[Jensson]

Nah, the inductive step is simply incorrect. For it to be correct it needs to say "for n > 1 the following is true", but it doesn't, the statement isn't true since they forgot to add that part. And if you added "for n > 1" to make the inductive statement correct then it is obvious where the reasoning goes wrong.

In other words, the base case is correct, in all sets of 1 horse all horses has the same color. The inductive step is not correct. Even if you changed the where you put the base case to N=2, the inductive steps reasoning is still wrong as long as it doesn't include the "for n > 1" part.

[mcphage]

You wouldn't need to add "for n > 2" to the inductive step, any more than you need to include "for n > 1" for arguments where the base step is 1, or "for n > 20" for arguments where the base step is 20.

[Jensson]

You need to do it because you prove the statements separately. Currently the inductive step statement is false, it says that applies for all sets. It doesn't.

Note that you can use false intermediate steps like that and still reach a correct conclusion. Then the conclusion is correct but the proof is still wrong since the steps you used were wrong.