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### SOME FUN PARADOXES, RANKED

- 1.Zeno's Tortoise ParadoxHercules challenges a tortoise to a foot race. Seeing as he is Hercules he gives the tortoise a 100 yard head start. The race begins with the tortoise walking slowly and Hercules in an all out sprint. Soon Hercules has run 100 yards and reached the tortoise's starting point. But by now the tortoise has walked ahead some. Hercules must now catch up to that point. When he does, obviously the tortoise has walked further still. It is clear Hercules can never catch up.
- 2.Galileo's Paradox9 squared is 81. 10 squared is 100. All the numbers from 82 to 99 are not perfect squares. In this way it can easily be shown that for every square there are many numbers that are not squares. But any number can be squared. So there is clearly one square for every non-square number. So there are both more non squares than squares and an equal number of both.
- 3.The Raven ParadoxThe claim: all ravens are black. As evidence we have all the black ravens we've seen in our lives. How about this claim: all non-black things are not ravens. Again we have as evidence all the non-black things we've ever seen, none of which were ravens. These two claims are logically identical. Therefore, the existence of your white refrigerator is evidence that all ravens are black. And your red car. And your blue shoes. Still more evidence that all ravens are black. ?!
- 4.The Unexpected Hanging ParadoxYou've been sentenced to hang. The judge promises that the executioner will knock on your door one day next week at noon and it will be a surprise. You return to your cell and think a bit. If Thursday noon passes safely then you'll know it must be Friday, and then the knock won't be a surprise so it *can't* be Friday. But then by the same logic it can't be Thursday either. Or Wednesday… You conclude logically that you can't be hung. The executioner shows up Thursday and you are indeed surprised.
- 5.The Interesting Number ParadoxWhat is the smallest number that is not interesting? One is interesting because it is the first. Two is the smallest even number. Three is the first odd prime. Four is a perfect square. But if we keep going up, there will certainly be some numbers that are not interesting in any way. Which one is the smallest? But wait! Being the smallest uninteresting number is kind of interesting! So it isn't the smallest uninteresting number anymore. But that was the only thing that made it interesting…
- 6.Berry ParadoxSimilar to the one above. What is the smallest number not nameable in under ten words? Whatever it is, you can certainly name it "the smallest number not nameable in under ten words" which is nine words.

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