how to point out errors in proof by induction I have searched for an answer to my question but no one seems to be talking about this particular matter..
I will use the all horses are the same color paradox as an example.
Everyone points out that the statement is false for n=2 and that if we want to prove the propositions we should use 2 as the base case for this proof.
But, (as I see it..), you have to use reason to figure that out. 
My question is, is there anything wrong with induction itself? 
except for the fact that we can use reason to understand why the proof is faulty.
This goes for all these problematic propositions..
Thank you all!

The induction principle is expressed in formal terms as follows :

$[P(1) \land \forall n(P(n) \rightarrow P(n+1))] \rightarrow \forall nP(n)$

Note : I'm starting from $1$ instead of $0$ in order to comply with the "horses example".
Consider now this "fake" application of it; let $P(n) := 2 \times n = 2$.
Clearly : $P(1)$ holds; thus, we have the base case.
What about the induction step ? $P(2) = 2 \times 2 = 4 \ne 2$; thus, $P(2)$ does not hold.
What we can infer about induction ? 
$P(1)$ is true while $P(2)=P(1+1)$ is false. Thus, by the truth-conditions for the conditional : $P(1) \rightarrow P(1+1)$ is false.
But this implies that $P(n) \rightarrow P(n+1)$ is not true for any $n$.
So, the antecedent of the induction principle : $P(1) \land \forall x(P(x) \rightarrow P(x+1))$ is false and the principle does not license us to infer anything about "all" $P(x)$'s, because a conditional with false antecedent is always true, irrespective of the consequent.

In conclusion, we have not "falsified" the induction principle ...
In the specific case, induction is not an issue here : having showed that $P(2)$ does not hold, we have already showed that $\forall nP(n)$ is false.

What we have learned from this "fake" example regarding the horses paradox ?

That the purported proof of the induction step is flawed, because we cannot correctly deduce, from the (true) fact that all the horses belonging to a single-members set of horses have the same color, the (false) fact that all the horses belonging to a two-members set of horses have the same color.

