10 Mind-bending Paradoxes 10 Achilles and the Tortoise In one of a set of paradoxes devised by Greek philosopher Zeno of Elea, Achilles challenges a tortoise to a race

He decides to give it a 500 meter head start, knowing that he can easily make up the distance But by the time Achilles has run 500 meters the tortoise has travelled 50 meters, so Achilles still needs to catch up But while Achilles sprints this next 50 meters, the tortoise will travel a further 5, so Achilles travels 5 meters more, in which time the tortoise is half a meter further along Therefore, as far as Achilles runs, a moving tortoise will always have travelled a fraction of that distance So mathematically, as long as the tortoise continues to move, he will stay ahead of Achilles by the distance he’s travelled while Achilles caught up

Obviously we know this won’t be the case in practice, but it demonstrates that any finite value can always be divided an infinite number of times 9 The Unexpected Hanging In this conundrum, first published in 1948 in the book Pragmatic Paradoxes, a death row prisoner is informed by a particularly sadistic judge that he will be hanged on a weekday, but won’t be told which From this, he deduces that he won’t be hanged on Friday, because if by the end of Thursday he’s still alive, he knows he will be hanged on Friday Having eliminated Friday, he reasons that he can’t then be hanged on Thursday, because if he was alive by Wednesday night, he would know he was to be hanged on Thursday by the same logic that he eliminated Friday

This follows this logic until he realises that no day would come as a surprise to him As the story follows, despite the logic suggesting that there would be no date for the hanging that will surprise him, the hangman arrives on Wednesday morning to his complete surprise 8 The Liar You’re holding a piece of paper On one side reads the statement “The statement on the other side of this paper is true”

You turn it over and the other side reads “The statement on the other side of this paper is false” If the first statement is true, it means that the second is also true But for the second statement to be true, the first must be false This pattern will continue infinitely with no resolution, with the only real resolution being the acceptance that a statement can simultaneously be true and false 7

The Buridan’s Ass Proposed by Aristotle, the Buridan’s ass is named after a philosopher, satirizing his belief that all events are predetermined It outlines a situation in which a donkey, equally as hungry as it is thirsty, is placed the same distance from a pile of hay as it is a pail of water Unable to make a rational decision on which craving to satiate first, the donkey dies of thirst and hunger between two perfect sources of sustenance While it seems ridiculous, the situation is virtually impossible to create in practice and so may never be disproven The solution may seem simple, but a purely logical being would have no ability to differentiate the two options

This makes the situation a paradox where the only solution is the presence of free will, and has been used to refute the philosophy of determinism 6 The Omnipotence Paradox Dating back to the 12th Century, The Omnipotence Paradox questions the parameters of power possessed by an all powerful being, like a God It manifests in several forms, the most common of which is known as the paradox of the stone Assuming that a being is all powerful, it follows that they would possess the power to do anything, including disproving themself

This is commonly phrased in the question ""Could a God create a stone so heavy that even they could not lift it?"" If a God can create the stone as described, then they won’t possess the power to lift it meaning they cease to be all powerful, as they don’t possess the power to lift the stone Similarly, if there is no stone an omnipotent God couldn’t lift, then they don’t possess the power to create one, and so can’t be omnipotent 5 The Crocodile Dilemma In an anecdotal paradox originating from Ancient Greece, a crocodile snatches a girl from the bank of a river and the mother begs the reptile to let her go Acting more reasonably and significantly more verbally than the average crocodile, he offers her a deal

The mother will get her child back if she can guess correctly whether or not he will return her If she guesses that the crocodile will return her daughter, then he either will or won’t and it proposes no logical problems But if the mother guesses that the crocodile won’t return her daughter, then an impossible situation arises The crocodile can either keep the girl, making the mother’s guess correct, which obliges him to return her Alternatively, he can return her, but this would render the mother’s guess incorrect, obliging him to keep her

Just another reason children and crocodiles don’t mix 4 The Bootstrap Paradox The name bootstrap comes from the physical impossibility of the common phrase ‘pulling yourself up by your bootstraps’ The paradox was laid out in a story by sci-fi author Robert Heinlein where a time traveller goes back in time and, for whatever reason, kills her own grandmother From there, her grandmother never meets her grandfather and her mother is never born

This means, as you may have guessed, that the time traveller herself can’t be born This means the time traveller won’t be able to go back and kill her grandmother to begin with If the grandmother isn’t killed, then the time traveller is born and can travel back to start the cycle again This repeats indefinitely until a version of the time traveller decides not to kill her grandmother, but seeing as each version of the time traveller is identical, this decision will never occur to her 3

The Dichotomy You want to cross the road, but to cross the road, you first have to travel half the distance across the road and to do that, you must first travel half of that distance and so on Logically, any task can be halved to make two smaller tasks, and so to travel any distance, there are an infinite number of half distances that must first be travelled Therefore to start travelling, we have to consider the first part of the journey as no distance whatsoever, but to travel no distance at all, you would have to stay still Similarly to Achilles and the tortoise, this logical paradox presents something we know to be physically possible as impossible based on the infinite fractions of finite values 2

The Barber The Barber is an anecdotal illustration of Russell’s Paradox, named after its discoverer, the nobel prize winner Bertrand Russell It questions whether a collection would contain itself if its purpose was to contain collections of things that do not contain themselves To make that slightly clearer, the story is translated to a barber, who says he will shave every man who does not shave himself But by that logic, does the barber shave himself? If he doesn’t, then he is a man who doesn’t shave himself, and by his own rule he has an obligation to shave himself But if he does, then he has shaved a man who shaves himself, thus breaking his only rule

1 Theseus’ Ship The paradox of Theseus’ ship examines the loose definition of identity and what constitutes it The paradox refers to the ship of Athens’ mythical founder, most famous for his legendary defeat of the Minotaur After defeating the creature, Theseus returned from Crete to Athens, where his ship remained in the harbour for centuries To keep the ship seaworthy, any broken or rotting parts of it were replaced, until the point that no single part of the original ship remained

If what remained was Theseus’ ship, then which piece of it ensured the title? Or, if it wasn’t then at what point had it ceased to be? Few would contest that it was the same ship when only one plank of wood was replaced, so deciding at which point the ship ceases to be the same becomes a paradox, by asking you to quantify the definitively unquantifiable So that was 10 Mindbending Paradoxes Does your head hurt? Do you know any more? Let us know in the comments and make sure to like and subscribe While you’re at it, check out this great Alltime10s video on screen now"

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