complete infinitely many tasks in a finite amount of time. But it’s about quickly adding and removing balls from a vase. For the sake of time, I’m just going to direct you to an old Infinite Series video that I made about a random version of the Ross-Littlewood paradox. The St. Petersburg paradox is about a very peculiar casino game. The amount of money you win is determined by how long it takes to flip “heads” with a fair coin. If the first flip is heads, you win $2. If it’s tails then heads, you win $4.

15:38 If it’s tails twice, then heads, you win $8. To put it more mathy, if the first heads is on the Nth flip, you win 2^N dollars. How much would you be willing to pay to play this game? Well, by some logic, you should be willing to spend what you will — on average — earn back. For example, if the average winnings in a particular lottery are $5 and the tickets cost less than $5 — it makes sense to buy one. In the St. Petersburg game, how much do you expect to win? Just like our other probability paradoxes, you can use some rather simple calculations to determine that it’s infinity. Your expected winnings are infinity dollars. So, given that, how much would you be willing to pay to play this game?

16:24 Well, if you apply the same logic — paying anything that’s less than the expected earning — then you should be willing to pay any fixed price — say, $10,000,000. The paradox is that — even though the calculation tells you to pay whatever the fee is — most people are not willing to pay very much, because usually — on most turns — you don’t win very much. It’s not really a paradox in probability theory, like the birthday paradox. It’s more a paradox about decision making. Self-referential paradoxes are popular — they’re compelling and usually pretty easy to state. Here’s the Berry paradox: “The smallest positive integer not definable in under sixty letters.”

17:11 Lots of numbers can be defined in under 60 letters — ”three”, “one-hundred and twelve”. But, the alphabet is finite — it’s only possible to define finitely many things in under 60 letters. Since there are infinitely many numbers, they can’t all be definable in under 60 letters. So, out of all the numbers which are not definable in under 60 letters, we’ll call the smallest of that group X. In other words, X is “the smallest positive integer not definable in under sixty letters”. But, that sentence has 57 letters, which means that it’s a definition of X with less than 60 letters. That’s the problem — the paradox. Another self-referential paradox — ranked slightly higher — is the Grelling-Nelson paradox.

18:01 “Autological” words describe themselves — “English” is autological, because it’s an English word; “word” is autological, because it’s a word. “Heterological” words do not describe themselves — ”German” is heterological, because it’s not German; “truck” is heterological, because it’s not a truck. Here’s the paradox: Which category should we put the word “heterological” in? Is it autological or heterological? Well, if it’s autological, then it describes itself — so the word “heterological” is heterological. But if it’s heterological, then it doesn’t describe itself — but that’s the definition of heterological, so it fits its own definition, which makes it autological.

18:51 Moving on. Here’s a part of M.C. Escher’s “Ascending and Descending.” The monk-like figures are marching on a never-ending staircase. You might say, “That’s an optical illusion. Not a paradox.” That’s fine. It’s not very high on my student’s list. But paradoxes are an incongruity — they’re a form of dissonance. When I look at this, I find it hard to resolve the clash between my intuition — the staircase is endless — with my knowledge that that’s not possible. *This* is the Müller-Lyer illusion. Most people perceive the horizontal line on the bottom shape as longer than the horizontal line on the top shape. But, they’re the same length. What’s cool about it is that different populations are more

19:37 or less susceptible to the optical illusion. Visual artists — people with lots of experience drawing lines and considering perspective — are more likely to immediately recognize that the lines are the same length. Similarly, your background in mathematics or philosophy probably changes your reaction to the other paradoxes. Anyway. Here’s the pop quiz paradox. Your statistics teacher says there will be a pop quiz next week. The class meets every day at noon. You want to figure out when the quiz will be, so you can fake sick on that day. You think: It can’t be on Friday, because then it wouldn’t be a surprise. By Thursday afternoon — after class, when you still haven’t taken the quiz — you would know it’s on Friday, making it not an actual surprise.

20:26 So, you cross off Friday. But then, you reason, it can’t be on Thursday. By Wednesday afternoon, you would know that it’s on Thursday — you already know it’s not on Friday — so a Thursday quiz would not be a surprise. Cross off Thursday. This pattern of logic repeats. A Wednesday quiz would not be a surprise — you would know on Tuesday afternoon. So, no pop quiz on Wednesday. Similarly, no pop quiz on Tuesday or Monday. Great news, you figure — there’s no quiz next week! But then, on Tuesday, the teacher gives you a pop quiz — which comes as quite a surprise. And that pop quiz is about Simpson’s paradox. Let’s say that Dude 1 has a higher batting average — that’s a baseball thing — than Dude 2 in both 1995 and 1996.

21:13 But, when you combine the two years, Dude 2 has a higher batting average than Dude 1. Maybe that seems impossible, but it’s totally possible. This is a simple, classic — and real — example, but there are lots of more interesting, modern examples that illustrate this statistical phenomenon. For example, *this* paper compared the Covid case fatality rate — that’s the percentage of confirmed Covid patients who died — between Italy and China. Within every age group, this number is lower in Italy. But overall, it’s lower in China. Within statistics, Simpson’s paradox is not really an isolated phenomenon. Statistics provides many different perspectives on data and they can often seem in conflict with each other.

22:01 On to our most paradoxical paradoxes. Third from the top is the grandfather paradox. Briefly summarized: A kid goes back in time and kills her grandfather. But without the grandfather, the kid wouldn’t have been born. But if the kid wasn’t born, she wouldn’t have killed her grandfather. So then she would have been born. But then she would have gone back in time to kill her grandfather. And so on and so on. Physicists and philosophers use this — and other time travel paradoxes — to debate possible theories of time and time travel. Second from the top, we have the Banach-Tarski paradox. To be very brief about it, the Banach-Tarski paradox is a way to take a three-dimensional solid ball, break it into a few pieces,

22:47 and reassemble those pieces to create two balls, each identical with the original. I’m not going to explain it here — it requires some time and there’s plenty of great explanations out there — but I want to point out a few things: First, it uses pretty standard — albeit advanced — mathematics. In that sense, it’s a veridical paradox. But it also uses the axiom of choice. There are a bunch of foundational rules — axioms — in mathematics, and a hundred years ago, these were heavily debated. In particular, this one — the axiom of choice — was quite controversial. One argument against the axiom of choice was that it led to some strange consequences, like the Banach-Tarski paradox. Now, it’s substantially less controversial.

23:34 Paradoxes are all relative to the assumptions we make. Sometimes those assumptions are subconscious — or otherwise difficult to identify. But in this case, the debate — the tension between the Banach-Tarski paradox and the axiom of choice — is very clear. The number one top paradox was — unsurprisingly — the self-referential Liar’s Paradox. “This sentence is false.” Is that true or false? Well, if it’s false, then it’s true. If it’s true, then it’s false. It’s classic. It’s fun. It’s a very good paradox. There you have it: 24 paradoxes, ranked from least to most paradoxical by my 12 year old students. I’m sure you have your own reactions to these paradoxes and many others.

24:20 Which is great. I think paradoxes are meant to be the start of a conversation. If there’s a particular paradox that you’d like me to make another video about, let me know in the comments. Here’s just a quick note to say that I’ve started a Patreon. I’ve decided to make this video without sponsorship — and I hope to do the same going forward. These are my videos — free of the creative constraints and ethical questions that arise from commercial sponsorship. But I have to pay the bills somehow. So, if you have the resources, your Patreon membership would be greatly appreciated. Or, if you’d like to support Chalk Talk in other ways: like, subscribe, share with friends, write nice comments. Everything helps. Thanks!