14 Brain-melting paradoxes that will destroy you

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There are many silly paradoxes. For example, buttered toast always lands butter side down, and cats always land on their feet, so what would happen if you stuck some buttered toast, buttered side up, to the back of a cat and dropped it?


This is silly because we all know the cat would just land on its feet and casually walk away as if nothing had happened, so we have a good laugh at it and get on with our lives. But some paradoxes aren’t so easy to address, in fact, some lead you into recursive chains of thought that don’t let up until you either quit or your brain melts away into nothingness. These are some of the latter, and although possible solutions may exist, most can’t be explained away quite so simply. Prepare to enter brain-melting territory...

\#1 – The grandfather paradox

This is a classic time travel example to get us started: if you go back in time and kill your grandfather before he had kids, you could never have been born, thus meaning that you couldn’t have killed your grandfather. But if you were born, then you actually could go back in time and kill him, and so it continues until you decide that time travel probably isn’t worth the hassle. The same basic issue can arise in other ways too, for example, if you went back in time to kill Hitler before he rose to power, then in your present time Hitler would no longer be a known figure, so you’d have never bothered killing him in the first place.

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\#2 – Zeno’s paradox

Right, there’s a smart-mouthed tortoise that’s just challenged you to a race. He says you’ll never beat him if you grant him just a ten metre head-start, and he can prove it without even racing. By the time you cover the ten metres, he argues, he’ll have covered some distance too, say one metre. Then you cover that metre, but he’ll have moved another small distance, and when you cover that small distance, he’ll have gone a little further, so you’ll never catch up. In another way, if you want to walk ten metres to punch the tortoise in the face, first you have to travel half that distance, so you’ll be five metres away, then half that distance, so you’re 2.5 m away, then half that to 1.25 m, half that to 0.625 m and so on infinitely, so you’ll never get there. In real life, though, you’d just get there and punch the damn tortoise, probably while screaming “feel the wrath of my summable infinite series!”

Related: Platonic Realms: Zeno’s paradox of the tortoise and Achilles

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\#3 – The liar’s paradox

This sentence is false. Take that, brain! If it’s false, then the sentence must be true, which means it’s actually false, which means it’s true, and so on. The same happens the other way: if it’s true, then the sentence itself is actually false, which means it’s true and so on. To rectify the paradox, some have asserted that it’s neither true nor false, but then if it’s simply changed to “this sentence is not true,” meaning that if it’s neither true nor false, then it’s actually true and the problem begins again.

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\#4 – The interesting number paradox

Many numbers have interesting properties, an idea famously supported by the mathematician Ramanujan and a conversation with G. H. Hardy. Hardy had just been in a cab with the number 1729, and he remarked was a boring number, but Ramanugan responded that it’s actually interesting, because it’s the smallest number expressible as the sum of two positive cubes in two different ways. Oddly impressive as that may be, imagine that we knew the smallest non-interesting number: that in itself makes the number interesting. In fact, if you grouped all non-interesting numbers together, the first would always be interesting because it’s the smallest non-interesting number, so therefore all numbers must be interesting. The only problem with this is that it’s self-referential; if you impose an objective standard of interestingness the problem disappears. The answer continually changes, but as of 2014, the smallest number not known to be interesting is 14,228. The fact it’s so large, though, is pretty interesting...

Related: Nathaniel Johnson: 11630 is the first uninteresting number

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\#5 – Omnipotence paradox

If there was an omnipotent god, could he (or she or it) create a boulder so large that he couldn’t lift it? Since an all-powerful god could do anything, he should obviously have the power to lift any rock, and he should also have the power to make something big enough that he can’t lift. This can be stated more generally: if god can do everything, this would include preventing himself from doing something. If he could do it, then he would be limited and not omnipotent, but if he couldn’t then he wouldn’t be omnipotent anyway.

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\#6 – Galileo’s paradox

Some numbers are squares (meaning another number multiplied by itself) and some are not. If you accept this, it seems that if you collected all squares and non-squares, there should be more of these than there are squares alone. But each number has a square root, and each has its own square, so there must also be an equal amount. Galileo therefore argued that infinite sets of numbers aren’t compatible with notions of equal, greater or smaller, but future work showed that the restriction wasn’t needed. A similar idea is the paradox of odd and natural numbers: if you write down the natural integers (1, 2, 3, 4, ...) with the odd numbers only underneath (1, 3, 5, 7, ...), both groups will contain the same (although infinite) amount of numbers, suggesting that there are as many odd numbers as there are odd and even ones combined.

Related: Suitcase of Dreams: Odd, even and natural numbers

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\#7 – Bootstrap paradox

This is another time-travel paradox in which information or objects from the future are sent back into the past and lead to the information or object existing in the present. There are many film examples: in Terminator, the remains of the T-800 from the first film were salvaged and used to create Skynet, which means the knowledge (originally) didn’t really come from anywhere – the T-800 had to already exist in the past, created by Skynet, in order for Skynet itself to be created. In Back to the Future, Marty McFly plays Chuck Berry’s “Johnny B. Goode” when he’s back in 1955, which is heard by Chuck Berry, causing him to release the same song three years later. Another example is from Bill and Ted’s Excellent Adventure: Rufus, the pair’s guide on their time-travel journey, was only introduced by name to the pair by Ted from the future, who himself must have also learned it from a future Ted, so the information actually has no original source.

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\#8 – The ship of Theseus

If you build a ship and give it a name – let’s just use your own name, or “Me” – and then gradually start replacing it, one piece at a time, with new material, when does it cease being the same ship? Once it’s entirely replaced with new material, how can you really say it’s still “Me?” The choice of your own name or “Me” as the ship’s name isn’t an accident, because the cells (or indeed, the atoms) that make up your body are gradually replaced, to the point where you get a whole new “me” every seven years. So, who are you, really? If you can’t accept the replacement ship as the same ship, can you really accept the replacement you as the same you?

Related: UTNE: The ship of Theseus and the question of identity

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\#9 – The predestination paradox

This is another time-travel paradox, closely related to the bootstrap paradox. It’s well-illustrated by the Futurama episode “Roswell That Ends Well,” in which Fry travels back in time and is warned not to do something stupid like kill his own grandfather. He locates his grandfather, who is accidently killed, but then he doesn’t cease to exist. He reasons that this means he couldn’t have been his grandfather, and in consoling his grandad’s fiancé, who he previously assumed was his grandmother, ends up having sex with her. It turns out, however, that she was his grandmother, and Fry has just made himself his own grandfather. His whole existence is contingent upon his later going back in time and impregnating his grandmother, and so he always had to take that action when he went back in time. It’s a causality loop, because he couldn’t have existed if he didn’t already take this action in the past, so he was pre-destined to do it.

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\#10 – Sorites’ paradox

This paradox revolves around a simple argument. If a pile of one million grains of sand is a heap of sand, and a heap of sand remains a heap of sand if you remove a single grain, then you can assert that a single grain of sand is a heap. By continually removing one grain, the “heap” is defined to be a smaller and smaller amount until it obviously fails to meet the definition of heap we all accept. You could even take it further and argue a heap of no grains of sand or a negative number of sand grains, but this is another example where a more objective definition of the core term (heap) would ultimately render the paradox non-existent.

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\#11 – The barber paradox

This is a less abstract version of Russell’s paradox. Imagine a town in which there was one barber, who is a man, and all of the men in the town keep themselves clean-shaven by either shaving themselves or going to the barber. This can equivalently be stated as the barber shaving those people and only those people who don’t shave themselves. In this situation, who shaves the barber? Clearly it can’t be himself, because he is the barber, who only shaves men who don’t shave themselves, but it can’t be the barber either for the same reason.

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\#12 – The crocodile’s dilemma

Imagine that your child is stolen by a (bizarrely dextrous and intelligent) crocodile, who stipulates that he’ll give your kid back if you correctly guess what he will do with him or her. What happens, then, if you guess that the crocodile won’t return your child? If he was going to return the child, then you’re wrong, so the crocodile will have to keep the child, which means you’re right. If he was going to keep it, then you’re right, which means that the crocodile has to give the child back, which makes you wrong. So what can the crocodile do? We’re guessing it would probably just eat the child and have done with it, because crocodiles hate paradoxes.

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\#13 – The unexpected hanging paradox

A judge tells a prisoner that he will be hanged at noon one weekday in the following week, and it will be a surprise, meaning that he will only know which day it is going to be when the guard knocks on his door on that day to take him there. However, when the prisoner reflects on his situation, he realises that the hanging can’t possibly be Friday, because that’s the last available day and therefore it wouldn’t be a surprise after he failed to be hanged on Thursday. Now, though, the same is true of Thursday, because that’s the last day eligible for the hanging, so it wouldn’t be a surprise by noon on Wednesday. The logic is followed through until he concludes that he won’t be hung at all, and starts to rest easy. He is genuinely surprised when it comes to noon on Wednesday and the guard knocks on his door to take him to be hung.

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\#14 – The arrow paradox

Zeno proposed this paradox, which basically asserts that nothing is capable of moving. If you have an arrow in flight but look at it only in a single instant, it can’t possibly be moving. This is because movement is travel to a different position in space, which is not possible in a single snapshot in time. So if something can’t move in any single instant, it can’t move in any other instant either, meaning that it can’t really move at all when these successive snapshots of time are combined. The paradox is obvious, because things do move – as Zeno would have probably concluded too if the arrow was flying directly towards his head.

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