[sidedishes]

I don't understand why it's more or less weird to say the base case is wrong vs. the inductive case is wrong. If H(n) is 'all sets of horses of size n contain horses of the same colour', the linked argument seems like

H(1) is true

H(n) => H(n+1) (with a sleight of hand that it's only for n >= 2)

It seems to me like changing either the argument to assert H(2) is true, or the scope of the second statement to include n = 1, would be enough. It seems the fault of both statements equally to not fit with the other, so why is it weirder to say the base case is wrong?

[amitkgupta84]

You just said it. The base case is true. The inductive case is wrong. You called it true “with sleight of hand”, but that’s more of a poetic statement; mathematically it’s simply wrong.

One could write a completely different proof where the base case(s) cover n=1,2 and the inductive step is as in the post. In such a proof, the base case would be wrong and the inductive step would be correct. But that’s a different proof, not the one in the post.

[Dylan16807]

> You just said it. The base case is true. The inductive case is wrong.

Not really. H(1) is true. H(2) is false. But either one could be your base case.

The inductive step could be true or false, again depending on what you call your base case.