[dwohnitmok]

> The argument doesn't apply in the base cases

No I'm still with eperdew on this. You don't apply an inductive argument to base cases, so that statement doesn't really make sense. From a pedagogical point of view, I think it is very important to drive home the point that there is nothing special about your choice of base case, other than that it affects the argument in your inductive step, otherwise it's too easy to make the choice of base case seem "magical."

In fact I'd go further. Whenever you have an error in an inductive argument, it is always in the inductive step and never the base case, since you can always choose whatever base case you want (you might just find it makes your inductive step impossible to prove, so if you do prove something, you've made a logical error in your inductive step).

At worst, an "error" in choice of base case leads to a proof that cannot be completed, not an erroneous proof.

EDIT: "it is always in the inductive step and never the base case" except of course if you've somehow made an error in the initial proof of the base case, but that is distinct from choosing the "wrong" base case.

EDIT 2: I just understood what you meant by "apply," (doesn't fulfill the preconditions of the inductive step, vs I thought you meant relevant to proving the base case) sure that's a reasonable way of looking at it too, but again I'm very wary of saying that it's the wrong base case that leads to proving a falsehood as I lay out elsewhere.

[shkkmo]

> you might just find it makes your inductive step impossible to prove, so if you do prove something, you've made a logical error in your inductive step

The inductive step should be proven completely independently of proving a base case. Indeed, if the inductive step is proven correctly you see that it is limited in scope to n>1.

You can then use any base case you can prove where n>1. Let's say you pick a base case of one trillion. Easy enough to show thay every set of one trillion horses is the same color (since no such sets can exist). You can then easily use induction to show that every set larger than one trillion horses is also the same color!

So really, both base and induction were wrong in different ways but if the induction had been proven correctly, the inapplicable base case would have been clear.

[poetically]

So what would say is the error in the inductive step here?

[dwohnitmok]

The inductive step is being obscured (intentionally of course in the article to illustrate how the error can hide) behind the n.

The inductive step here is not true for all n, n + 1 pairs.

It is only true for all n, n + 1 pairs given that n is greater than or equal to 2.

The error is to drop that >= condition. Hence the inductive step is wrong as stated in the article (for all n), and its proof is erroneous.

Again while changing the base case can "fix" the argument in that it no longer proves an falsehood, without altering the statement of the inductive step it is still not a valid argument and only happens to prove a true statement "by chance."

If the base case was the error in the sense of proving a falsehood, then it would be possible to have an unsound inductive argument by choosing the wrong base case, but that's impossible. Rather the wrong base case (but correct inductive step) simply means you end up unable to complete your proof.