This one (but in Italian) was the introductory chapter of my algebra textbook in 1st year uni. Scattered in the book there were more such "wrong theorems" without an explanation on why they were wrong (the explanation was left as an exercise for the student) 
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Nice! I never had a textbook with "wrong theorems" in it. Sometimes in my calculus class I deliberately make mistakes. Sometimes I tell the students ahead of time "Okay, somewhere in this calculation I'm going to make a mistake. See if you can spot it!"
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We can also prove by induction that all natural numbers are small: n=1 is small, ok! Suppose now that n is small. It is obvious that in this case n+1 is still small. Therefore, by induction, every natural number is small.
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In philosophy they call this the "sorites paradox". How many grains of sand make a heap?
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how do we know that 1st and 2nd horses have the same color?
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For that case you just have to go out to the stable and look.
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It's funny but I don't think it's an actual paradox, but more like a word game. The sentence should read "All pair of horses have the same color", with the induction starting from the base case n=2. (In n example below with collinear points, it should start with n=3) :D
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It's not a paradox - it's a joke, and a lesson that one needs to be careful in writing proofs. Details here: https://en.wikipedia.org/wiki/All_horses_are_the_same_color …
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I believe this is Exercise 2, Chapter 1 of "Concrete Mathematics".
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It's in a lot of texts. It's a pretty classical fake proof to warn people about picking your base case carefully.
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Čini se da učitavanje traje već neko vrijeme.
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