The Liar’s Paradox

Apologies for the short one today- I usually write these in a burst of motivation, and I haven’t really had one of those this week.

The Liar’s Paradox is the simplest paradox there is. It’s just one sentence long, and it’s a total mindblow.
Here it is.
“This sentence is a lie.”
Well, is it? Because if you say that it is a lie, then the statement must be true, since that’s what it claims. And if you say that the sentence is true, then it must be a lie, because it says that it is a lie. Either way, you find yourself saying that the sentence is true and false at the same time- which is impossible! Paradox!
The cool thing about the Liar’s Paradox is that it can be remixed and twisted into tons of other paradoxes.
For example, there’s the Card Paradox, where the front of a card says “The back of this card is true,” and the back of the card says “The front of this card is a lie”.
Or there’s the Cretan Paradox, where someone from Crete says “all Cretans are liars” or “Cretans never tell the truth”.
Or there’s the Pinocchio Paradox, where no one can decide what happens if Pinocchio says “My nose will grow now”.
Or, of course, there is heterological, which I covered in an earlier post because I love how small it is.
All of these paradoxes are slow-acting, by which I mean, they don’t seem paradoxical immediately, but if you stare at them for a moment and think it through, it hits you. For that reason, when I need to tell someone a quick, easy-to-get paradox, I go for a version of the Liar’s paradox.

Hilbert’s Paradox of the Grand Hotel

Guys, this is a public service announcement: infinity is CRAZY. Infinity is BIG. In fact it is SO BIG that when you’re talking about it, you TRIP over PARADOXES like they are LEGO pieces and BOXES in your house in the middle of the night when you just want to PEE.

One of the many paradoxes associated with infinity is Hilbert’s Paradox of the Grand Hotel *flourishes dramatically, throws sparkles in the air*. This grandiosely named conundrum takes place in a completely normal, everyday sort of building: a hotel with infinite rooms.

Ok, I lied about it being normal. Obviously, such a building could not exist. I mean, acquiring the building permits alone would be an impossible goal. But this is math, and one of the best things about math is that if you say an axiom is true, it might as well be, until you prove yourself wrong. Anything can happen! Except for the things that can’t! Don’t quote me on this! I am not a mathematician! That picture of my calculus notes is as far as my math education goes, and I’ve forgotten most of it. 

Back to the hotel with infinite rooms. As it just so happens, the hotel is pretty busy today. In fact, every single room is booked! Every numbered room has exactly one person staying in it. (I bet the person in room Googol felt pretty special, 42 was reserved early by a massive nerd, and whoever got the room with the smallest uninteresting number was really confused.)

So, the entire hotel is full. Then, a weary traveler shows up with his suitcase. Even though the hotel owner has infinite dollars now, from all these bookings, he wants to fit one more in. But he can’t, right? The hotel is full, and ‘full’ literally means ‘cannot fit any more’.  

But remember, this is infinity we’re talking about here. And as I mentioned, infinity is CRAZY. The hotel owner decides that it’s worth a little inconvenience on everyone’s part to get one more guest in, so here’s what he does: the person in room 1 is moved to room 2, that person is moved to room 3, the person in room 3 is moved to room 4, and so on and so forth.

Now, if you tried to do this with a finite amount of rooms, eventually you’d get to the end of the rooms, and someone would be kicked out. But there is no end to the rooms. They go on forever. There will always be another room to move the person to, no matter how far along you go. And so, with room 1 left vacant, the newcomer can slide right in!

This doesn’t just work for one person. For a family of five, you’d just move everyone up by five. For a group of one million, you’d simply ask everyone to walk past one million doors, and stay there instead. Easy!

But what if it’s not a group of one million? What if it’s a group of… infinity?

A HUGE bus carrying an infinite number of people drives up to the hotel. The hotel owner’s eyes are bugging out of his head. He could double his money! Of course, two times infinity is still infinity, but a savvy businessman would never turn such an offer down!

But how can the hotel owner fit these new guests? Your immediate guess might be that he moves everyone up an infinity of rooms, but of course, you can’t do that, because there is no end to infinity. They’d never reach their rooms, not in all of eternity, and that leads to bad Yelp reviews. That simply won’t do. So, here’s what the hotel owner does: the person in room 1 is moved to room 2. So far, it’s the same as before. But then, the person in room 2 is moved to room 4. The person in room 3 is moved to room 6. The person in room 10 is moved to room 20. Everyone is moved to the room with the number that is twice their current number. This is pretty easy for the person in room 25, but for the person in room 28,738,789,723,478,932, it’s a bit harder. Many- in fact, almost all of the people will need to walk or take the elevator for longer than the universe has existed. But if they are immortal, and very determined, they will, eventually, get there.

And the neat thing is, since everyone multiplied their number by 2, everyone is in an even-numbered room. This leaves all the odd-numbered rooms open, ready for the bus full of infinite people to check in!

I don’t think I’ll try to draw this into a larger context today. I think it’s amazing enough as it is.

 

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FOLLOW UP: Paradox of Choice

I was delighted yesterday to find this comment from Jose underneath my Paradox of Choice description. It is at least 100X more insightful and interesting than the post itself, and I thought I might post it up here, so all three of you can see it, lol.

“Grocery stores have researched this quite a bit. It’s how they survive (and thrive) on slim margins. They have figured out how to steer us towards what they want us to want to buy, and how to manipulate the paradox of choice in their favor.

That aside, we want choices, but we don’t want to have to =make= choices. That is, we want our choices to be curated. But we want to choose who curates them. This is part of the idea of branding and belonging – once we choose a brand or group, we stick with it in part because it relieves the need to keep choosing, so we can enjoy the thing we’ve chosen.

So, why =do= we want choices? To help think about this, imagine going to a store with a friend, and have that friend choose for you. You will have no say in the matter – you give her the generic shopping list and she fills your cart right in front of you, making the choices out of the zillions of options. You can look over the options, but cannot tell her what to put in or what to avoid (allergies aside).

How does it feel? Are you happy afterwards? Does it matter which friend you do this with?

I posit that eventually this will be the way life goes, if it’s not already doing this. Software is already choosing what news you see, who you go out with, what ads are plopped in front of you, and even literally what goes in your shopping cart (peapod anybody?) As software gets to know us better, it can make better choices for us and we’ll happily let it… until we realize that the “better” choices it is making benefits not us but them. But by then you’ll be used to robots refilling your fridge, and your not being able to fine-tune the algorithm. You won’t want to go back to manually buying every single thing you need.

Jose”

Paradox of Choice

Ok, here’s the deal. I want to cover the Paradox of Choice, because I think it’s a really cool concept to have in your head. However, in the course of researching it, I discovered that there isn’t a ton of experimental evidence supporting it. So, take this with a grain of salt. I’ll do a real, math-containing paradox this Wednesday, I promise!

 

So, having more options is good, right? The more types of bread, or movies, or insurance plans are made, the more likely it is you’ll get exactly what you’re looking for.

But… but just look at this picture.

Are you feeling a teensy bit stressed? Because I’m feeling a teEENSY bit STRESSED. That is SO much YOGURT.

The idea behind the paradox of choice is that while people always say they want more options when you ask them, if you have too many options, it can cause stress and worry that you chose wrong, and it takes much longer to choose.

For example, look again at the picture of yogurt from my local Fred Meyer. Imagine trying to choose one yogurt out of this mix- you’d have to sort it out by type (Greek? Whipped? Plain?) flavor (Caramel? Strawberry? Vanilla?) and price (by item, or by weight?). And even once you chose that yoplait, you might wonder if another type might taste better.   

That’s really all I’ve got for you today. I just think it’s a neat little concept to have, to remind you that when you’re making a decision, sometimes the stress of the decision making process itself can be a factor.

The Meat Paradox

I live near Seattle Washington, and this Saturday I attended Vegfest, a vegan/vegetarian event in Seattle Center’s Exhibition Hall.

While I am not a vegetarian (yet; one of these days I want to make the leap), it was fun to try all the samples and learn about different recipes. It also got me thinking about the meat paradox.
Now, I’ll be the first to admit, very little of what humans do makes any sort of logical sense. We make dumb, illogical decisions in love, economics, education, and pretty much everything. Today, though, I want to bring up a specific example of human illogic, one that results in a lot of suffering. Meat eating.
The paradox is distressingly simple. We love some animals- dogs, cats, parrots, and goldfish for examples, and yet other animals, we kill for food. Cows, pigs, and chickens, we slaughter by the billions, while we treat dogs like part of the family.
Why? We don’t need to eat animals- if we did, vegetarians would all starve to death. We, unlike carnivores, have a choice of whether we eat meat or not. (This point is also sometimes called the Omnivore’s Dilemma).
Are any of you vegetarian? And for those that aren’t, how do you resolve this paradox?

Zeno’s Paradox (Christmas Edition)

I originally posted this article on the Imzy community around Christmas. I planned to post it here after I edited the Christmas stuff out of it, but I think it’s more fun this way. Brand-new paradox coming this Wednesday!

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Zeno’s paradoxes are some of the oldest in the book; they were written around 450 bc. Traditionally, the motion paradoxes have been presented with a tortoise and/or Achilles running a race. But, because this is Christmas Eve, I’m going to present it with a kid running to their Christmas tree!

At 6:00 a.m., the kid wakes up and jumps out of bed. Let’s call him Jimmy. Let’s also say he lives in a really big house, and it will take him 10 seconds to get to the Christmas tree, because that makes the math easy.

Before Jimmy gets to his tree, he needs to get to the halfway point, obviously. The teleporter kit is under the tree- he doesn’t have it yet. In this thought experiment, Jimmy gets halfway to his Christmas tree, no problem. After five seconds, he is just starting to go down the stairs to the living room. But, before he gets to his Christmas tree, now Jimmy has to get 3/4th of the way to his Christmas tree. Obviously. So, 2.5 seconds later, Jimmy is stepping off the stairs. Now, he’s got to get 7/8th of the way to the tree. 1.25 seconds later, he can see the tree; he’s mostly through the living room. Now, he just has to get 15/16ths of the way… then 35/36ths… then 71/72ths… forever. There will always be another fraction to go: there is no limit to how small you can make a step. And if there is always another step, how could he ever get to the tree? You can’t do infinite things in a limited amount of time, right? It would have to take eternity to move anywhere… or so the Greeks thought. For millennia, this paradox stood, an awkwardly unsolved guest at the logic party. But then, an unlikely hero stepped forward: integral calculus.

(Sidebar: I’ve known about this paradox since middle school, and for years, I couldn’t understand the solution. Then, last year, I took BC Calculus. When I realized that the problem on the board was the solution to Zeno’s paradox, I literally gasped. My classmates thought I was weird when I tried to explain, but it was worth it.)

Now, I’m not here to explain Calculus to you. Check out Khan Academy for that. All you need to know is that integrals allow you to add up an infinite number of things, and if they behave themselves and converge, you can get a finite answer. Luckily for anything that wants to move, Zeno’s sequence of half, then half again, then half again, forever, converges quite nicely. In our case, it converges to a 10 second journey. Jimmy gets to his present with ease, unaware that, theoretically, he just did infinite things.

The story could stop there. Math had stepped in and given philosophy a helping hand, with the bonus of making every journey an infinite one. Sadly, physics showed up and ruined everything. Well, not really. But it is a bit of a bummer. Turns out, you can’t keep dividing distances forever, like I said you could. Sorry. Turns out, there is a ‘smallest distance’, called the Planck length. If you try to divide a Planck length in half, it straight-up doesn’t work.

This paradox is about as resolved as it gets. Both math and physics have beaten it to a pulp. Still, though, I’ve found that Zeno’s paradoxes are the most fun to explain to non-mathematicians, because they are the only paradoxes that allow you to move around a bit, adding some drama. If you describe this paradox to someone unfamiliar with its solutions, you can make every walk across the room a little magical. And in this season especially, we can all do with a little more magic.  

 I am electrified by the number of people interested in my writing; more than 500 people visited today! I’m so glad to be helping you understand amazing paradoxes, and I hope we can keep this momentum going! Please share this site with your friends, however you can.

Let’s make the whole world go, “Wait, what? Hmmmm…”