Horse paradox
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The horse paradox is the following (invalid) proof of the statement All horses are the same colour.
We use the principle of mathematical induction. As the basis case, we note that in a set containing a single horse, all horses are clearly the same colour. Now assume the truth of the statement for all sets of at most n horses. Let there be n+1 horses in a set. Remove the first horse to get a set of n horses. By the induction assumption, all horses in this set are the same colour. It remains to show that this colour is the same as that of the horse we removed. But this is easy: put back the first horse, take out a different horse and apply the induction principle to this set of n horses. Thus all horses in any set of n+1 horses are the same colour. By the principle of induction, we have established that all horses are the same colour.
The hole in the above "proof" is easy to spot with a little thought: it makes the implicit assumption that the two subsets of horses to which we apply the induction assumption have a common element, but this fails when n=1.
Thus this is not a paradox, but merely the result of flawed reasoning; it exposes the pitfalls arising from failure to consider special cases for which a general statement may be false.es:Paradoja del caballo pl:paradoks koni hu:Ló-paradoxon he:פרדוקס הסוס