All horses are the same color

All horses are the same color is a falsidical paradox that arises from a flawed use of mathematical induction to prove the statement All horses are the same color.

[1] There is no actual contradiction, as these arguments have a crucial flaw that makes them incorrect.

This example was originally raised by George Pólya in a 1954 book in different terms: "Are any n numbers equal?"

or "Any n girls have eyes of the same color", as an exercise in mathematical induction.

[3] The "horses" version of the paradox was presented in 1961 in a satirical article by Joel E. Cohen.

It was stated as a lemma, which in particular allowed the author to "prove" that Alexander the Great did not exist, and he had an infinite number of limbs.

First, we establish a base case for one horse (

Likewise, exclude some other horse (not identical to the one first removed) and look only at the other

The inductive step proved here implies that since the rule is valid for

, which in turn implies that the rule is valid for

[2][5] The argument above makes the implicit assumption that the set of

horses has the size at least 3,[3] so that the two proper subsets of horses to which the induction assumption is applied would necessarily share a common element.

Therefore, the above proof has a logical link broken.

The proof forms a falsidical paradox; it seems to show by valid reasoning something that is manifestly false, but in fact the reasoning is flawed.