About: All horses have the same colour   Sponge Permalink

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The horse paradox is an apparent paradox which arises from the following supposed demonstration, using mathematical induction, of the statement All horses are the same colour: As the basis case, 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 hypothesis, all horses in this set are the same colour.

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  • All horses have the same colour
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  • The horse paradox is an apparent paradox which arises from the following supposed demonstration, using mathematical induction, of the statement All horses are the same colour: As the basis case, 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 hypothesis, all horses in this set are the same colour.
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abstract
  • The horse paradox is an apparent paradox which arises from the following supposed demonstration, using mathematical induction, of the statement All horses are the same colour: As the basis case, 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 hypothesis, 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.
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