| There’s a lot of confusion in this thread, and the original post. To diagnose the problem with the proof, it’s worth knowing why proof by induction works, to thereby know how it actually works, and then diagnose which part of how it’s meant to be executed isn’t actually being executed in this blog post. We need a starting point. For natural numbers, that’s Peano arithmetic. For simplicity, Peano arithmetic says that for any unary predicate P, the following is axiomatic: If P(1) And For all n > 0, P(n) => P(n+1) Then For all n > 0, P(n) Proof by induction involves formulating your hypothesis in the form “For all n > 0, P(n)”, then proving the two conjuncts “P(1)” and “For all n > 0, P(n) => P(n+1)”, and finally concluding your hypothesis by modus applied to the inductive Peano axiom for P. In this case: P(n) is “in all sets of horses of size n, all the horses in the set are the same color” The first conjunct, P(1), generally referred to as the base case, is true, and correctly recognized as such in the blog post. The second conjunct, “For all n > 0, P(n) => P(n+1)”, generally referred to as eg inductive case, is false for this P. It’s not about “would the inductive case be true if the base case were different”. There is a single proof presented in the article. It has a base case which is a true statement, and an inductive case which is a false statement. So the problem is the inductive case. You could have a different proof which, reformulating the Peano axioms a little, could look like: Base: P(1) and P(2) Inductive: For all n > 1, P(n) => P(n+1) In this different proof, let’s call this Proof 2, the base case would be false and the inductive case would be true. Maybe the author is not intending to say the base case is “wrong” in that is false, but that is “wrong” in that it should have been formulated as in Proof 2, and that the inductive case is valid when interpreted in the sense of Proof 2. But this is completely confused. Why would it have been better for the proof presented in the article be interpreted as a proof (Proof 2) that’s not presented in the article? Both proofs fail, neither is a “better” formulation. Rather than saying: the proof presented in the article should instead have be formulated in a superficially different but fundamentally equally invalid way, under which formulation the base case would be wrong. Let’s just say: the inductive case of the proof presented in the article is false. |