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# Paradoxes and Fallacies

(Alex and Charlie were walking when they met their friend Simon)

Alex: Hi, Simon!

Simon: Hi, Alex! I have recently disproved the big bang theory.

Charlie: How did you manage that?

Simon: It means that the universe is finite at all times. But then you can reach the edge! What happens once you reach the edge? You can’t just “fall off”.

Charlie: Hang on! The Earth is finite. If you walk on it for as long as you want, you will never reach the edge because there isn’t one!

Alex: That’s mostly correct but some cosmologists believe that the universe is connected like a giant dodecahedron. If you go through one face you come out of the opposite one. But you wouldn’t even notice. There is another answer. If the universe expands faster than the speed of light, you can never reach the edge because it is expanding faster than you can travel.

Charlie: Here’s another one. Mrs Jones once tells you that she has two children. At least one is a girl. What is the chance that they are both girls?

Simon: ½. The other one is either a boy or a girl.

Alex: No. That is a common fallacy. The answer is one third. Consider the elder child. The elder child could be a boy or a girl. The younger child could also be a boy or a girl. There are four combinations. Rule out the one where both are boys and that leaves three equally likely cases. In one of them both children are girls.

Simon: There is one that I can’t understand. There is a barber who shaves inhabitants of a certain village. He shaves everyone who does not shave themselves but does not shave anyone who does. It seems like a perfectly natural case but who shaves the barber?

Charlie: If the barber shaves himself then he is shaved by the barber so he doesn’t. If he doesn’t then he is shaved by the barber so he does. Uh oh. We have a paradox.

Alex: This is Bertrand Russell’s barber paradox. The only solution I can think of is that the barber is not an inhabitant of the village. Russell originally explained it in terms of sets. His barber was the set of all sets which do not contain themselves as members. My solution means that that is not a set.

Simon: Let x2 = x + x + x + x... with x xs. Taking the derivative, 2x = 1 + 1+ 1+ 1... with x 1s so 2x = x and 2 = 1.

Alex: That one is easy. You forgot to account for the change in the number of xs. Accounting for this is enough to make it 2x.

Charlie: There are probably many more paradoxes and fallacies but I have to go. Bye!

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