All of us know that if you ever travel back in time, you should definitely not kill
your own grandfather. I mean, that’s generally good advice even for those of us who remain
fixed to the linear timeline, but it’s especially good advice for those who don’t want to create
some kind of temporal paradox-slash-rift in the space-time continuum. This problem, known as the
Grandfather paradox, presents the main problem of time travel: if you go back and prevent yourself
from being born, how would you ever have been able to go back in time in the first place?
Other time travel paradoxes, such as the bootstrap paradox, take the question in a different
direction. Imagine you go to a book store and purchase yourself a copy of Hamlet. Then, you
step into a time machine, head back to Elizabethan England, and find a young William Shakespeare who
has yet to write his opus. You present to him the book, which he copies, and distributes as his own.
Centuries later, copies of his copy still exist, ending up in the bookstore that YOU purchased
Hamlet from. Who, in this case, actually wrote the play? If your brain is starting to
itch in a funny way, you’ve got the right idea. Hi I’m Erin McCarthy and this is the List Show.
These time travel paradoxes have been the subject of countless sci-fi plots,
and are actually a hot topic of debate in the field of quantum physics.
But they remain unresolved—as is evidenced by how many different answers get proposed for them.
While possibly frustrating, this is the essence of paradoxes: self-contradictory,
logically impossible, or simply unanswerable questions that are likely to keep you up at night.
Today I’m going to share with you a number of paradoxes that are sure to melt your brain
and mine, from a very conflicted Pinocchio to a hotel with infinite rooms. Let’s get started.
One of the more famous paradoxes, thanks in part to the Marvel show WandaVision, is the ship of
Theseus paradox. Here’s a brief summary. Theseus was a mythical king and the hero of
Athens. He was the guy who slayed the Minotaur, amongst other feats. He did a lot of sailing,
and his famed ship was eventually kept in an Athenian harbor as a sort of memorial-slash-museum
piece. As time went on, the ship’s wood began to rot in various places. Those wooden pieces
would be replaced, one by one. As time went on, more pieces needed replacing. The process of
replacing rotten planks with new ones continued, at least in modern versions, until the entire ship
was made up of new pieces of wood. This thought experiment asks the question: Is this completely
refurbished vessel still the ship of Theseus? Let’s take it a step further: What if someone
else took all of the discarded, original pieces of wood and reassembled them into a ship. Would THIS
object be Theseus’s ship? And if so, what do we make of the restored ship sitting in the harbor?
This paradox is all about the nature of identity over time, and has been the subject of
philosophical discussions for thousands of years. It appears in other forms, such as the question of
the grandfather’s axe and Trigger’s broom, both of which ask whether an object remains the same after
all the aggregate parts have been replaced. The idea even expands to questions of personal
identity. If a person changes drastically over time, so much so that who they are no
longer matches any part of who they once were, are they still the same person?
Another paradox about the vague nature of identity is the sorites paradox. The premise is fairly
simple. It generally involves a heap of sand. If you don’t have a spare heap of sand on you,
it’ll suffice to simply imagine one in your head. Now, let’s take away a single grain of sand from
the heap. It’s still, almost certainly, a heap of sand. Now take away another grain.
Still a heap. If we continue this enough times, eventually it will be down to one grain of sand,
which is, almost certainly, NOT a heap anymore. When did the sand cease being
a heap and start being… something else? The sorites paradox is all about the vagueness
of language. Because the word heap doesn’t have a specific quantity assigned to it,
the nature of a heap is subjective. It also leads to false premises. For example,
if you try the paradox in reverse, you start with a single grain of sand, which is not a heap. Then,
one could argue that one grain of sand plus another grain of sand is also not a heap. Then,
two grains of sand plus another grain of sand is also not a heap. This continues until even
the statement “a million grains of sand is not a heap” which, as we know, does not make sense.
The name of the paradox, Sorites, comes from the Greek word soros, which means heap or
pile. It’s often attributed to Eubulides of Miletus, a logician from the 4th century BCE
who was basically a paradox machine. Most of his paradoxes deal with semantic fallacies,
like the Horn paradox. If we accept the idea that ‘What you have not lost, you have,’
then consider the fact that you have not lost your horns. Therefore, you must have horns. And yes,
most of his paradoxes are just as infuriating. However, one of his more famous paradoxes,
the Liar paradox, is still discussed today. The Liar paradox has a very simple premise
but a very mind-boggling result. Here it is. This sentence is false.
That’s it. Just think about it for a moment. If the statement is true, then that means that
the sentence is in fact false, as it claims. But that would then mean that the sentence
is false. And if the sentence “this sentence is false” is false, then that means it’s true. But,
if it’s true that it’s false then… OK, you get the picture. It goes on and on, forever.
The liar’s paradox has been discussed and adapted many times through history, eventually
leading us to my favorite version: the Pinocchio paradox. It follows the same general structure,
but with an added visual component for flair. Imagine Pinocchio uttering the statement “My nose
grows longer now.” If he’s telling the truth, then his nose should grow longer, like he said. But
as we know, Pinocchio’s nose only grows if he’s telling a lie. Which means that if his nose DID
grow longer, then the statement would have been false. But if “my nose grows longer now” is false,
then it should not have grown in the first place… Has your brain exploded yet? This version of the
paradox was created in 2001 by philosopher Peter Eldridge-Smith’s 11-year-old daughter.
Peter summarized it neatly like this: "Pinocchio’s nose will grow if and only if it does not.”
Another variant of the Liar paradox actually helped shape language in the 16th century.
A crocodile grabs a child from the banks of a river. His horrified mother is told by the
croc that he will return the child only if the mother can correctly guess whether the crocodile
will return him. If she guesses that he will be returned and she is correct, the croc will
return him, no problem. If she is not correct, the crocodile will keep him. However, if she
guesses that the crocodile will not return him, an interesting paradox occurs. If she is correct,
then the crocodile must return the boy, but in doing so, breaks his word and now contradicts
her answer. If she is incorrect and the crocodile actually was going to return the boy,
he will now have to keep him, which means he’s now breaking his word. Due to the popularity of this
paradox, the word "crocodilite" came to be used to refer to any similarly paradoxical dilemma.
The interesting number paradox is debatably not a paradox at all, though it’s often called one,
but we stumbled on it while researching this script and really wanted to share it with you.
It basically goes to prove that all numbers are “interesting”—even the boring ones … which are
actually interesting, of course, and not boring at all … because they’re boring.
Interesting, in this case, means it has something unique to it. For example, 1 is the first non-zero
natural number. 2 is the smallest prime number. 3 is the first odd prime number. The list can go on
and on, until you reach the first “uninteresting” number. It doesn’t have anything special or
fascinating about it. But, being the first uninteresting number you stumbled upon, it is,
in fact, unique, and therefore interesting. This process can be repeated, hypothetically,
indefinitely. This idea was born out of a discussion between the mathematicians Srinivasa
Ramanujan and GH Hardy. Hardy remarked that the number of the taxicab he had recently ridden in,
1729, was “rather a dull one.” Ramanujan responded that it actually was interesting,
being the smallest number that is the sum of two cubes in two different ways. Ah yes, of course,
that’s what I was about to say as well. While most paradoxes are presented through
a spoken or written philosophical prompt, some are visual in nature. Take for example,
the Penrose triangle. Just a warning: if you stare at it for too long, your brain WILL melt.
It’s an object that is described by one of its creators as “impossibility … in
its purest form.” But the weird part is you can build one and show it to people.
Obviously it’s a trick of proportions and viewing angles, but
even after you reveal the trick people will still see it as an impossible triangle.
Let’s end today with a dizzying mathematical paradox that takes place in, of all places,
a fancy hotel. Hilbert’s paradox of the Grand Hotel is a famous thought experiment that is meant
to show the counterintuitive nature of infinity. Imagine walking into a big, beautiful, hotel,
looking for a room. How big? Infinitely big. This hotel has a countably infinite number of rooms.
However, all the rooms are currently occupied by a countably infinite number of guests. (Countably
infinite means you can one-to-one attach a natural number to everything in the set.) One might assume
that the hotel would not be able to accommodate you, let alone more guests, but Hilbert’s
paradox proves that this is not the case. In order to accommodate you, the hotel could,
hypothetically, move the guest in room 1 to room 2. Simultaneously, the guest in room 2
could be moved to 3, and so on, which would move every guest from their current room,
x, to a new room, x+1. As there are infinite rooms, everyone would get a new room, and now,
room 1 is totally vacant. Enjoy your stay. Let’s say we wanted to apply this idea to
any number of finite guests. Let’s say 3000 people arrive and want rooms. No problem,
just the repeat process but instead of x+1, simply do x+y, y, in this case, being 3000.
What if a countably infinite number of people line up behind you, each of which wants a room? We have
a solution to that, too. The pattern would now be 2x. Simply move the guest in room 1 to room 2,
the guest in room 2 to room 4, and the guest in room 3 to room 6, and so on. This would
leave all the odd-numbered rooms open, so each new guest could take one of the newly vacated
odd-numbered rooms and the previous patrons would all be moved to the next even room.
The basis of the Grand Hotel paradox is the idea of counterintuitive results that are
still provably true. In this example, the statements “there is a guest in every room”
and “no more guests can be accommodated” are not the same thing because of the nature of infinity.
In a normal set of numbers, such as the number of rooms in a normal hotel,
the number of odd-numbered rooms would obviously be smaller than the total number of rooms.
But in the case of infinity, this is not the case, as there are an infinite number of odd numbers,
and an infinite number of total numbers. This paradox was first introduced by philosopher
David Hilbert in a 1924 lecture and has been used to demonstrate various principles of infinity ever
since. Personally, the concept of infinity hurts my brain, and if I walked into the Grand Hotel
and they told me all the rooms were occupied, I would simply go down the street to the
nearest Holiday Inn, thank you very much. Thanks for watching this episode of the List
Show. I hope your brain hasn’t melted into a big pile of goop like mine has.
Let us know your favorite mind-boggling puzzle in the comments below, and I’ll see you next time.